fundamentals of computational fluid dynamics.pdf

Fundamentals of Computational Fluid

Dynamics

Harvard Lomax and Thomas H. Pulliam

NASA Ames Research Center

David W. Zingg

University of Toronto Institute for Aerospace Studies

August 26, 1999

Contents

1 INTRODUCTION 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Problem Speci�cation and Geometry Preparation . . . . . . . 2

1.2.2 Selection of Governing Equations and Boundary Conditions . 3

1.2.3 Selection of Gridding Strategy and Numerical Method . . . . 3

1.2.4 Assessment and Interpretation of Results . . . . . . . . . . . . 4

1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 CONSERVATION LAWS AND THE MODEL EQUATIONS 7

2.1 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 The Navier-Stokes and Euler Equations . . . . . . . . . . . . . . . . . 8

2.3 The Linear Convection Equation . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Di�erential Form . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.2 Solution in Wave Space . . . . . . . . . . . . . . . . . . . . . . 13

2.4 The Di�usion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.1 Di�erential Form . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.2 Solution in Wave Space . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Linear Hyperbolic Systems . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 FINITE-DIFFERENCE APPROXIMATIONS 21

3.1 Meshes and Finite-Di�erence Notation . . . . . . . . . . . . . . . . . 21

3.2 Space Derivative Approximations . . . . . . . . . . . . . . . . . . . . 24

3.3 Finite-Di�erence Operators . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.1 Point Di�erence Operators . . . . . . . . . . . . . . . . . . . . 25

3.3.2 Matrix Di�erence Operators . . . . . . . . . . . . . . . . . . . 25

3.3.3 Periodic Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.4 Circulant Matrices . . . . . . . . . . . . . . . . . . . . . . . . 30

iii

3.4 Constructing Di�erencing Schemes of Any Order . . . . . . . . . . . . 31

3.4.1 Taylor Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4.2 Generalization of Di�erence Formulas . . . . . . . . . . . . . . 34

3.4.3 Lagrange and Hermite Interpolation Polynomials . . . . . . . 35

3.4.4 Practical Application of Pad�e Formulas . . . . . . . . . . . . . 37

3.4.5 Other Higher-Order Schemes . . . . . . . . . . . . . . . . . . . 38

3.5 Fourier Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5.1 Application to a Spatial Operator . . . . . . . . . . . . . . . . 39

3.6 Di�erence Operators at Boundaries . . . . . . . . . . . . . . . . . . . 43

3.6.1 The Linear Convection Equation . . . . . . . . . . . . . . . . 44

3.6.2 The Di�usion Equation . . . . . . . . . . . . . . . . . . . . . . 46

3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 THE SEMI-DISCRETE APPROACH 51

4.1 Reduction of PDE's to ODE's . . . . . . . . . . . . . . . . . . . . . . 52

4.1.1 The Model ODE's . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1.2 The Generic Matrix Form . . . . . . . . . . . . . . . . . . . . 53

4.2 Exact Solutions of Linear ODE's . . . . . . . . . . . . . . . . . . . . 54

4.2.1 Eigensystems of Semi-Discrete Linear Forms . . . . . . . . . . 54

4.2.2 Single ODE's of First- and Second-Order . . . . . . . . . . . . 55

4.2.3 Coupled First-Order ODE's . . . . . . . . . . . . . . . . . . . 57

4.2.4 General Solution of Coupled ODE's with Complete Eigensystems 59

4.3 Real Space and Eigenspace . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3.1 De�nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3.2 Eigenvalue Spectrums for Model ODE's . . . . . . . . . . . . . 62

4.3.3 Eigenvectors of the Model Equations . . . . . . . . . . . . . . 63

4.3.4 Solutions of the Model ODE's . . . . . . . . . . . . . . . . . . 65

4.4 The Representative Equation . . . . . . . . . . . . . . . . . . . . . . 67

4.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5 FINITE-VOLUME METHODS 71

5.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2 Model Equations in Integral Form . . . . . . . . . . . . . . . . . . . . 73

5.2.1 The Linear Convection Equation . . . . . . . . . . . . . . . . 73

5.2.2 The Di�usion Equation . . . . . . . . . . . . . . . . . . . . . . 74

5.3 One-Dimensional Examples . . . . . . . . . . . . . . . . . . . . . . . 74

5.3.1 A Second-Order Approximation to the Convection Equation . 75

5.3.2 A Fourth-Order Approximation to the Convection Equation . 77

5.3.3 A Second-Order Approximation to the Di�usion Equation . . 78

5.4 A Two-Dimensional Example . . . . . . . . . . . . . . . . . . . . . . 80

5.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6 TIME-MARCHING METHODS FOR ODE'S 85

6.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.2 Converting Time-Marching Methods to O�E's . . . . . . . . . . . . . 87

6.3 Solution of Linear O�E's With Constant Coe�cients . . . . . . . . . 88

6.3.1 First- and Second-Order Di�erence Equations . . . . . . . . . 89

6.3.2 Special Cases of Coupled First-Order Equations . . . . . . . . 90

6.4 Solution of the Representative O�E's . . . . . . . . . . . . . . . . . 91

6.4.1 The Operational Form and its Solution . . . . . . . . . . . . . 91

6.4.2 Examples of Solutions to Time-Marching O�E's . . . . . . . . 92

6.5 The �� Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.5.1 Establishing the Relation . . . . . . . . . . . . . . . . . . . . . 93

6.5.2 The Principal �-Root . . . . . . . . . . . . . . . . . . . . . . . 95

6.5.3 Spurious �-Roots . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.5.4 One-Root Time-Marching Methods . . . . . . . . . . . . . . . 96

6.6 Accuracy Measures of Time-Marching Methods . . . . . . . . . . . . 97

6.6.1 Local and Global Error Measures . . . . . . . . . . . . . . . . 97

6.6.2 Local Accuracy of the Transient Solution (er�; j�j ; er!) . . . . 98

6.6.3 Local Accuracy of the Particular Solution (er�) . . . . . . . . 99

6.6.4 Time Accuracy For Nonlinear Applications . . . . . . . . . . . 100

6.6.5 Global Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.7 Linear Multistep Methods . . . . . . . . . . . . . . . . . . . . . . . . 102

6.7.1 The General Formulation . . . . . . . . . . . . . . . . . . . . . 102

6.7.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.7.3 Two-Step Linear Multistep Methods . . . . . . . . . . . . . . 105

6.8 Predictor-Corrector Methods . . . . . . . . . . . . . . . . . . . . . . . 106

6.9 Runge-Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.10 Implementation of Implicit Methods . . . . . . . . . . . . . . . . . . . 110

6.10.1 Application to Systems of Equations . . . . . . . . . . . . . . 110

6.10.2 Application to Nonlinear Equations . . . . . . . . . . . . . . . 111

6.10.3 Local Linearization for Scalar Equations . . . . . . . . . . . . 112

6.10.4 Local Linearization for Coupled Sets of Nonlinear Equations . 115

6.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7 STABILITY OF LINEAR SYSTEMS 121

7.1 Dependence on the Eigensystem . . . . . . . . . . . . . . . . . . . . . 122

7.2 Inherent Stability of ODE's . . . . . . . . . . . . . . . . . . . . . . . 123

7.2.1 The Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.2.2 Complete Eigensystems . . . . . . . . . . . . . . . . . . . . . . 123

7.2.3 Defective Eigensystems . . . . . . . . . . . . . . . . . . . . . . 123

7.3 Numerical Stability of O�E 's . . . . . . . . . . . . . . . . . . . . . . 124

7.3.1 The Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.3.2 Complete Eigensystems . . . . . . . . . . . . . . . . . . . . . . 125

7.3.3 Defective Eigensystems . . . . . . . . . . . . . . . . . . . . . . 125

7.4 Time-Space Stability and Convergence of O�E's . . . . . . . . . . . . 125

7.5 Numerical Stability Concepts in the Complex �-Plane . . . . . . . . . 128

7.5.1 �-Root Traces Relative to the Unit Circle . . . . . . . . . . . 128

7.5.2 Stability for Small �t . . . . . . . . . . . . . . . . . . . . . . . 132

7.6 Numerical Stability Concepts in the Complex �h Plane . . . . . . . . 135

7.6.1 Stability for Large h. . . . . . . . . . . . . . . . . . . . . . . . 135

7.6.2 Unconditional Stability, A-Stable Methods . . . . . . . . . . . 136

7.6.3 Stability Contours in the Complex �h Plane. . . . . . . . . . . 137

7.7 Fourier Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . 141

7.7.1 The Basic Procedure . . . . . . . . . . . . . . . . . . . . . . . 141

7.7.2 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 142

7.7.3 Relation to Circulant Matrices . . . . . . . . . . . . . . . . . . 143

7.8 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

7.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

8 CHOICE OF TIME-MARCHING METHODS 149

8.1 Sti�ness De�nition for ODE's . . . . . . . . . . . . . . . . . . . . . . 149

8.1.1 Relation to �-Eigenvalues . . . . . . . . . . . . . . . . . . . . 149

8.1.2 Driving and Parasitic Eigenvalues . . . . . . . . . . . . . . . . 151

8.1.3 Sti�ness Classi�cations . . . . . . . . . . . . . . . . . . . . . . 151

8.2 Relation of Sti�ness to Space Mesh Size . . . . . . . . . . . . . . . . 152

8.3 Practical Considerations for Comparing Methods . . . . . . . . . . . 153

8.4 Comparing the E�ciency of Explicit Methods . . . . . . . . . . . . . 154

8.4.1 Imposed Constraints . . . . . . . . . . . . . . . . . . . . . . . 154

8.4.2 An Example Involving Di�usion . . . . . . . . . . . . . . . . . 154

8.4.3 An Example Involving Periodic Convection . . . . . . . . . . . 155

8.5 Coping With Sti�ness . . . . . . . . . . . . . . . . . . . . . . . . . . 158

8.5.1 Explicit Methods . . . . . . . . . . . . . . . . . . . . . . . . . 158

8.5.2 Implicit Methods . . . . . . . . . . . . . . . . . . . . . . . . . 159

8.5.3 A Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

8.6 Steady Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

8.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

9 RELAXATION METHODS 163

9.1 Formulation of the Model Problem . . . . . . . . . . . . . . . . . . . 164

9.1.1 Preconditioning the Basic Matrix . . . . . . . . . . . . . . . . 164

9.1.2 The Model Equations . . . . . . . . . . . . . . . . . . . . . . . 166

9.2 Classical Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

9.2.1 The Delta Form of an Iterative Scheme . . . . . . . . . . . . . 168

9.2.2 The Converged Solution, the Residual, and the Error . . . . . 168

9.2.3 The Classical Methods . . . . . . . . . . . . . . . . . . . . . . 169

9.3 The ODE Approach to Classical Relaxation . . . . . . . . . . . . . . 170

9.3.1 The Ordinary Di�erential Equation Formulation . . . . . . . . 170

9.3.2 ODE Form of the Classical Methods . . . . . . . . . . . . . . 172

9.4 Eigensystems of the Classical Methods . . . . . . . . . . . . . . . . . 173

9.4.1 The Point-Jacobi System . . . . . . . . . . . . . . . . . . . . . 174

9.4.2 The Gauss-Seidel System . . . . . . . . . . . . . . . . . . . . . 176

9.4.3 The SOR System . . . . . . . . . . . . . . . . . . . . . . . . . 180

9.5 Nonstationary Processes . . . . . . . . . . . . . . . . . . . . . . . . . 182

9.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

10 MULTIGRID 191

10.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

10.1.1 Eigenvector and Eigenvalue Identi�cation with Space Frequencies191

10.1.2 Properties of the Iterative Method . . . . . . . . . . . . . . . 192

10.2 The Basic Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

10.3 A Two-Grid Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

10.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

11 NUMERICAL DISSIPATION 203

11.1 One-Sided First-Derivative Space Di�erencing . . . . . . . . . . . . . 204

11.2 The Modi�ed Partial Di�erential Equation . . . . . . . . . . . . . . . 205

11.3 The Lax-Wendro� Method . . . . . . . . . . . . . . . . . . . . . . . . 207

11.4 Upwind Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

11.4.1 Flux-Vector Splitting . . . . . . . . . . . . . . . . . . . . . . . 210

11.4.2 Flux-Di�erence Splitting . . . . . . . . . . . . . . . . . . . . . 212

11.5 Arti�cial Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

11.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

12 SPLIT AND FACTORED FORMS 217

12.1 The Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

12.2 Factoring Physical Representations | Time Splitting . . . . . . . . . 218

12.3 Factoring Space Matrix Operators in 2{D . . . . . . . . . . . . . . . . 220

12.3.1 Mesh Indexing Convention . . . . . . . . . . . . . . . . . . . . 220

12.3.2 Data Bases and Space Vectors . . . . . . . . . . . . . . . . . . 221

12.3.3 Data Base Permutations . . . . . . . . . . . . . . . . . . . . . 221

12.3.4 Space Splitting and Factoring . . . . . . . . . . . . . . . . . . 223

12.4 Second-Order Factored Implicit Methods . . . . . . . . . . . . . . . . 226

12.5 Importance of Factored Forms in 2 and 3 Dimensions . . . . . . . . . 226

12.6 The Delta Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

12.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

13 LINEAR ANALYSIS OF SPLIT AND FACTORED FORMS 233

13.1 The Representative Equation for Circulant Operators . . . . . . . . . 233

13.2 Example Analysis of Circulant Systems . . . . . . . . . . . . . . . . . 234

13.2.1 Stability Comparisons of Time-Split Methods . . . . . . . . . 234

13.2.2 Analysis of a Second-Order Time-Split Method . . . . . . . . 237

13.3 The Representative Equation for Space-Split Operators . . . . . . . . 238

13.4 Example Analysis of 2-D Model Equations . . . . . . . . . . . . . . . 242

13.4.1 The Unfactored Implicit Euler Method . . . . . . . . . . . . . 242

13.4.2 The Factored Nondelta Form of the Implicit Euler Method . . 243

13.4.3 The Factored Delta Form of the Implicit Euler Method . . . . 243

13.4.4 The Factored Delta Form of the Trapezoidal Method . . . . . 244

13.5 Example Analysis of the 3-D Model Equation . . . . . . . . . . . . . 245

13.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

A USEFUL RELATIONS AND DEFINITIONS FROM LINEAR AL-

GEBRA 249

A.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

A.2 De�nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

A.3 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

A.4 Eigensystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

A.5 Vector and Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . 254

B SOME PROPERTIES OF TRIDIAGONAL MATRICES 257

B.1 Standard Eigensystem for Simple Tridiagonals . . . . . . . . . . . . . 257

B.2 Generalized Eigensystem for Simple Tridiagonals . . . . . . . . . . . . 258

B.3 The Inverse of a Simple Tridiagonal . . . . . . . . . . . . . . . . . . . 259

B.4 Eigensystems of Circulant Matrices . . . . . . . . . . . . . . . . . . . 260

B.4.1 Standard Tridiagonals . . . . . . . . . . . . . . . . . . . . . . 260

B.4.2 General Circulant Systems . . . . . . . . . . . . . . . . . . . . 261

B.5 Special Cases Found From Symmetries . . . . . . . . . . . . . . . . . 262

B.6 Special Cases Involving Boundary Conditions . . . . . . . . . . . . . 263

C THEHOMOGENEOUS PROPERTYOF THE EULER EQUATIONS265

Chapter 1

INTRODUCTION

1.1 Motivation

The material in this book originated from attempts to understand and systemize nu-

merical solution techniques for the partial di�erential equations governing the physics

of uid ow. As time went on and these attempts began to crystallize, underlying

constraints on the nature of the material began to form. The principal such constraint

was the demand for uni�cation. Was there one mathematical structure which could

be used to describe the behavior and results of most numerical methods in common

use in the �eld of uid dynamics? Perhaps the answer is arguable, but the authors

believe the answer is a�rmative and present this book as justi�cation for that be-

lief. The mathematical structure is the theory of linear algebra and the attendant

eigenanalysis of linear systems.

The ultimate goal of the �eld of computational uid dynamics (CFD) is to under-

stand the physical events that occur in the ow of uids around and within designated

objects. These events are related to the action and interaction of phenomena such

as dissipation, di�usion, convection, shock waves, slip surfaces, boundary layers, and

turbulence. In the �eld of aerodynamics, all of these phenomena are governed by

the compressible Navier-Stokes equations. Many of the most important aspects of

these relations are nonlinear and, as a consequence, often have no analytic solution.

This, of course, motivates the numerical solution of the associated partial di�erential

equations. At the same time it would seem to invalidate the use of linear algebra for

the classi�cation of the numerical methods. Experience has shown that such is not

the case.

As we shall see in a later chapter, the use of numerical methods to solve partial

di�erential equations introduces an approximation that, in e�ect, can change the

form of the basic partial di�erential equations themselves. The new equations, which

1

2 CHAPTER 1. INTRODUCTION

are the ones actually being solved by the numerical process, are often referred to as

the modi�ed partial di�erential equations. Since they are not precisely the same as

the original equations, they can, and probably will, simulate the physical phenomena

listed above in ways that are not exactly the same as an exact solution to the basic

partial di�erential equation. Mathematically, these di�erences are usually referred to

as truncation errors. However, the theory associated with the numerical analysis of

uid mechanics was developed predominantly by scientists deeply interested in the

physics of uid ow and, as a consequence, these errors are often identi�ed with a

particular physical phenomenon on which they have a strong e�ect. Thus methods are

said to have a lot of \arti�cial viscosity" or said to be highly dispersive. This means

that the errors caused by the numerical approximation result in a modi�ed partial

di�erential equation having additional terms that can be identi�ed with the physics

of dissipation in the �rst case and dispersion in the second. There is nothing wrong,

of course, with identifying an error with a physical process, nor with deliberately

directing an error to a speci�c physical process, as long as the error remains in some

engineering sense \small". It is safe to say, for example, that most numerical methods

in practical use for solving the nondissipative Euler equations create a modi�ed partial

di�erential equation that produces some form of dissipation. However, if used and

interpreted properly, these methods give very useful information.

Regardless of what the numerical errors are called, if their e�ects are not thor-

oughly understood and controlled, they can lead to serious di�culties, producing

answers that represent little, if any, physical reality. This motivates studying the

concepts of stability, convergence, and consistency. On the other hand, even if the

errors are kept small enough that they can be neglected (for engineering purposes),

the resulting simulation can still be of little practical use if ine�cient or inappropriate

algorithms are used. This motivates studying the concepts of sti�ness, factorization,

and algorithm development in general. All of these concepts we hope to clarify in

this book.

1.2 Background

The �eld of computational uid dynamics has a broad range of applicability. Indepen-

dent of the speci�c application under study, the following sequence of steps generally

must be followed in order to obtain a satisfactory solution.

1.2.1 Problem Speci�cation and Geometry Preparation

The �rst step involves the speci�cation of the problem, including the geometry, ow

conditions, and the requirements of the simulation. The geometry may result from

1.2. BACKGROUND 3

measurements of an existing con�guration or may be associated with a design study.

Alternatively, in a design context, no geometry need be supplied. Instead, a set

of objectives and constraints must be speci�ed. Flow conditions might include, for

example, the Reynolds number and Mach number for the ow over an airfoil. The

requirements of the simulation include issues such as the level of accuracy needed, the

turnaround time required, and the solution parameters of interest. The �rst two of

these requirements are often in con ict and compromise is necessary. As an example

of solution parameters of interest in computing the ow�eld about an airfoil, one may

be interested in i) the lift and pitching moment only, ii) the drag as well as the lift

and pitching moment, or iii) the details of the ow at some speci�c location.

1.2.2 Selection of Governing Equations and Boundary Con-

ditions

Once the problem has been speci�ed, an appropriate set of governing equations and

boundary conditions must be selected. It is generally accepted that the phenomena of

importance to the �eld of continuum uid dynamics are governed by the conservation

of mass, momentum, and energy. The partial di�erential equations resulting from

these conservation laws are referred to as the Navier-Stokes equations. However, in

the interest of e�ciency, it is always prudent to consider solving simpli�ed forms

of the Navier-Stokes equations when the simpli�cations retain the physics which are

essential to the goals of the simulation. Possible simpli�ed governing equations include

the potential- ow equations, the Euler equations, and the thin-layer Navier-Stokes

equations. These may be steady or unsteady and compressible or incompressible.

Boundary types which may be encountered include solid walls, in ow and out ow

boundaries, periodic boundaries, symmetry boundaries, etc. The boundary conditions

which must be speci�ed depend upon the governing equations. For example, at a solid

wall, the Euler equations require ow tangency to be enforced, while the Navier-Stokes

equations require the no-slip condition. If necessary, physical models must be chosen

for processes which cannot be simulated within the speci�ed constraints. Turbulence

is an example of a physical process which is rarely simulated in a practical context (at

the time of writing) and thus is often modelled. The success of a simulation depends

greatly on the engineering insight involved in selecting the governing equations and

physical models based on the problem speci�cation.

1.2.3 Selection of Gridding Strategy and Numerical Method

Next a numerical method and a strategy for dividing the ow domain into cells, or

elements, must be selected. We concern ourselves here only with numerical meth-

ods requiring such a tessellation of the domain, which is known as a grid, or mesh.


Many di�erent gridding strategies exist, including structured, unstructured, hybrid,

composite, and overlapping grids. Furthermore, the grid can be altered based on

the solution in an approach known as solution-adaptive gridding. The numerical

methods generally used in CFD can be classi�ed as �nite-di�erence, �nite-volume,

�nite-element, or spectral methods. The choices of a numerical method and a grid-

ding strategy are strongly interdependent. For example, the use of �nite-di�erence

methods is typically restricted to structured grids. Here again, the success of a sim-

ulation can depend on appropriate choices for the problem or class of problems of

interest.

1.2.4 Assessment and Interpretation of Results

Finally, the results of the simulation must be assessed and interpreted. This step can

require post-processing of the data, for example calculation of forces and moments,

and can be aided by sophisticated ow visualization tools and error estimation tech-

niques. It is critical that the magnitude of both numerical and physical-model errors

be well understood.

1.3 Overview

It should be clear that successful simulation of uid ows can involve a wide range of

issues from grid generation to turbulence modelling to the applicability of various sim-

pli�ed forms of the Navier-Stokes equations. Many of these issues are not addressed

in this book. Some of them are presented in the books by Anderson, Tannehill, and

Pletcher [1] and Hirsch [2]. Instead we focus on numerical methods, with emphasis

on �nite-di�erence and �nite-volume methods for the Euler and Navier-Stokes equa-

tions. Rather than presenting the details of the most advanced methods, which are

still evolving, we present a foundation for developing, analyzing, and understanding

such methods.

Fortunately, to develop, analyze, and understand most numerical methods used to

�nd solutions for the complete compressible Navier-Stokes equations, we can make use

of much simpler expressions, the so-called \model" equations. These model equations

isolate certain aspects of the physics contained in the complete set of equations. Hence

their numerical solution can illustrate the properties of a given numerical method

when applied to a more complicated system of equations which governs similar phys-

ical phenomena. Although the model equations are extremely simple and easy to

solve, they have been carefully selected to be representative, when used intelligently,

of di�culties and complexities that arise in realistic two- and three-dimensional uid

ow simulations. We believe that a thorough understanding of what happens when

1.4. NOTATION 5

numerical approximations are applied to the model equations is a major �rst step in

making con�dent and competent use of numerical approximations to the Euler and

Navier-Stokes equations. As a word of caution, however, it should be noted that,

although we can learn a great deal by studying numerical methods as applied to the

model equations and can use that information in the design and application of nu-

merical methods to practical problems, there are many aspects of practical problems

which can only be understood in the context of the complete physical systems.

1.4 Notation

The notation is generally explained as it is introduced. Bold type is reserved for real

physical vectors, such as velocity. The vector symbol~ is used for the vectors (or

column matrices) which contain the values of the dependent variable at the nodes

of a grid. Otherwise, the use of a vector consisting of a collection of scalars should

be apparent from the context and is not identi�ed by any special notation. For

example, the variable u can denote a scalar Cartesian velocity component in the Euler

and Navier-Stokes equations, a scalar quantity in the linear convection and di�usion

equations, and a vector consisting of a collection of scalars in our presentation of

hyperbolic systems. Some of the abbreviations used throughout the text are listed

and de�ned below.

PDE Partial di�erential equation

ODE Ordinary di�erential equation

O�E Ordinary di�erence equation

RHS Right-hand side

P.S. Particular solution of an ODE or system of ODE's

S.S. Fixed (time-invariant) steady-state solution

k-D k-dimensional space�~bc

�Boundary conditions, usually a vector

O(�) A term of order (i.e., proportional to) �

Chapter 2

CONSERVATION LAWS AND

THE MODEL EQUATIONS

We start out by casting our equations in the most general form, the integral conserva-

tion-law form, which is useful in understanding the concepts involved in �nite-volume

schemes. The equations are then recast into divergence form, which is natural for

�nite-di�erence schemes. The Euler and Navier-Stokes equations are brie y discussed

in this Chapter. The main focus, though, will be on representative model equations,

in particular, the convection and di�usion equations. These equations contain many

of the salient mathematical and physical features of the full Navier-Stokes equations.

The concepts of convection and di�usion are prevalent in our development of nu-

merical methods for computational uid dynamics, and the recurring use of these

model equations allows us to develop a consistent framework of analysis for consis-

tency, accuracy, stability, and convergence. The model equations we study have two

properties in common. They are linear partial di�erential equations (PDE's) with

coe�cients that are constant in both space and time, and they represent phenomena

of importance to the analysis of certain aspects of uid dynamic problems.

2.1 Conservation Laws

Conservation laws, such as the Euler and Navier-Stokes equations and our model

equations, can be written in the following integral form:

ZV (t2)

QdV �ZV (t1)

QdV +

Z t2

t1

IS(t)

n:FdSdt =Z t2

t1

ZV (t)

PdV dt (2.1)

In this equation, Q is a vector containing the set of variables which are conserved,

e.g., mass, momentum, and energy, per unit volume. The equation is a statement of

7

8 CHAPTER 2. CONSERVATION LAWS AND THE MODEL EQUATIONS

the conservation of these quantities in a �nite region of space with volume V (t) and

surface area S(t) over a �nite interval of time t2 � t1. In two dimensions, the region

of space, or cell, is an area A(t) bounded by a closed contour C(t). The vector n is

a unit vector normal to the surface pointing outward, F is a set of vectors, or tensor,

containing the ux of Q per unit area per unit time, and P is the rate of production

of Q per unit volume per unit time. If all variables are continuous in time, then Eq.

2.1 can be rewritten as

d

dt

ZV (t)

QdV +

IS(t)

n:FdS =

ZV (t)

PdV (2.2)

Those methods which make various numerical approximations of the integrals in Eqs.

2.1 and 2.2 and �nd a solution for Q on that basis are referred to as �nite-volume

methods. Many of the advanced codes written for CFD applications are based on the

�nite-volume concept.

On the other hand, a partial derivative form of a conservation law can also be

derived. The divergence form of Eq. 2.2 is obtained by applying Gauss's theorem to

the ux integral, leading to@Q

@t+r:F = P (2.3)

where r: is the well-known divergence operator given, in Cartesian coordinates, by

r: �

i@

@x+ j

@

@y+ k

@

@z

!: (2.4)

and i; j, and k are unit vectors in the x; y, and z coordinate directions, respectively.

Those methods which make various approximations of the derivatives in Eq. 2.3 and

�nd a solution for Q on that basis are referred to as �nite-di�erence methods.

2.2 The Navier-Stokes and Euler Equations

The Navier-Stokes equations form a coupled system of nonlinear PDE's describing

the conservation of mass, momentum and energy for a uid. For a Newtonian uid

in one dimension, they can be written as

@Q

@t+@E

@x= 0 (2.5)

with

Q =

2666664

�

�u

e

3777775 ; E =

2666664

�u

�u2 + p

u(e+ p)

3777775�

2666664

0

43�@u@x

43�u@u

@x+ �@T

@x

3777775 (2.6)

2.2. THE NAVIER-STOKES AND EULER EQUATIONS 9

where � is the uid density, u is the velocity, e is the total energy per unit volume, p is

the pressure, T is the temperature, � is the coe�cient of viscosity, and � is the thermal

conductivity. The total energy e includes internal energy per unit volume �� (where

� is the internal energy per unit mass) and kinetic energy per unit volume �u2=2.

These equations must be supplemented by relations between � and � and the uid

state as well as an equation of state, such as the ideal gas law. Details can be found

in Anderson, Tannehill, and Pletcher [1] and Hirsch [2]. Note that the convective

uxes lead to �rst derivatives in space, while the viscous and heat conduction terms

involve second derivatives. This form of the equations is called conservation-law or

conservative form. Non-conservative forms can be obtained by expanding derivatives

of products using the product rule or by introducing di�erent dependent variables,

such as u and p. Although non-conservative forms of the equations are analytically

the same as the above form, they can lead to quite di�erent numerical solutions in

terms of shock strength and shock speed, for example. Thus the conservative form is

appropriate for solving ows with features such as shock waves.

Many ows of engineering interest are steady (time-invariant), or at least may be

treated as such. For such ows, we are often interested in the steady-state solution of

the Navier-Stokes equations, with no interest in the transient portion of the solution.

The steady solution to the one-dimensional Navier-Stokes equations must satisfy

@E

@x= 0 (2.7)

If we neglect viscosity and heat conduction, the Euler equations are obtained. In

two-dimensional Cartesian coordinates, these can be written as

@Q

@t+@E

@x+@F

@y= 0 (2.8)

with

Q =

26664q1q2q3q4

37775 =

26664�

�u

�v

e

37775 ; E =

26664

�u

�u2 + p

�uv

u(e+ p)

37775 ; F =

26664

�v

�uv

�v2 + p

v(e+ p)

37775 (2.9)

where u and v are the Cartesian velocity components. Later on we will make use of

the following form of the Euler equations as well:

@Q

@t+ A

@Q

@x+B

@Q

@y= 0 (2.10)

The matrices A = @E

@Qand B = @F

@Qare known as the ux Jacobians. The ux vectors

given above are written in terms of the primitive variables, �, u, v, and p. In order


to derive the ux Jacobian matrices, we must �rst write the ux vectors E and F in

terms of the conservative variables, q1, q2, q3, and q4, as follows:

E =

2666666666664

E1

E2

E3

E4

3777777777775=

2666666666666664

q2

( � 1)q4 +3� 2

q22

q1�

�12

q23

q1

q3q2q1

q4q2q1�

�12

�q32

q21

+q23q2

q21

�

3777777777777775

(2.11)

F =

2666666666664

F1

F2

F3

F4

3777777777775=

266666666666664

q3

q3q2q1

( � 1)q4 +3� 2

q23

q1�

�12

q22

q1

q4q3q1�

�12

�q22q3

q21

+q33

q21

�

377777777777775

(2.12)

We have assumed that the pressure satis�es p = ( � 1)[e � �(u2 + v2)=2] from the

ideal gas law, where is the ratio of speci�c heats, cp=cv. From this it follows that

the ux Jacobian of E can be written in terms of the conservative variables as

A =@Ei

@qj=

266666666666664

0 1 0 0

a21 (3� )�q2q1

�(1� )

�q3q1

� � 1

�

�q2q1

��q3q1

� �q3q1

� �q2q1

�0

a41 a42 a43 �q2q1

�

377777777777775

(2.13)

where

a21 = � 1

2

q3

q1

!2

�

3�

2

q2

q1

!2

2.2. THE NAVIER-STOKES AND EULER EQUATIONS 11

a41 = ( � 1)

24 q2

q1

!3

+

q3

q1

!2 q2

q1

!35�

q4

q1

! q2

q1

!

a42 =

q4

q1

!�

� 1

2

243 q2

q1

!2

+

q3

q1

!235

a43 = �( � 1)

q2

q1

! q3

q1

!(2.14)

and in terms of the primitive variables as

A =

2666666666664

0 1 0 0

a21 (3� )u (1� )v ( � 1)

�uv v u 0

a41 a42 a43 u

3777777777775

(2.15)

where

a21 = � 1

2v2 �

3�

2u2

a41 = ( � 1)u(u2 + v2)� ue

�

a42 = e

��

� 1

2(3u2 + v2)

a43 = (1� )uv (2.16)

Derivation of the two forms of B = @F=@Q is similar. The eigenvalues of the ux

Jacobian matrices are purely real. This is the de�ning feature of hyperbolic systems

of PDE's, which are further discussed in Section 2.5. The homogeneous property of

the Euler equations is discussed in Appendix C.

The Navier-Stokes equations include both convective and di�usive uxes. This

motivates the choice of our two scalar model equations associated with the physics

of convection and di�usion. Furthermore, aspects of convective phenomena associ-

ated with coupled systems of equations such as the Euler equations are important in

developing numerical methods and boundary conditions. Thus we also study linear

hyperbolic systems of PDE's.


2.3 The Linear Convection Equation

2.3.1 Di�erential Form

The simplest linear model for convection and wave propagation is the linear convection

equation given by the following PDE:

@u

@t+ a

@u

@x= 0 (2.17)

Here u(x; t) is a scalar quantity propagating with speed a, a real constant which may

be positive or negative. The manner in which the boundary conditions are speci�ed

separates the following two phenomena for which this equation is a model:

(1) In one type, the scalar quantity u is given on one boundary, corresponding

to a wave entering the domain through this \in ow" boundary. No bound-

ary condition is speci�ed at the opposite side, the \out ow" boundary. This

is consistent in terms of the well-posedness of a 1st-order PDE. Hence the

wave leaves the domain through the out ow boundary without distortion or

re ection. This type of phenomenon is referred to, simply, as the convection

problem. It represents most of the \usual" situations encountered in convect-

ing systems. Note that the left-hand boundary is the in ow boundary when

a is positive, while the right-hand boundary is the in ow boundary when a is

negative.

(2) In the other type, the ow being simulated is periodic. At any given time,

what enters on one side of the domain must be the same as that which is

leaving on the other. This is referred to as the biconvection problem. It is

the simplest to study and serves to illustrate many of the basic properties of

numerical methods applied to problems involving convection, without special

consideration of boundaries. Hence, we pay a great deal of attention to it in

the initial chapters.

Now let us consider a situation in which the initial condition is given by u(x; 0) =

u0(x), and the domain is in�nite. It is easy to show by substitution that the exact

solution to the linear convection equation is then

u(x; t) = u0(x� at) (2.18)

The initial waveform propagates unaltered with speed jaj to the right if a is positive

and to the left if a is negative. With periodic boundary conditions, the waveform

travels through one boundary and reappears at the other boundary, eventually re-

turning to its initial position. In this case, the process continues forever without any

2.3. THE LINEAR CONVECTION EQUATION 13

change in the shape of the solution. Preserving the shape of the initial condition

u0(x) can be a di�cult challenge for a numerical method.

2.3.2 Solution in Wave Space

We now examine the biconvection problem in more detail. Let the domain be given

by 0 � x � 2�. We restrict our attention to initial conditions in the form

u(x; 0) = f(0)ei�x (2.19)

where f(0) is a complex constant, and � is the wavenumber. In order to satisfy the

periodic boundary conditions, � must be an integer. It is a measure of the number of

wavelengths within the domain. With such an initial condition, the solution can be

written as

u(x; t) = f(t)ei�x (2.20)

where the time dependence is contained in the complex function f(t). Substituting

this solution into the linear convection equation, Eq. 2.17, we �nd that f(t) satis�es

the following ordinary di�erential equation (ODE)

df

dt= �ia�f (2.21)

which has the solution

f(t) = f(0)e�ia�t (2.22)

Substituting f(t) into Eq. 2.20 gives the following solution

u(x; t) = f(0)ei�(x�at) = f(0)ei(�x�!t) (2.23)

where the frequency, !, the wavenumber, �, and the phase speed, a, are related by

! = �a (2.24)

The relation between the frequency and the wavenumber is known as the dispersion

relation. The linear relation given by Eq. 2.24 is characteristic of wave propagation

in a nondispersive medium. This means that the phase speed is the same for all

wavenumbers. As we shall see later, most numerical methods introduce some disper-

sion; that is, in a simulation, waves with di�erent wavenumbers travel at di�erent

speeds.

An arbitrary initial waveform can be produced by summing initial conditions of

the form of Eq. 2.19. For M modes, one obtains

u(x; 0) =MXm=1

fm(0)ei�mx (2.25)


where the wavenumbers are often ordered such that �1 � �2 � � � � � �M . Since the

wave equation is linear, the solution is obtained by summing solutions of the form of

Eq. 2.23, giving

u(x; t) =MXm=1

fm(0)ei�m(x�at) (2.26)

Dispersion and dissipation resulting from a numerical approximation will cause the

shape of the solution to change from that of the original waveform.

2.4 The Di�usion Equation

2.4.1 Di�erential Form

Di�usive uxes are associated with molecular motion in a continuum uid. A simple

linear model equation for a di�usive process is

@u

@t= �

@2u

@x2(2.27)

where � is a positive real constant. For example, with u representing the tempera-

ture, this parabolic PDE governs the di�usion of heat in one dimension. Boundary

conditions can be periodic, Dirichlet (speci�ed u), Neumann (speci�ed @u=@x), or

mixed Dirichlet/Neumann.

In contrast to the linear convection equation, the di�usion equation has a nontrivial

steady-state solution, which is one that satis�es the governing PDE with the partial

derivative in time equal to zero. In the case of Eq. 2.27, the steady-state solution

must satisfy

@2u

@x2= 0 (2.28)

Therefore, u must vary linearly with x at steady state such that the boundary con-

ditions are satis�ed. Other steady-state solutions are obtained if a source term g(x)

is added to Eq. 2.27, as follows:

@u

@t= �

"@2u

@x2� g(x)

#(2.29)

giving a steady state-solution which satis�es

@2u

@x2� g(x) = 0 (2.30)

2.4. THE DIFFUSION EQUATION 15

In two dimensions, the di�usion equation becomes

@u

@t= �

"@2u

@x2+@2u

@y2� g(x; y)

#(2.31)

where g(x; y) is again a source term. The corresponding steady equation is

@2u

@x2+@2u

@y2� g(x; y) = 0 (2.32)

While Eq. 2.31 is parabolic, Eq. 2.32 is elliptic. The latter is known as the Poisson

equation for nonzero g, and as Laplace's equation for zero g.

2.4.2 Solution in Wave Space

We now consider a series solution to Eq. 2.27. Let the domain be given by 0 � x � �

with boundary conditions u(0) = ua, u(�) = ub. It is clear that the steady-state

solution is given by a linear function which satis�es the boundary conditions, i.e.,

h(x) = ua + (ub � ua)x=�. Let the initial condition be

u(x; 0) =MXm=1

fm(0) sin�mx+ h(x) (2.33)

where � must be an integer in order to satisfy the boundary conditions. A solution

of the form

u(x; t) =MXm=1

fm(t) sin �mx + h(x) (2.34)

satis�es the initial and boundary conditions. Substituting this form into Eq. 2.27

gives the following ODE for fm:

dfm

dt= ��2m�fm (2.35)

and we �nd

fm(t) = fm(0)e��2m�t (2.36)

Substituting fm(t) into equation 2.34, we obtain

u(x; t) =MXm=1

fm(0)e��2

m�t sin�mx + h(x) (2.37)

The steady-state solution (t!1) is simply h(x). Eq. 2.37 shows that high wavenum-

ber components (large �m) of the solution decay more rapidly than low wavenumber

components, consistent with the physics of di�usion.


2.5 Linear Hyperbolic Systems

The Euler equations, Eq. 2.8, form a hyperbolic system of partial di�erential equa-

tions. Other systems of equations governing convection and wave propagation phe-

nomena, such as the Maxwell equations describing the propagation of electromagnetic

waves, are also of hyperbolic type. Many aspects of numerical methods for such sys-

tems can be understood by studying a one-dimensional constant-coe�cient linear

system of the form@u

@t+ A

@u

@x= 0 (2.38)

where u = u(x; t) is a vector of length m and A is a real m � m matrix. For

conservation laws, this equation can also be written in the form

@u

@t+@f

@x= 0 (2.39)

where f is the ux vector and A = @f

@uis the ux Jacobian matrix. The entries in the

ux Jacobian are

aij =@fi

@uj(2.40)

The ux Jacobian for the Euler equations is derived in Section 2.2.

Such a system is hyperbolic if A is diagonalizable with real eigenvalues.1 Thus

� = X�1AX (2.41)

where � is a diagonal matrix containing the eigenvalues of A, and X is the matrix

of right eigenvectors. Premultiplying Eq. 2.38 by X�1, postmultiplying A by the

product XX�1, and noting that X and X�1 are constants, we obtain

@X�1u

@t+@

�z }| {X�1AX X�1u

@x= 0 (2.42)

With w = X�1u, this can be rewritten as

@w

@t+ �

@w

@x= 0 (2.43)

When written in this manner, the equations have been decoupled into m scalar equa-

tions of the form@wi

@t+ �i

@wi

@x= 0 (2.44)

1See Appendix A for a brief review of some basic relations and de�nitions from linear algebra.

2.6. PROBLEMS 17

The elements of w are known as characteristic variables. Each characteristic variable

satis�es the linear convection equation with the speed given by the corresponding

eigenvalue of A.

Based on the above, we see that a hyperbolic system in the form of Eq. 2.38 has a

solution given by the superposition of waves which can travel in either the positive or

negative directions and at varying speeds. While the scalar linear convection equation

is clearly an excellent model equation for hyperbolic systems, we must ensure that

our numerical methods are appropriate for wave speeds of arbitrary sign and possibly

widely varying magnitudes.

The one-dimensional Euler equations can also be diagonalized, leading to three

equations in the form of the linear convection equation, although they remain non-

linear, of course. The eigenvalues of the ux Jacobian matrix, or wave speeds, are

u; u + c, and u � c, where u is the local uid velocity, and c =q p=� is the local

speed of sound. The speed u is associated with convection of the uid, while u + c

and u � c are associated with sound waves. Therefore, in a supersonic ow, where

juj > c, all of the wave speeds have the same sign. In a subsonic ow, where juj < c,

wave speeds of both positive and negative sign are present, corresponding to the fact

that sound waves can travel upstream in a subsonic ow.

The signs of the eigenvalues of the matrix A are also important in determining

suitable boundary conditions. The characteristic variables each satisfy the linear con-

vection equation with the wave speed given by the corresponding eigenvalue. There-

fore, the boundary conditions can be speci�ed accordingly. That is, characteristic

variables associated with positive eigenvalues can be speci�ed at the left boundary,

which corresponds to in ow for these variables. Characteristic variables associated

with negative eigenvalues can be speci�ed at the right boundary, which is the in-

ow boundary for these variables. While other boundary condition treatments are

possible, they must be consistent with this approach.

2.6 Problems

1. Show that the 1-D Euler equations can be written in terms of the primitive

variables R = [�; u; p]T as follows:

@R

@t+M

@R

@x= 0

where

M =

264u � 0

0 u ��1

0 p u

375


Assume an ideal gas, p = ( � 1)(e� �u2=2).

2. Find the eigenvalues and eigenvectors of the matrix M derived in question 1.

3. Derive the ux Jacobian matrix A = @E=@Q for the 1-D Euler equations result-

ing from the conservative variable formulation (Eq. 2.5). Find its eigenvalues

and compare with those obtained in question 2.

4. Show that the two matricesM and A derived in questions 1 and 3, respectively,

are related by a similarity transform. (Hint: make use of the matrix S =

@Q=@R.)

5. Write the 2-D di�usion equation, Eq. 2.31, in the form of Eq. 2.2.

6. Given the initial condition u(x; 0) = sinx de�ned on 0 � x � 2�, write it in the

form of Eq. 2.25, that is, �nd the necessary values of fm(0). (Hint: use M = 2

with �1 = 1 and �2 = �1.) Next consider the same initial condition de�ned

only at x = 2�j=4, j = 0; 1; 2; 3. Find the values of fm(0) required to reproduce

the initial condition at these discrete points using M = 4 with �m = m� 1.

7. Plot the �rst three basis functions used in constructing the exact solution to

the di�usion equation in Section 2.4.2. Next consider a solution with boundary

conditions ua = ub = 0, and initial conditions from Eq. 2.33 with fm(0) = 1

for 1 � m � 3, fm(0) = 0 for m > 3. Plot the initial condition on the domain

0 � x � �. Plot the solution at t = 1 with � = 1.

8. Write the classical wave equation @2u=@t2 = c2@2u=@x2 as a �rst-order system,

i.e., in the form

@U

@t+ A

@U

@x= 0

where U = [@u=@x; @u=@t]T . Find the eigenvalues and eigenvectors of A.

9. The Cauchy-Riemann equations are formed from the coupling of the steady

compressible continuity (conservation of mass) equation

@�u

@x+@�v

@y= 0

and the vorticity de�nition

! = �

@v

@x+@u

@y= 0

2.6. PROBLEMS 19

where ! = 0 for irrotational ow. For isentropic and homenthalpic ow, the

system is closed by the relation

� =

�1�

� 1

2

�u2 + v2 � 1

�� 1

�1

Note that the variables have been nondimensionalized. Combining the two

PDE's, we have

@f(q)

@x+@g(q)

@y= 0

where

q =

�u

v

�; f =

��u

v

�; g =

��v

�u

�

One approach to solving these equations is to add a time-dependent term and

�nd the steady solution of the following equation:

@q

@t+@f

@x+@g

@y= 0

(a) Find the ux Jacobians of f and g with respect to q.

(b) Determine the eigenvalues of the ux Jacobians.

(c) Determine the conditions (in terms of � and u) under which the system is

hyperbolic, i.e., has real eigenvalues.

(d) Are the above uxes homogeneous? (See Appendix C.)

��

� � ��

� ��

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Chapter 4

THE SEMI-DISCRETE

APPROACH

One strategy for obtaining �nite-di�erence approximations to a PDE is to start by

di�erencing the space derivatives only, without approximating the time derivative.

In the following chapters, we proceed with an analysis making considerable use of

this concept, which we refer to as the semi-discrete approach. Di�erencing the space

derivatives converts the basic PDE into a set of coupled ODE's. In the most general

notation, these ODE's would be expressed in the form

d~u

dt= ~F (~u; t) (4.1)

which includes all manner of nonlinear and time-dependent possibilities. On occasion,

we use this form, but the rest of this chapter is devoted to a more specialized matrix

notation described below.

Another strategy for constructing a �nite-di�erence approximation to a PDE is

to approximate all the partial derivatives at once. This generally leads to a point

di�erence operator (see Section 3.3.1) which, in turn, can be used for the time advance

of the solution at any given point in the mesh. As an example let us consider the

model equation for di�usion@u

@t= �

@2u

@x2

Using three-point central-di�erencing schemes for both the time and space derivatives,

we �ndu(n+1)j � u

(n�1)j

2h= �

24u(n)j+1 � 2u

(n)j + u

(n)j�1

�x2

35

or

u(n+1)j = u

(n�1)j +

2h�

�x2

hu(n)j+1 � 2u

(n)j + u

(n)j�1

i(4.2)

51

52 CHAPTER 4. THE SEMI-DISCRETE APPROACH

Clearly Eq. 4.2 is a di�erence equation which can be used at the space point j to

advance the value of u from the previous time levels n and n � 1 to the level n + 1.

It is a full discretization of the PDE. Note, however, that the spatial and temporal

discretizations are separable. Thus, this method has an intermediate semi-discrete

form and can be analyzed by the methods discussed in the next few chapters.

Another possibility is to replace the value of u(n)j in the right hand side of Eq. 4.2

with the time average of u at that point, namely (u(n+1)j + u

(n�1)j )=2. This results in

the formula

u(n+1)j = u

(n�1)j +

2h�

�x2

24u(n)j+1 � 2

0@u(n+1)j + u

(n�1)j

2

1A+ u

(n)j�1

35 (4.3)

which can be solved for u(n+1) and time advanced at the point j. In this case, the

spatial and temporal discretizations are not separable, and no semi-discrete form

exists.

Equation 4.2 is sometimes called Richardson's method of overlapping steps and

Eq. 4.3 is referred to as the DuFort-Frankel method. As we shall see later on, there

are subtle points to be made about using these methods to �nd a numerical solution

to the di�usion equation. There are a number of issues concerning the accuracy,

stability, and convergence of Eqs. 4.2 and 4.3 which we cannot comment on until we

develop a framework for such investigations. We introduce these methods here only to

distinguish between methods in which the temporal and spatial terms are discretized

separately and those for which no such separation is possible. For the time being, we

shall separate the space di�erence approximations from the time di�erencing. In this

approach, we reduce the governing PDE's to ODE's by discretizing the spatial terms

and use the well-developed theory of ODE solutions to aid us in the development of

an analysis of accuracy and stability.

4.1 Reduction of PDE's to ODE's

4.1.1 The Model ODE's

First let us consider the model PDE's for di�usion and biconvection described in

Chapter 2. In these simple cases, we can approximate the space derivatives with

di�erence operators and express the resulting ODE's with a matrix formulation. This

is a simple and natural formulation when the ODE's are linear.

4.1. REDUCTION OF PDE'S TO ODE'S 53

Model ODE for Di�usion

For example, using the 3-point central-di�erencing scheme to represent the second

derivative in the scalar PDE governing di�usion leads to the following ODE di�usion

model

d~u

dt=

�

�x2B(1;�2; 1)~u+ ~(bc) (4.4)

with Dirichlet boundary conditions folded into the ~(bc) vector.

Model ODE for Biconvection

The term biconvection was introduced in Section 2.3. It is used for the scalar con-

vection model when the boundary conditions are periodic. In this case, the 3-point

central-di�erencing approximation produces the ODE model given by

d~u

dt= �

a

2�xBp(�1; 0; 1)~u (4.5)

where the boundary condition vector is absent because the ow is periodic.

Eqs. 4.4 and 4.5 are the model ODE's for di�usion and biconvection of a scalar in

one dimension. They are linear with coe�cient matrices which are independent of x

and t.

4.1.2 The Generic Matrix Form

The generic matrix form of a semi-discrete approximation is expressed by the equation

d~u

dt= A~u� ~f(t) (4.6)

Note that the elements in the matrix A depend upon both the PDE and the type of

di�erencing scheme chosen for the space terms. The vector ~f(t) is usually determined

by the boundary conditions and possibly source terms. In general, even the Euler

and Navier-Stokes equations can be expressed in the form of Eq. 4.6. In such cases

the equations are nonlinear, that is, the elements of A depend on the solution ~u and

are usually derived by �nding the Jacobian of a ux vector. Although the equations

are nonlinear, the linear analysis presented in this book leads to diagnostics that are

surprisingly accurate when used to evaluate many aspects of numerical methods as

they apply to the Euler and Navier-Stokes equations.


4.2 Exact Solutions of Linear ODE's

In order to advance Eq. 4.1 in time, the system of ODE's must be integrated using a

time-marching method. In order to analyze time-marching methods, we will make use

of exact solutions of coupled systems of ODE's, which exist under certain conditions.

The ODE's represented by Eq. 4.1 are said to be linear if F is linearly dependent on

u (i.e., if @F=@u = A where A is independent of u). As we have already pointed out,

when the ODE's are linear they can be expressed in a matrix notation as Eq. 4.6 in

which the coe�cient matrix, A, is independent of u. If A does depend explicitly on

t, the general solution cannot be written; whereas, if A does not depend explicitly on

t, the general solution to Eq. 4.6 can be written. This holds regardless of whether or

not the forcing function, ~f , depends explicitly on t.

As we shall soon see, the exact solution of Eq. 4.6 can be written in terms of

the eigenvalues and eigenvectors of A. This will lead us to a representative scalar

equation for use in analyzing time-marching methods. These ideas are developed in

the following sections.

4.2.1 Eigensystems of Semi-Discrete Linear Forms

Complete Systems

An M �M matrix is represented by a complete eigensystem if it has a complete set

of linearly independent eigenvectors (see Appendix A) . An eigenvector, ~xm, and its

corresponding eigenvalue, �m, have the property that

A~xm = �m~xm or

[A� �mI]~xm = 0 (4.7)

The eigenvalues are the roots of the equation

det[A� �I] = 0

We form the right-hand eigenvector matrix of a complete system by �lling its columns

with the eigenvectors ~xm:

X =h~x1 ; ~x2 : : : ; ~xM

iThe inverse is the left-hand eigenvector matrix, and together they have the property

that

X�1AX = � (4.8)

where � is a diagonal matrix whose elements are the eigenvalues of A.

4.2. EXACT SOLUTIONS OF LINEAR ODE'S 55

Defective Systems

If anM�M matrix does not have a complete set of linearly independent eigenvectors,

it cannot be transformed to a diagonal matrix of scalars, and it is said to be defective.

It can, however, be transformed to a diagonal set of blocks, some of which may be

scalars (see Appendix A). In general, there exists some S which transforms any

matrix A such that

S�1AS = J

where

J =

26666664

J1J2

. . .

Jm. . .

37777775

and

J (n)m

=

266664�m 1

�m. . .. . . 1

�m

377775

1......

n

The matrix J is said to be in Jordan canonical form, and an eigenvalue with multi-

plicity n within a Jordan block is said to be a defective eigenvalue. Defective systems

play a role in numerical stability analysis.

4.2.2 Single ODE's of First- and Second-Order

First-Order Equations

The simplest nonhomogeneous ODE of interest is given by the single, �rst-order

equation

du

dt= �u+ ae�t (4.9)

where �, a, and � are scalars, all of which can be complex numbers. The equation

is linear because � does not depend on u, and has a general solution because � does

not depend on t. It has a steady-state solution if the right-hand side is independent

of t, i.e., if � = 0, and is homogeneous if the forcing function is zero, i.e., if a = 0.

Although it is quite simple, the numerical analysis of Eq. 4.9 displays many of the

fundamental properties and issues involved in the construction and study of most

popular time-marching methods. This theme will be developed as we proceed.


The exact solution of Eq. 4.9 is, for � 6= �,

u(t) = c1e�t +

ae�t

��

where c1 is a constant determined by the initial conditions. In terms of the initial

value of u, it can be written

u(t) = u(0)e�t + ae�t � e�t

��

The interesting question can arise: What happens to the solution of Eq. 4.9 when

� = �? This is easily found by setting � = � + �, solving, and then taking the limit

as �! 0. Using this limiting device, we �nd that the solution to

du

dt= �u+ ae�t (4.10)

is given by

u(t) = [u(0) + at]e�t

As we shall soon see, this solution is required for the analysis of defective systems.

Second-Order Equations

The homogeneous form of a second-order equation is given by

d2u

dt2+ a1

du

dt+ a0u = 0 (4.11)

where a1 and a0 are complex constants. Now we introduce the di�erential operator

D such that

D �

d

dt

and factor u(t) out of Eq. 4.11, giving

(D2 + a1D + a0) u(t) = 0

The polynomial in D is referred to as a characteristic polynomial and designated

P (D). Characteristic polynomials are fundamental to the analysis of both ODE's and

O�E's, since the roots of these polynomials determine the solutions of the equations.

For ODE's, we often label these roots in order of increasing modulus as �1, �2, � � �, �m,


� � �, �M . They are found by solving the equation P (�) = 0. In our simple example,

there would be two roots, �1 and �2, determined from

P (�) = �2 + a1�+ a0 = 0 (4.12)

and the solution to Eq. 4.11 is given by

u(t) = c1e�1t + c2e

�2t (4.13)

where c1 and c2 are constants determined from initial conditions. The proof of this

is simple and is found by substituting Eq. 4.13 into Eq. 4.11. One �nds the result

c1e�1t(�21 + a1�1 + a0) + c2e

�2t(�22 + a1�2 + a0)

which is identically zero for all c1, c2, and t if and only if the �'s satisfy Eq. 4.12.

4.2.3 Coupled First-Order ODE's

A Complete System

A set of coupled, �rst-order, homogeneous equations is given by

u01 = a11 u1 + a12 u2

u02 = a21 u1 + a22 u2 (4.14)

which can be written

~u0

= A~u ; ~u = [u1; u2]T ; A = (aij) =

"a11 a12a21 a22

#

Consider the possibility that a solution is represented by

u1 = c1 x11e�1t + c2 x12e

�2t

u2 = c1 x21e�1t + c2 x22e

�2t (4.15)

By substitution, these are indeed solutions to Eq. 4.14 if and only if"a11 a12a21 a22

# "x11x21

#= �1

"x11x21

#;

"a11 a12a21 a22

# "x12x22

#= �2

"x12x22

#(4.16)

Notice that a higher-order equation can be reduced to a coupled set of �rst-order

equations by introducing a new set of dependent variables. Thus, by setting

u1 = u0 ; u2 = u ;

we �nd Eq. 4.11 can be written

u01 = �a1u1 � a0u2

u02 = u1 (4.17)

which is a subset of Eq. 4.14.


A Derogatory System

Eq. 4.15 is still a solution to Eq. 4.14 if �1 = �2 = �, provided two linearly independent

vectors exist to satisfy Eq. 4.16 with A = �. In this case

�� 0

0 �

� �1

0

�= �

�1

0

�and

�� 0

0 �

� �0

1

�= �

�0

1

�

provide such a solution. This is the case where A has a complete set of eigenvectors

and is not defective.

A Defective System

If A is defective, then it can be represented by the Jordan canonical form

"u01u02

#=

"� 0

1 �

# "u1u2

#(4.18)

whose solution is not obvious. However, in this case, one can solve the top equation

�rst, giving u1(t) = u1(0)e�t. Then, substituting this result into the second equation,

one �ndsdu2

dt= �u2 + u1(0)e

�t

which is identical in form to Eq. 4.10 and has the solution

u2(t) = [u2(0) + u1(0)t]e�t

From this the reader should be able to verify that

u3(t) =�a + bt+ ct2

�e�t

is a solution to 264u01u02u03

375 =

264�

1 �

1 �

375264u1u2u3

375

if

a = u3(0) ; b = u2(0) ; c =1

2u1(0) (4.19)

The general solution to such defective systems is left as an exercise.


4.2.4 General Solution of Coupled ODE's with Complete Eigen-

systems

Let us consider a set of coupled, nonhomogeneous, linear, �rst-order ODE's with

constant coe�cients which might have been derived by space di�erencing a set of

PDE's. Represent them by the equation

d~u

dt= A~u� ~f(t) (4.20)

Our assumption is that the M �M matrix A has a complete eigensystem1 and can

be transformed by the left and right eigenvector matrices, X�1 and X, to a diagonal

matrix � having diagonal elements which are the eigenvalues of A, see Section 4.2.1.

Now let us multiply Eq. 4.20 from the left by X�1 and insert the identity combination

XX�1 = I between A and ~u. There results

X�1d~u

dt= X�1AX �X�1~u�X�1~f(t) (4.21)

Since A is independent of both ~u and t, the elements in X�1 and X are also indepen-

dent of both ~u and t, and Eq. 4.21 can be modi�ed to

d

dtX�1~u = �X�1~u�X�1~f(t)

Finally, by introducing the new variables ~w and ~g such that

~w = X�1~u ; ~g(t) = X�1~f(t) (4.22)

we reduce Eq. 4.20 to a new algebraic form

d~w

dt= �~w � ~g(t) (4.23)

It is important at this point to review the results of the previous paragraph. Notice

that Eqs. 4.20 and 4.23 are expressing exactly the same equality. The only di�erence

between them was brought about by algebraic manipulations which regrouped the

variables. However, this regrouping is crucial for the solution process because Eqs.

1In the following, we exclude defective systems, not because they cannot be analyzed (the example

at the conclusion of the previous section proves otherwise), but because they are only of limited

interest in the general development of our theory.


4.23 are no longer coupled. They can be written line by line as a set of independent,

single, �rst-order equations, thus

w0

1 = �1w1 � g1(t)...

w0

m= �mwm � gm(t)...

w0

M= �MwM � gM(t) (4.24)

For any given set of gm(t) each of these equations can be solved separately and then

recoupled, using the inverse of the relations given in Eqs. 4.22:

~u(t) = X~w(t)

=MXm=1

wm(t)~xm (4.25)

where ~xm is the m'th column of X, i.e., the eigenvector corresponding to �m.

We next focus on the very important subset of Eq. 4.20 when neither A nor ~f has

any explicit dependence on t. In such a case, the gm in Eqs. 4.23 and 4.24 are also

time invariant and the solution to any line in Eq. 4.24 is

wm(t) = cme�mt +

1

�mgm

where the cm are constants that depend on the initial conditions. Transforming back

to the u-system gives

~u(t) = X~w(t)

=MXm=1

wm(t)~xm

=MXm=1

cme�mt ~xm +

MXm=1

1

�mgm~xm

=MXm=1

cme�mt ~xm +X��1X�1~f

=MXm=1

cme�mt ~xm + A�1~f

| {z }Transient

| {z }Steady-state

(4.26)

4.3. REAL SPACE AND EIGENSPACE 61

Note that the steady-state solution is A�1~f , as might be expected.

The �rst group of terms on the right side of this equation is referred to classically

as the complementary solution or the solution of the homogeneous equations. The

second group is referred to classically as the particular solution or the particular

integral. In our application to uid dynamics, it is more instructive to refer to these

groups as the transient and steady-state solutions, respectively. An alternative, but

entirely equivalent, form of the solution is

~u(t) = c1e�1t ~x1 + � � �+ cme

�mt ~xm + � � �+ cMe�M t ~xM + A�1~f (4.27)

4.3 Real Space and Eigenspace

4.3.1 De�nition

Following the semi-discrete approach discussed in Section 4.1, we reduce the partial

di�erential equations to a set of ordinary di�erential equations represented by the

generic form

d~u

dt= A~u� ~f (4.28)

The dependent variable ~u represents some physical quantity or quantities which relate

to the problem of interest. For the model problems on which we are focusing most

of our attention, the elements of A are independent of both u and t. This permits

us to say a great deal about such problems and serves as the basis for this section.

In particular, we can develop some very important and fundamental concepts that

underly the global properties of the numerical solutions to the model problems. How

these relate to the numerical solutions of more practical problems depends upon the

problem and, to a much greater extent, on the cleverness of the relator.

We begin by developing the concept of \spaces". That is, we identify di�erent

mathematical reference frames (spaces) and view our solutions from within each.

In this way, we get di�erent perspectives of the same solution, and this can add

signi�cantly to our understanding.

The most natural reference frame is the physical one. We say

If a solution is expressed in terms of ~u, it is said

to be in real space.

There is, however, another very useful frame. We saw in Sections 4.2.1 and 4.2 that

pre- and post-multiplication of A by the appropriate similarity matrices transforms A

into a diagonal matrix, composed, in the most general case, of Jordan blocks or, in the


simplest nondefective case, of scalars. Following Section 4.2 and, for simplicity, using

only complete systems for our examples, we found that Eq. 4.28 had the alternative

formd~w

dt= �~w � ~g

which is an uncoupled set of �rst-order ODE's that can be solved independently for

the dependent variable vector ~w. We say

If a solution is expressed in terms of ~w, it is said to

be in eigenspace (often referred to as wave space).

The relations that transfer from one space to the other are:

~w = X�1~u

~g = X�1~f

~u = X~w

~f = X~g

The elements of ~u relate directly to the local physics of the problem. However, the

elements of ~w are linear combinations of all of the elements of ~u, and individually

they have no direct local physical interpretation.

When the forcing function ~f is independent of t, the solutions in the two spaces

are represented by~u(t) =

Xcme

�mt ~xm +X��1X�1~f

and

wm(t) = cme�mt +

1

�mgm ; m = 1; 2; � � � ;M

for real space and eigenspace, respectively. At this point we make two observations:

1. the transient portion of the solution in real space consists of a linear combination

of contributions from each eigenvector, and

2. the transient portion of the solution in eigenspace provides the weighting of

each eigenvector component of the transient solution in real space.

4.3.2 Eigenvalue Spectrums for Model ODE's

It is instructive to consider the eigenvalue spectrums of the ODE's formulated by

central di�erencing the model equations for di�usion and biconvection. These model

ODE's are presented in Section 4.1. Equations for the eigenvalues of the simple


tridiagonals,B(M : a; b; c) and Bp(M : a; b; c), are given in Appendix B. From Section

B.1, we �nd for the model di�usion equation with Dirichlet boundary conditions

�m =�

�x2

��2 + 2 cos

�m�

M + 1

��

=�4�

�x2sin2

m�

2(M + 1)

!; m = 1; 2; � � � ;M (4.29)

and, from Section B.4, for the model biconvection equation

�m =�ia

�xsin

�2m�

M

�m = 0; 1; � � � ;M � 1

= �i��ma m = 0; 1; � � � ;M � 1 (4.30)

where

��m=

sin�m�x

�xm = 0; 1; � � � ;M � 1 (4.31)

is the modi�ed wavenumber from Section 3.5, �m = m, and �x = 2�=M . Notice

that the di�usion eigenvalues are real and negative while those representing periodic

convection are all pure imaginary. The interpretation of this result plays a very

important role later in our stability analysis.

4.3.3 Eigenvectors of the Model Equations

Next we consider the eigenvectors of the two model equations. These follow as special

cases from the results given in Appendix B.

The Di�usion Model

Consider Eq. 4.4, the model ODE's for di�usion. First, to help visualize the matrix

structure, we present results for a simple 4-point mesh and then we give the general

case. The right-hand eigenvector matrix X is given by26664sin (x1) sin (2x1) sin (3x1) sin (4x1)

sin (x2) sin (2x2) sin (3x2) sin (4x2)



37775

The columns of the matrix are proportional to the eigenvectors. Recall that xj =

j�x = j�=(M + 1), so in general the relation ~u = X ~w can be written as

uj =MXm=1

wm sinmxj ; j = 1; 2; � � � ;M (4.32)


For the inverse, or left-hand eigenvector matrix X�1, we �nd

26664sin (x1) sin (x2) sin (x3) sin (x4)

sin (2x1) sin (2x2) sin (2x3) sin (2x4)



37775

The rows of the matrix are proportional to the eigenvectors. In general ~w = X�1~u

gives

wm =MXj=1

uj sinmxj ; m = 1; 2; � � � ;M (4.33)

In the �eld of harmonic analysis, Eq. 4.33 represents a sine transform of the func-

tion u(x) for an M -point sample between the boundaries x = 0 and x = � with the

condition u(0) = u(�) = 0. Similarly, Eq. 4.32 represents the sine synthesis that

companions the sine transform given by Eq. 4.33. In summary,

For the model di�usion equation:

~w = X�1~u is a sine transform from real space to (sine) wave

space.

~u = X~w is a sine synthesis from wave space back to real

space.

The Biconvection Model

Next consider the model ODE's for periodic convection, Eq. 4.5. The coe�cient

matrices for these ODE's are always circulant. For our model ODE, the right-hand

eigenvectors are given by

~xm = ei j (2�m=M);j = 0; 1; � � � ;M � 1

m = 0; 1; � � � ;M � 1

With xj = j ��x = j � 2�=M , we can write ~u = X ~w as

uj =M�1Xm=0

wmeimxj ; j = 0; 1; � � � ;M � 1 (4.34)


For a 4-point periodic mesh, we �nd the following left-hand eigenvector matrix

from Appendix B.4:

26664w1

w2

w3

w4

37775 =

1

4

266641 1 1 1

1 e�2i�=4 e�4i�=4 e�6i�=4

1 e�4i�=4 e�8i�=4 e�12i�=4

1 e�6i�=4 e�12i�=4 e�18i�=4

3777526664u1u2u3u4

37775 = X�1~u

In general

wm =1

M

M�1Xj=0

uje�imxj ; m = 0; 1; � � � ;M � 1

This equation is identical to a discrete Fourier transform of the periodic dependent

variable ~u using an M -point sample between and including x = 0 and x = 2� ��x.

For circulant matrices, it is straightforward to establish the fact that the relation~u = X~w represents the Fourier synthesis of the variable ~w back to ~u. In summary,

For any circulant system:

~w = X�1~u is a complex Fourier transform from real space to

wave space.

~u = X~w is a complex Fourier synthesis from wave space

back to real space.

4.3.4 Solutions of the Model ODE's

We can now combine the results of the previous sections to write the solutions of our

model ODE's.

The Di�usion Equation

For the di�usion equation, Eq. 4.27 becomes

uj(t) =MXm=1

cme�mt sinmxj + (A�1f)j; j = 1; 2; � � � ;M (4.35)

where

�m =�4�

�x2sin2

m�

2(M + 1)

!(4.36)


With the modi�ed wavenumber de�ned as

��m=

2

�xsin

��m�x

2

�(4.37)

and using �m = m, we can write the ODE solution as

uj(t) =MXm=1

cme��

�

m

2t sin�mxj + (A�1f)j; j = 1; 2; � � � ;M (4.38)

This can be compared with the exact solution to the PDE, Eq. 2.37, evaluated at the

nodes of the grid:

uj(t) =MXm=1

cme��m

2t sin�mxj + h(xj); j = 1; 2; � � � ;M (4.39)

We see that the solutions are identical except for the steady solution and the

modi�ed wavenumber in the transient term. The modi�ed wavenumber is an approx-

imation to the actual wavenumber. The di�erence between the modi�ed wavenumber

and the actual wavenumber depends on the di�erencing scheme and the grid resolu-

tion. This di�erence causes the various modes (or eigenvector components) to decay

at rates which di�er from the exact solution. With conventional di�erencing schemes,

low wavenumber modes are accurately represented, while high wavenumber modes (if

they have signi�cant amplitudes) can have large errors.

The Convection Equation

For the biconvection equation, we obtain

uj(t) =M�1Xm=0

cme�mtei�mxj ; j = 0; 1; � � � ;M � 1 (4.40)

where

�m = �i��ma (4.41)

with the modi�ed wavenumber de�ned in Eq. 4.31. We can write this ODE solution

as

uj(t) =M�1Xm=0

cme�i�

�

matei�mxj ; j = 0; 1; � � � ;M � 1 (4.42)

4.4. THE REPRESENTATIVE EQUATION 67

and compare it to the exact solution of the PDE, Eq. 2.26, evaluated at the nodes of

the grid:

uj(t) =M�1Xm=0

fm(0)e�i�matei�mxj ; j = 0; 1; � � � ;M � 1 (4.43)

Once again the di�erence appears through the modi�ed wavenumber contained in

�m. As discussed in Section 3.5, this leads to an error in the speed with which various

modes are convected, since �� is real. Since the error in the phase speed depends on

the wavenumber, while the actual phase speed is independent of the wavenumber,

the result is erroneous numerical dispersion. In the case of non-centered di�erencing,

discussed in Chapter 11, the modi�ed wavenumber is complex. The form of Eq. 4.42

shows that the imaginary portion of the modi�ed wavenumber produces nonphysical

decay or growth in the numerical solution.

4.4 The Representative Equation

In Section 4.3, we pointed out that Eqs. 4.20 and 4.23 express identical results but in

terms of di�erent groupings of the dependent variables, which are related by algebraic

manipulation. This leads to the following important concept:

The numerical solution to a set of linear ODE's (in which A is

not a function of t) is entirely equivalent to the solution obtained

if the equations are transformed to eigenspace, solved there in

their uncoupled form, and then returned as a coupled set to real

space.

The importance of this concept resides in its message that we can analyze time-

marching methods by applying them to a single, uncoupled equation and our con-

clusions will apply in general. This is helpful both in analyzing the accuracy of

time-marching methods and in studying their stability, topics which are covered in

Chapters 6 and 7.

Our next objective is to �nd a \typical" single ODE to analyze. We found the

uncoupled solution to a set of ODE's in Section 4.2. A typical member of the family

is

dwm

dt= �mwm � gm(t) (4.44)

The goal in our analysis is to study typical behavior of general situations, not partic-

ular problems. For such a purpose Eq. 4.44 is not quite satisfactory. The role of �m is

clear; it stands for some representative eigenvalue in the original A matrix. However,


the question is: What should we use for gm(t) when the time dependence cannot be

ignored? To answer this question, we note that, in principle, one can express any one

of the forcing terms gm(t) as a �nite Fourier series. For example

�g(t) =Xk

akeikt

for which Eq. 4.44 has the exact solution:

w(t) = ce�t +Xk

akeikt

ik � �

From this we can extract the k'th term and replace ik with �. This leads to

The Representative ODE

dw

dt= �w + ae�t

(4.45)

which can be used to evaluate all manner of time-marching methods. In such evalua-

tions the parameters � and � must be allowed to take the worst possible combination

of values that might occur in the ODE eigensystem. The exact solution of the repre-

sentative ODE is (for � 6= �):

w(t) = ce�t +ae�t

�� (4.46)

4.5 Problems

1. Consider the �nite-di�erence operator derived in question 1 of Chapter 3. Using

this operator to approximate the spatial derivative in the linear convection equa-

tion, write the semi-discrete form obtained with periodic boundary conditions

on a 5-point grid (M = 5).

2. Consider the application of the operator given in Eq. 3.52 to the 1-D di�usion

equation with Dirichlet boundary conditions. Write the resulting semi-discrete

ODE form. Find the entries in the boundary-condition vector.

3. Write the semi-discrete form resulting from the application of second-order cen-

tered di�erences to the following equation on the domain 0 � x � 1 with

boundary conditions u(0) = 0, u(1) = 1:

@u

@t=

@2u

@x2� 6x

4.5. PROBLEMS 69

4. Consider a grid with 10 interior points spanning the domain 0 � x � �. For

initial conditions u(x; 0) = sin(mx) and boundary conditions u(0; t) = u(�; t) =

0, plot the exact solution of the di�usion equation with � = 1 at t = 1 with

m = 1 and m = 3. (Plot the solution at the grid nodes only.) Calculate the

corresponding modi�ed wavenumbers for the second-order centered operator

from Eq. 4.37. Calculate and plot the corresponding ODE solutions.

5. Consider the matrix

A = �Bp(10;�1; 0; 1)=(2�x)

corresponding to the ODE form of the biconvection equation resulting from the

application of second-order central di�erencing on a 10-point grid. Note that

the domain is 0 � x � 2� and �x = 2�=10. The grid nodes are given by

xj = j�x; j = 0; 1; : : : 9. The eigenvalues of the above matrix A, as well as the

matrices X and X�1, can be found from Appendix B.4. Using these, compute

and plot the ODE solution at t = 2� for the initial condition u(x; 0) = sinx.

Compare with the exact solution of the PDE. Calculate the numerical phase

speed from the modi�ed wavenumber corresponding to this initial condition

and show that it is consistent with the ODE solution. Repeat for the initial

condition u(x; 0) = sin 2x.

Chapter 5

FINITE-VOLUME METHODS

In Chapter 3, we saw how to derive �nite-di�erence approximations to arbitrary

derivatives. In Chapter 4, we saw that the application of a �nite-di�erence approxi-

mation to the spatial derivatives in our model PDE's produces a coupled set of ODE's.

In this Chapter, we will show how similar semi-discrete forms can be derived using

�nite-volume approximations in space. Finite-volume methods have become popu-

lar in CFD as a result, primarily, of two advantages. First, they ensure that the

discretization is conservative, i.e., mass, momentum, and energy are conserved in a

discrete sense. While this property can usually be obtained using a �nite-di�erence

formulation, it is obtained naturally from a �nite-volume formulation. Second, �nite-

volume methods do not require a coordinate transformation in order to be applied on

irregular meshes. As a result, they can be applied on unstructured meshes consisting

of arbitrary polyhedra in three dimensions or arbitrary polygons in two dimensions.

This increased exibility can be used to great advantage in generating grids about

arbitrary geometries.

Finite-volume methods are applied to the integral form of the governing equations,

either in the form of Eq. 2.1 or Eq. 2.2. Consistent with our emphasis on semi-discrete

methods, we will study the latter form, which is

d

dt

ZV (t)

QdV +

IS(t)

n:FdS =

ZV (t)

PdV (5.1)

We will begin by presenting the basic concepts which apply to �nite-volume strategies.

Next we will give our model equations in the form of Eq. 5.1. This will be followed

by several examples which hopefully make these concepts clear.

71

72 CHAPTER 5. FINITE-VOLUME METHODS

5.1 Basic Concepts

The basic idea of a �nite-volume method is to satisfy the integral form of the con-

servation law to some degree of approximation for each of many contiguous control

volumes which cover the domain of interest. Thus the volume V in Eq. 5.1 is that of a

control volume whose shape is dependent on the nature of the grid. In our examples,

we will consider only control volumes which do not vary with time. Examining Eq.

5.1, we see that several approximations must be made. The ux is required at the

boundary of the control volume, which is a closed surface in three dimensions and a

closed contour in two dimensions. This ux must then be integrated to �nd the net

ux through the boundary. Similarly, the source term P must be integrated over the

control volume. Next a time-marching method1 can be applied to �nd the value of

ZVQdV (5.2)

at the next time step.

Let us consider these approximations in more detail. First, we note that the

average value of Q in a cell with volume V is

�Q � 1

V

ZVQdV (5.3)

and Eq. 5.1 can be written as

Vd

dt�Q +

ISn:FdS =

ZVPdV (5.4)

for a control volume which does not vary with time. Thus after applying a time-

marching method, we have updated values of the cell-averaged quantities �Q. In order

to evaluate the uxes, which are a function of Q, at the control-volume boundary, Q

can be represented within the cell by some piecewise approximation which produces

the correct value of �Q. This is a form of interpolation often referred to as recon-

struction. As we shall see in our examples, each cell will have a di�erent piecewise

approximation to Q. When these are used to calculate F(Q), they will generally

produce di�erent approximations to the ux at the boundary between two control

volumes, that is, the ux will be discontinuous. A nondissipative scheme analogous

to centered di�erencing is obtained by taking the average of these two uxes. Another

approach known as ux-di�erence splitting is described in Chapter 11.

The basic elements of a �nite-volume method are thus the following:

1Time-marching methods will be discussed in the next chapter.

5.2. MODEL EQUATIONS IN INTEGRAL FORM 73

1. Given the value of �Q for each control volume, construct an approximation to

Q(x; y; z) in each control volume. Using this approximation, �nd Q at the

control-volume boundary. Evaluate F(Q) at the boundary. Since there is a

distinct approximation to Q(x; y; z) in each control volume, two distinct values

of the ux will generally be obtained at any point on the boundary between two

control volumes.

2. Apply some strategy for resolving the discontinuity in the ux at the control-

volume boundary to produce a single value of F(Q) at any point on the bound-

ary. This issue is discussed in Section 11.4.2.

3. Integrate the ux to �nd the net ux through the control-volume boundary

using some sort of quadrature.

4. Advance the solution in time to obtain new values of �Q.

The order of accuracy of the method is dependent on each of the approximations.

These ideas should be clari�ed by the examples in the remainder of this chapter.

In order to include di�usive uxes, the following relation between rQ and Q is

sometimes used: ZVrQdV =

ISnQdS (5.5)

or, in two dimensions, ZArQdA =

ICnQdl (5.6)

where the unit vector n points outward from the surface or contour.

5.2 Model Equations in Integral Form

5.2.1 The Linear Convection Equation

A two-dimensional form of the linear convection equation can be written as

@u

@t+ a cos �

@u

@x+ a sin �

@u

@y= 0 (5.7)

This PDE governs a simple plane wave convecting the scalar quantity, u(x; y; t) with

speed a along a straight line making an angle � with respect to the x-axis. The

one-dimensional form is recovered with � = 0.


For unit speed a, the two-dimensional linear convection equation is obtained from

the general divergence form, Eq. 2.3, with

Q = u (5.8)

F = iu cos � + ju sin � (5.9)

P = 0 (5.10)

Since Q is a scalar, F is simply a vector. Substituting these expressions into a two-

dimensional form of Eq. 2.2 gives the following integral form

d

dt

ZAudA+

ICn:(iu cos � + ju sin �)ds = 0 (5.11)

where A is the area of the cell which is bounded by the closed contour C.

5.2.2 The Di�usion Equation

The integral form of the two-dimensional di�usion equation with no source term and

unit di�usion coe�cient � is obtained from the general divergence form, Eq. 2.3, with

Q = u (5.12)

F = �ru (5.13)

= � i@u

@x+ j

@u

@y

!(5.14)

P = 0 (5.15)

Using these, we �ndd

dt

ZAudA =

ICn:

i@u

@x+ j

@u

@y

!ds (5.16)

to be the integral form of the two-dimensional di�usion equation.

5.3 One-Dimensional Examples

We restrict our attention to a scalar dependent variable u and a scalar ux f , as in

the model equations. We consider an equispaced grid with spacing �x. The nodes of

the grid are located at xj = j�x as usual. Control volume j extends from xj ��x=2

to xj +�x=2, as shown in Fig. 5.1. We will use the following notation:

xj�1=2 = xj ��x=2; xj+1=2 = xj +�x=2 (5.17)

5.3. ONE-DIMENSIONAL EXAMPLES 75

j j+1j-1 j+2j-2

j-1/2 j+1/2

∆ x

L RL R

Figure 5.1: Control volume in one dimension.

uj�1=2 = u(xj�1=2); fj�1=2 = f(uj�1=2) (5.18)

With these de�nitions, the cell-average value becomes

�uj(t) �1

�x

Z xj+1=2

xj�1=2

u(x; t)dx (5.19)

and the integral form becomes

d

dt(�x�uj) + fj+1=2 � fj�1=2 =

Z xj+1=2

xj�1=2

Pdx (5.20)

Now with � = x� xj, we can expand u(x) in Eq. 5.19 in a Taylor series about xj(with t �xed) to get

�uj � 1

�x

Z �x=2

��x=2

24uj + �

@u

@x

!j

+�2

2

@2u

@x2

!j

+�3

6

@3u

@x3

!j

+ : : :

35 d�

= uj +�x2

24

@2u

@x2

!j

+�x4

1920

@4u

@x4

!j

+O(�x6) (5.21)

or

�uj = uj +O(�x2) (5.22)

where uj is the value at the center of the cell. Hence the cell-average value and the

value at the center of the cell di�er by a term of second order.

5.3.1 A Second-Order Approximation to the Convection Equa-

tion

In one dimension, the integral form of the linear convection equation, Eq. 5.11, be-

comes

�xd�uj

dt+ fj+1=2 � fj�1=2 = 0 (5.23)


with f = u. We choose a piecewise constant approximation to u(x) in each cell such

that

u(x) = �uj xj�1=2 � x � xj+1=2 (5.24)

Evaluating this at j + 1=2 gives

fLj+1=2 = f(uLj+1=2) = uLj+1=2 = �uj (5.25)

where the L indicates that this approximation to fj+1=2 is obtained from the approx-

imation to u(x) in the cell to the left of xj+1=2, as shown in Fig. 5.1. The cell to the

right of xj+1=2, which is cell j + 1, gives

fRj+1=2 = �uj+1 (5.26)

Similarly, cell j is the cell to the right of xj�1=2, giving

fRj�1=2 = �uj (5.27)

and cell j � 1 is the cell to the left of xj�1=2, giving

fLj�1=2 = �uj�1 (5.28)

We have now accomplished the �rst step from the list in Section 5.1; we have de�ned

the uxes at the cell boundaries in terms of the cell-average data. In this example,

the discontinuity in the ux at the cell boundary is resolved by taking the average of

the uxes on either side of the boundary. Thus

fj+1=2 =1

2(fLj+1=2 + fRj+1=2) =

1

2(�uj + �uj+1) (5.29)

and

fj�1=2 =1

2(fLj�1=2 + fRj�1=2) =

1

2(�uj�1 + �uj) (5.30)

where f denotes a numerical ux which is an approximation to the exact ux.

Substituting Eqs. 5.29 and 5.30 into the integral form, Eq. 5.23, we obtain

�xd�uj

dt+

1

2(�uj + �uj+1)�

1

2(�uj�1 + �uj) = �x

d�uj

dt+

1

2(�uj+1 � �uj�1) = 0 (5.31)

With periodic boundary conditions, this point operator produces the following semi-

discrete form:

d~�u

dt= � 1

2�xBp(�1; 0; 1)~�u (5.32)


This is identical to the expression obtained using second-order centered di�erences,

except it is written in terms of the cell average ~�u, rather than the nodal values, ~u.

Hence our analysis and understanding of the eigensystem of the matrix Bp(�1; 0; 1)is relevant to �nite-volume methods as well as �nite-di�erence methods. Since the

eigenvalues of Bp(�1; 0; 1) are pure imaginary, we can conclude that the use of the

average of the uxes on either side of the cell boundary, as in Eqs. 5.29 and 5.30, can

lead to a nondissipative �nite-volume method.

5.3.2 A Fourth-Order Approximation to the Convection Equa-

tion

Let us replace the piecewise constant approximation in Section 5.3.1 with a piecewise

quadratic approximation as follows

u(�) = a�2 + b� + c (5.33)

where � is again equal to x � xj. The three parameters a, b, and c are chosen to

satisfy the following constraints:

1

�x

Z��x=2

�3�x=2u(�)d� = �uj�1

1

�x

Z �x=2

��x=2u(�)d� = �uj (5.34)

1

�x

Z 3�x=2

�x=2u(�)d� = �uj+1

These constraints lead to

a =�uj+1 � 2�uj + �uj�1

2�x2

b =�uj+1 � �uj�1

2�x(5.35)

c =��uj�1 + 26�uj � �uj+1

24

With these values of a, b, and c, the piecewise quadratic approximation produces

the following values at the cell boundaries:

uLj+1=2 =1

6(2�uj+1 + 5�uj � �uj�1) (5.36)


uRj�1=2 =1

6(��uj+1 + 5�uj + 2�uj�1) (5.37)

uRj+1=2 =1

6(��uj+2 + 5�uj+1 + 2�uj) (5.38)

uLj�1=2 =1

6(2�uj + 5�uj�1 � �uj�2) (5.39)

using the notation de�ned in Section 5.3.1. Recalling that f = u, we again use the

average of the uxes on either side of the boundary to obtain

fj+1=2 =1

2[f(uLj+1=2) + f(uRj+1=2)]

=1

12(��uj+2 + 7�uj+1 + 7�uj � �uj�1) (5.40)

and

fj�1=2 =1

2[f(uLj�1=2) + f(uRj�1=2)]

=1

12(��uj+1 + 7�uj + 7�uj�1 � �uj�2) (5.41)

Substituting these expressions into the integral form, Eq. 5.23, gives

�xd�uj

dt+

1

12(��uj+2 + 8�uj+1 � 8�uj�1 + �uj�2) = 0 (5.42)

This is a fourth-order approximation to the integral form of the equation, as can be

veri�ed using Taylor series expansions (see question 1 at the end of this chapter).

With periodic boundary conditions, the following semi-discrete form is obtained:

d~�u

dt= � 1

12�xBp(1;�8; 0; 8;�1)~�u (5.43)

This is a system of ODE's governing the evolution of the cell-average data.

5.3.3 A Second-Order Approximation to the Di�usion Equa-

tion

In this section, we describe two approaches to deriving a �nite-volume approximation

to the di�usion equation. The �rst approach is simpler to extend to multidimensions,

while the second approach is more suited to extension to higher order accuracy.


In one dimension, the integral form of the di�usion equation, Eq. 5.16, becomes

�xd�uj

dt+ fj+1=2 � fj�1=2 = 0 (5.44)

with f = �ru = �@u=@x. Also, Eq. 5.6 becomes

Z b

a

@u

@xdx = u(b)� u(a) (5.45)

We can thus write the following expression for the average value of the gradient of u

over the interval xj � x � xj+1:

1

�x

Z xj+1

xj

@u

@xdx =

1

�x(uj+1 � uj) (5.46)

From Eq. 5.22, we know that the value of a continuous function at the center of a given

interval is equal to the average value of the function over the interval to second-order

accuracy. Hence, to second-order, we can write

fj+1=2 = � @u

@x

!j+1=2

= � 1

�x(�uj+1 � �uj) (5.47)

Similarly,

fj�1=2 = � 1

�x(�uj � �uj�1) (5.48)

Substituting these into the integral form, Eq. 5.44, we obtain

�xd�uj

dt=

1

�x(�uj�1 � 2�uj + �uj+1) (5.49)

or, with Dirichlet boundary conditions,

d~�u

dt=

1

�x2B(1;�2; 1)~�u+

�~bc�

(5.50)

This provides a semi-discrete �nite-volume approximation to the di�usion equation,

and we see that the properties of the matrix B(1;�2; 1) are relevant to the study of

�nite-volume methods as well as �nite-di�erence methods.

For our second approach, we use a piecewise quadratic approximation as in Section

5.3.2. From Eq. 5.33 we have

@u

@x=

@u

@�= 2a� + b (5.51)


with a and b given in Eq. 5.35. With f = �@u=@x, this gives

fRj+1=2 = fLj+1=2 = � 1

�x(�uj+1 � �uj) (5.52)

fRj�1=2 = fLj�1=2 = � 1

�x(�uj � �uj�1) (5.53)

Notice that there is no discontinuity in the ux at the cell boundary. This produces

d�uj

dt=

1

�x2(�uj�1 � 2�uj + �uj+1) (5.54)

which is identical to Eq. 5.49. The resulting semi-discrete form with periodic bound-

ary conditions is

d~�u

dt=

1

�x2Bp(1;�2; 1)~�u (5.55)

which is written entirely in terms of cell-average data.

5.4 A Two-Dimensional Example

The above one-dimensional examples of �nite-volume approximations obscure some

of the practical aspects of such methods. Thus our �nal example is a �nite-volume

approximation to the two-dimensional linear convection equation on a grid consisting

of regular triangles, as shown in Figure 5.2. As in Section 5.3.1, we use a piecewise

constant approximation in each control volume and the ux at the control volume

boundary is the average of the uxes obtained on either side of the boundary. The

nodal data are stored at the vertices of the triangles formed by the grid. The control

volumes are regular hexagons with area A, � is the length of the sides of the triangles,

and ` is the length of the sides of the hexagons. The following relations hold between

`, �, and A.

` =1p3�

A =3p3

2`2

`

A=

2

3�(5.56)

The two-dimensional form of the conservation law is

d

dt

ZAQdA+

ICn:Fdl = 0 (5.57)

5.4. A TWO-DIMENSIONAL EXAMPLE 81

05

4

3

2

1

a

bc

d

e f

lp

∆

Figure 5.2: Triangular grid.

where we have ignored the source term. The contour in the line integral is composed

of the sides of the hexagon. Since these sides are all straight, the unit normals can

be taken outside the integral and the ux balance is given by

d

dt

ZAQ dA+

5X�=0

n� �Z�Fdl = 0

where � indexes a side of the hexagon, as shown in Figure 5.2. A list of the normals

for the mesh orientation shown is given in Table 5.1.

Side; � Outward Normal;n

0 (i�p3 j)=2

1 i

2 (i+p3 j)=2

3 (�i+p3 j)=2

4 �i5 (�i�

p3 j)=2

Table 5.1. Outward normals, see Fig. 5.2.

i and j are unit normals along x and y, respectively.


For Eq. 5.11, the two-dimensional linear convection equation, we have for side �

n� �Z�Fdl = n� � (i cos � + j sin �)

Z `=2

�`=2u�(�)d� (5.58)

where � is a length measured from the middle of a side �. Making the change of

variable z = �=`, one has the expression

Z `=2

�`=2u(�)d� = `

Z 1=2

�1=2u(z)dz (5.59)

Then, in terms of u and the hexagon area A, we have

d

dt

ZAu dA+

5X�=0

n� � (i cos � + j sin �)

"`Z 1=2

�1=2u(z)dz

#�

= 0 (5.60)

The values of n� � (i cos � + j sin �) are given by the expressions in Table 5.2. There

are no numerical approximations in Eq. 5.60. That is, if the integrals in the equation

are evaluated exactly, the integrated time rate of change of the integral of u over the

area of the hexagon is known exactly.

Side; � n� � (i cos � + j sin �)

0 (cos � �p3 sin �)=2

1 cos �

2 (cos � +p3 sin �)=2

3 (� cos � +p3 sin �)=2

4 � cos �

5 (� cos � �p3 sin �)=2

Table 5.2. Weights of ux integrals, see Eq. 5.60.

Introducing the cell average, ZAu dA = A�up (5.61)

and the piecewise-constant approximation u = �up over the entire hexagon, the ap-

proximation to the ux integral becomes trivial. Taking the average of the ux on

either side of each edge of the hexagon gives for edge 1:

Z1u(z)dz =

�up + �ua

2

Z 1=2

�1=2dz =

�up + �ua

2(5.62)

5.5. PROBLEMS 83

Similarly, we have for the other �ve edges:Z2u(z)dz =

�up + �ub

2(5.63)

Z3u(z)dz =

�up + �uc

2(5.64)

Z4u(z)dz =

�up + �ud

2(5.65)

Z5u(z)dz =

�up + �ue

2(5.66)

Z0u(z)dz =

�up + �uf

2(5.67)

Substituting these into Eq. 5.60, along with the expressions in Table 5.2, we obtain

Ad�up

dt+

`

2[(2 cos �)(�ua � �ud) + (cos � +

p3 sin �)(�ub � �ue)

+(� cos � +p3 sin �)(�uc � �uf)] = 0 (5.68)

or

d�up

dt+

1

3�[(2 cos �)(�ua � �ud) + (cos � +

p3 sin �)(�ub � �ue)

+(� cos � +p3 sin �)(�uc � �uf)] = 0 (5.69)

The reader can verify, using Taylor series expansions, that this is a second-order

approximation to the integral form of the two-dimensional linear convection equation.

5.5 Problems

1. Use Taylor series to verify that Eq. 5.42 is a fourth-order approximation to Eq.

5.23.

2. Find the semi-discrete ODE form governing the cell-average data resulting from

the use of a linear approximation in developing a �nite-volume method for the

linear convection equation. Use the following linear approximation:

u(�) = a� + b


where b = �uj and

a =�uj+1 � �uj�1

2�x

and use the average ux at the cell interface.

3. Using the �rst approach given in Section 5.3.3, derive a �nite-volume approx-

imation to the spatial terms in the two-dimensional di�usion equation on a

square grid.

4. Repeat question 3 for a grid consisting of equilateral triangles.

Chapter 6

TIME-MARCHING METHODS

FOR ODE'S

After discretizing the spatial derivatives in the governing PDE's (such as the Navier-Stokes equations), we obtain a coupled system of nonlinear ODE's in the form

d~u

dt= ~F (~u; t) (6.1)

These can be integrated in time using a time-marching method to obtain a time-

accurate solution to an unsteady ow problem. For a steady ow problem, spatialdiscretization leads to a coupled system of nonlinear algebraic equations in the form

~F (~u) = 0 (6.2)

As a result of the nonlinearity of these equations, some sort of iterative method is

required to obtain a solution. For example, one can consider the use of Newton'smethod, which is widely used for nonlinear algebraic equations (See Section 6.10.3.).

This produces an iterative method in which a coupled system of linear algebraic

equations must be solved at each iteration. These can be solved iteratively usingrelaxation methods, which will be discussed in Chapter 9, or directly using Gaussian

elimination or some variation thereof.Alternatively, one can consider a time-dependent path to the steady state and use

a time-marching method to integrate the unsteady form of the equations until the

solution is su�ciently close to the steady solution. The subject of the present chapter,

time-marching methods for ODE's, is thus relevant to both steady and unsteady owproblems. When using a time-marching method to compute steady ows, the goal is

simply to remove the transient portion of the solution as quickly as possible; time-accuracy is not required. This motivates the study of stability and sti�ness, topics

which are covered in the next two chapters.

85

86 CHAPTER 6. TIME-MARCHING METHODS FOR ODE'S

Application of a spatial discretization to a PDE produces a coupled system of

ODE's. Application of a time-marching method to an ODE produces an ordinary

di�erence equation (O�E ). In earlier chapters, we developed exact solutions to our

model PDE's and ODE's. In this chapter we will present some basic theory of linear

O�E's which closely parallels that for linear ODE's, and, using this theory, we will

develop exact solutions for the model O�E's arising from the application of time-

marching methods to the model ODE's.

6.1 Notation

Using the semi-discrete approach, we reduce our PDE to a set of coupled ODE'srepresented in general by Eq. 4.1. However, for the purpose of this chapter, we need

only consider the scalar case

du

dt= u0 = F (u; t) (6.3)

Although we use u to represent the dependent variable, rather than w, the readershould recall the arguments made in Chapter 4 to justify the study of a scalar ODE.Our �rst task is to �nd numerical approximations that can be used to carry out the

time integration of Eq. 6.3 to some given accuracy, where accuracy can be measuredeither in a local or a global sense. We then face a further task concerning the numerical

stability of the resulting methods, but we postpone such considerations to the nextchapter.

In Chapter 2, we introduced the convention that the n subscript, or the (n) su-

perscript, always points to a discrete time value, and h represents the time interval

�t. Combining this notation with Eq. 6.3 gives

u0n= Fn = F (un; tn) ; tn = nh

Often we need a more sophisticated notation for intermediate time steps involvingtemporary calculations denoted by ~u, �u, etc. For these we use the notation

~u0n+� =

~Fn+� = F (~un+�; tn + �h)

The choice of u0 or F to express the derivative in a scheme is arbitrary. They are

both commonly used in the literature on ODE's.The methods we study are to be applied to linear or nonlinear ODE's, but the

methods themselves are formed by linear combinations of the dependent variable and

its derivative at various time intervals. They are represented conceptually by

un+1 = f��1u

0

n+1; �0u0

n; ��1u

0

n�1; � � � ; �0un; ��1un�1; � � ��

(6.4)

6.2. CONVERTING TIME-MARCHING METHODS TO O�E'S 87

With an appropriate choice of the �0s and � 0s, these methods can be constructed

to give a local Taylor series accuracy of any order. The methods are said to be

explicit if �1 = 0 and implicit otherwise. An explicit method is one in which the new

predicted solution is only a function of known data, for example, u0n, u0

n�1, un; and

un�1 for a method using two previous time levels, and therefore the time advance is

simple. For an implicit method, the new predicted solution is also a function of the

time derivative at the new time level, that is, u0n+1. As we shall see, for systems of

ODE's and nonlinear problems, implicit methods require more complicated strategiesto solve for un+1 than explicit methods.

6.2 Converting Time-Marching Methods to O�E's

Examples of some very common forms of methods used for time-marching general

ODE's are:

un+1 = un + hu0n

(6.5)

un+1 = un + hu0n+1 (6.6)

and

~un+1 = un + hu0n

un+1 =1

2[un + ~un+1 + h~u0

n+1] (6.7)

According to the conditions presented under Eq. 6.4, the �rst and third of these are

examples of explicit methods. We refer to them as the explicit Euler method and the

MacCormack predictor-corrector method,1 respectively. The second is implicit andreferred to as the implicit (or backward) Euler method.

These methods are simple recipes for the time advance of a function in terms of its

value and the value of its derivative, at given time intervals. The material presentedin Chapter 4 develops a basis for evaluating such methods by introducing the concept

of the representative equation

du

dt= u0 = �u+ ae�t (6.8)

written here in terms of the dependent variable, u. The value of this equation arises

from the fact that, by applying a time-marching method, we can analytically convert

1Here we give only MacCormack's time-marching method. The method commonly referred to as

MacCormack's method, which is a fully-discrete method, will be presented in Section 11.3


such a linear ODE into a linear O�E . The latter are subject to a whole body of

analysis that is similar in many respects to, and just as powerful as, the theory of

ODE's. We next consider examples of this conversion process and then go into the

general theory on solving O�E's.

Apply the simple explicit Euler scheme, Eq. 6.5, to Eq. 6.8. There results

un+1 = un + h(�un + ae�hn)

or

un+1 � (1 + �h)un = hae�hn (6.9)

Eq. 6.9 is a linear O�E , with constant coe�cients, expressed in terms of the depen-dent variable un and the independent variable n. As another example, applying theimplicit Euler method, Eq. 6.6, to Eq. 6.8, we �nd

un+1 = un + h��un+1 + ae�h(n+1)

�or

(1� �h)un+1 � un = he�h � ae�hn (6.10)

As a �nal example, the predictor-corrector sequence, Eq. 6.7, gives

~un+1 � (1 + �h)un = ahe�hn

�1

2(1 + �h)~un+1 + un+1 �

1

2un =

1

2ahe�h(n+1) (6.11)

which is a coupled set of linear O�E's with constant coe�cients. Note that the �rstline in Eq. 6.11 is identical to Eq. 6.9, since the predictor step in Eq. 6.7 is simply

the explicit Euler method. The second line in Eq. 6.11 is obtained by noting that

~u0n+1 = F (~un+1; tn + h)

= �~un+1 + ae�h(n+1) (6.12)

Now we need to develop techniques for analyzing these di�erence equations so thatwe can compare the merits of the time-marching methods that generated them.

6.3 Solution of Linear O�E's With Constant Co-

e�cients

The techniques for solving linear di�erence equations with constant coe�cients is as

well developed as that for ODE's and the theory follows a remarkably parallel path.This is demonstrated by repeating some of the developments in Section 4.2, but for

di�erence rather than di�erential equations.

6.3. SOLUTION OF LINEAR O�E'S WITH CONSTANT COEFFICIENTS 89

6.3.1 First- and Second-Order Di�erence Equations

First-Order Equations

The simplest nonhomogeneous O�E of interest is given by the single �rst-order equa-

tion

un+1 = �un + abn (6.13)

where �, a, and b are, in general, complex parameters. The independent variable is

now n rather than t, and since the equations are linear and have constant coe�cients,

� is not a function of either n or u. The exact solution of Eq. 6.13 is

un = c1�n +

abn

b� �

where c1 is a constant determined by the initial conditions. In terms of the initialvalue of u it can be written

un = u0�n + a

bn � �n

b� �

Just as in the development of Eq. 4.10, one can readily show that the solution of thedefective case, (b = �),

un+1 = �un + a�n

is

un =hu0 + an��1

i�n

This can all be easily veri�ed by substitution.

Second-Order Equations

The homogeneous form of a second-order di�erence equation is given by

un+2 + a1un+1 + a0un = 0 (6.14)

Instead of the di�erential operator D �d

dtused for ODE's, we use for O�E's the

di�erence operator E (commonly referred to as the displacement or shift operator)and de�ned formally by the relations

un+1 = Eun ; un+k = Ekun

Further notice that the displacement operator also applies to exponents, thus

b� � bn = bn+� = E�� bn


where � can be any fraction or irrational number.

The roles of D and E are the same insofar as once they have been introduced to

the basic equations the value of u(t) or un can be factored out. Thus Eq. 6.14 can

now be re-expressed in an operational notion as

(E2 + a1E + a0)un = 0 (6.15)

which must be zero for all un. Eq. 6.15 is known as the operational form of Eq. 6.14.

The operational form contains a characteristic polynomial P (E) which plays the same

role for di�erence equations that P (D) played for di�erential equations; that is, its

roots determine the solution to the O�E. In the analysis of O�E's, we label these

roots �1, �2, � � �, etc, and refer to them as the �-roots. They are found by solving theequation P (�) = 0. In the simple example given above, there are just two � rootsand in terms of them the solution can be written

un = c1(�1)n + c2(�2)

n (6.16)

where c1 and c2 depend upon the initial conditions. The fact that Eq. 6.16 is asolution to Eq. 6.14 for all c1 ; c2 and n should be veri�ed by substitution.

6.3.2 Special Cases of Coupled First-Order Equations

A Complete System

Coupled, �rst-order, linear homogeneous di�erence equations have the form

u(n+1)

1 = c11u(n)

1 + c12u(n)

2

u(n+1)

2 = c21u(n)

1 + c22u(n)

2 (6.17)

which can also be written

~un+1 = C~un ; ~un =hu(n)

1 ; u(n)

2

iT; C =

�c11 c12c21 c22

�The operational form of Eq. 6.17 can be written

�(c11 � E) c12

c21 (c22 � E)

� �u1u2

�(n)= [C � E I ]~un = 0

which must be zero for all u1 and u2. Again we are led to a characteristic polynomial,

this time having the form P (E) = det [C � E I ]. The �-roots are found from

P (�) = det

�(c11 � �) c12

c21 (c22 � �)

�= 0

6.4. SOLUTION OF THE REPRESENTATIVE O�E'S 91

Obviously the �k are the eigenvalues of C and, following the logic of Section 4.2,

if ~x are its eigenvectors, the solution of Eq. 6.17 is

~un =2X

k=1

ck(�k)n~xk

where ck are constants determined by the initial conditions.

A Defective System

The solution of O�E's with defective eigensystems follows closely the logic in Section

4.2.2 for defective ODE's. For example, one can show that the solution to264 �un+1

un+1

un+1

375 =264 �1 �

1 �

375264 �ununun

375is

�un = �u0�n

un =hu0 + �u0n�

�1i�n

un =

"u0 + u0n�

�1 + �u0n(n� 1)

2��2

#�n (6.18)

6.4 Solution of the Representative O�E's

6.4.1 The Operational Form and its Solution

Examples of the nonhomogeneous, linear, �rst-order ordinary di�erence equations,produced by applying a time-marching method to the representative equation, aregiven by Eqs. 6.9 to 6.11. Using the displacement operator, E, these equations can

be written

[E � (1 + �h)]un = h � ae�hn (6.19)

[(1� �h)E � 1]un = h � E � ae�hn (6.20)

"E �(1 + �h)

�12(1 + �h)E E �

12

# "~u

u

#n

= h �

"112E

#� ae�hn (6.21)


All three of these equations are subsets of the operational form of the representative

O�E

P (E)un = Q(E) � ae�hn (6.22)

which is produced by applying time-marching methods to the representative ODE, Eq.

4.45. We can express in terms of Eq. 6.22 all manner of standard time-marching meth-

ods having multiple time steps and various types of intermediate predictor-corrector

families. The terms P (E) and Q(E) are polynomials in E referred to as the charac-

teristic polynomial and the particular polynomial, respectively.

The general solution of Eq. 6.22 can be expressed as

un =KXk=1

ck(�k)n + ae�hn �

Q(e�h)

P (e�h)(6.23)

where �k are the K roots of the characteristic polynomial, P (�) = 0. When determi-nants are involved in the construction of P (E) and Q(E), as would be the case for

Eq. 6.21, the ratio Q(E)=P (E) can be found by Kramer's rule. Keep in mind that

for methods such as in Eq. 6.21 there are multiple (two in this case) solutions, onefor un and ~un and we are usually only interested in the �nal solution un. Notice also,the important subset of this solution which occurs when � = 0, representing a time

invariant particular solution, or a steady state. In such a case

un =KXk=1

ck(�k)n+ a �

Q(1)

P (1)

6.4.2 Examples of Solutions to Time-Marching O�E's

As examples of the use of Eqs. 6.22 and 6.23, we derive the solutions of Eqs. 6.19 to

6.21. For the explicit Euler method, Eq. 6.19, we have

P (E) = E � 1� �h

Q(E) = h (6.24)

and the solution of its representative O�E follows immediately from Eq. 6.23:

un = c1(1 + �h)n + ae�hn �h

e�h � 1� �h

For the implicit Euler method, Eq. 6.20, we have

P (E) = (1� �h)E � 1

Q(E) = hE (6.25)

6.5. THE �� RELATION 93

so

un = c1

�1

1� �h

�n+ ae�hn �

he�h

(1� �h)e�h � 1

In the case of the coupled predictor-corrector equations, Eq. 6.21, one solves for the

�nal family un (one can also �nd a solution for the intermediate family ~u), and there

results

P (E) = det

"E �(1 + �h)

�12(1 + �h)E E �

12

#= E

�E � 1� �h�

1

2�2h2

�

Q(E) = det

�E h

�12(1 + �h)E 1

2hE

�=

1

2hE(E + 1 + �h)

The �-root is found from

P (�) = �

�� 1� �h�

1

2�2h2

�= 0

which has only one nontrivial root (� = 0 is simply a shift in the reference index).The complete solution can therefore be written

un = c1

�1 + �h+

1

2�2h2

�n+ ae�hn �

1

2h�e�h + 1 + �h

�e�h � 1� �h�

1

2�2h2

(6.26)

6.5 The �� Relation

6.5.1 Establishing the Relation

We have now been introduced to two basic kinds of roots, the �-roots and the �-roots.The former are the eigenvalues of the A matrix in the ODE's found by space di�er-

encing the original PDE, and the latter are the roots of the characteristic polynomial

in a representative O�E found by applying a time-marching method to the repre-sentative ODE. There is a fundamental relation between the two which can be used

to identify many of the essential properties of a time-march method. This relation is�rst demonstrated by developing it for the explicit Euler method.

First we make use of the semi-discrete approach to �nd a system of ODE's and

then express its solution in the form of Eq. 4.27. Remembering that t = nh, one can

write

u(t) = c1�e�1h

�n~x1 + � � �+ cm

�e�mh

�n~xm + � � �+ cM

�e�Mh

�n~xM + P:S: (6.27)


where for the present we are not interested in the form of the particular solution

(P:S:). Now the explicit Euler method produces for each �-root, one �-root, which

is given by � = 1 + �h. So if we use the Euler method for the time advance of the

ODE's, the solution2 of the resulting O�E is

un = c1(�1)n ~x1 + � � �+ cm(�m)

n ~xm + � � �+ cM(�M)n ~xM + P:S: (6.28)

where the cm and the ~xm in the two equations are identical and �m = (1 + �mh).Comparing Eq. 6.27 and Eq. 6.28, we see a correspondence between �m and e�mh.

Since the value of e�h can be expressed in terms of the series

e�h = 1 + �h +1

2�2h2 +

1

6�3h3 + � � �+

1

n!�nhn + � � �

the truncated expansion � = 1+ �h is a reasonable3 approximation for small enough�h.

Suppose, instead of the Euler method, we use the leapfrog method for the time

advance, which is de�ned by

un+1 = un�1 + 2hu0n

(6.29)

Applying Eq. 6.8 to Eq. 6.29, we have the characteristic polynomial P (E) = E2�

2�hE � 1, so that for every � the � must satisfy the relation

�2m� 2�mh�m � 1 = 0 (6.30)

Now we notice that each � produces two �-roots. For one of these we �nd

�m = �mh +

q1 + �2

mh2 (6.31)

= 1 + �mh +

1

2�2mh2 �

1

8�4mh4 + � � � (6.32)

This is an approximation to e�mh with an error O(�3h3). The other root, �mh �q

1 + �2mh2, will be discussed in Section 6.5.3.

2Based on Section 4.4.3The error is O(�2h2).

6.5. THE �� RELATION 95

6.5.2 The Principal �-Root

Based on the above we make the following observation:

Application of the same time-marching method to all of the equations in

a coupled system linear ODE's in the form of Eq. 4.6, always produces

one �-root for every �-root that satis�es the relation

� = 1 + �h+1

2�2h2 + � � �+

1

k!�khk +O

�hk+1

�where k is the order of the time-marching method.

(6.33)

We refer to the root that has the above property as the principal �-root, and designateit (�m)1. The above property can be stated regardless of the details of the time-

marching method, knowing only that its leading error is O�hk+1

�. Thus the principal

root is an approximation to e�h up to O�hk�.

Note that a second-order approximation to a derivative written in the form

(�tu)n =1

2h(un+1 � un�1) (6.34)

has a leading truncation error which is O(h2), while the second-order time-marching

method which results from this approximation, which is the leapfrog method:

un+1 = un�1 + 2hu0n

(6.35)

has a leading truncation error O(h3). This arises simply because of our notation

for the time-marching method in which we have multiplied through by h to get an

approximation for the function un+1 rather than the derivative as in Eq. 6.34. The

following example makes this clear. Consider a solution obtained at a given time T

using a second-order time-marching method with a time step h. Now consider the

solution obtained using the same method with a time step h=2. Since the error per

time step is O(h3), this is reduced by a factor of eight (considering the leading term

only). However, twice as many time steps are required to reach the time T . Therefore

the error at the end of the simulation is reduced by a factor of four, consistent with

a second-order approximation.

6.5.3 Spurious �-Roots

We saw from Eq. 6.30 that the �� relation for the leapfrog method produces two

�-roots for each �. One of these we identi�ed as the principal root which always


has the property given in 6.33. The other is referred to as a spurious �-root and

designated (�m)2. In general, the �� relation produced by a time-marching scheme

can result in multiple �-roots all of which, except for the principal one, are spurious.

All spurious roots are designated (�m)k where k = 2; 3; � � �. No matter whether a

�-root is principal or spurious, it is always some algebraic function of the product �h.

To express this fact we use the notation � = �(�h).

If a time-marching method produces spurious �-roots, the solution for the O�E in

the form shown in Eq. 6.28 must be modi�ed. Following again the message of Section4.4, we have

un = c11(�1)n

1~x1 + � � �+ cm1(�m)

n

1~xm + � � �+ cM1(�M)

n

1~xM + P:S:

+c12(�1)n

2~x1 + � � �+ cm2(�m)

n

2~xm + � � �+ cM2(�M)n

2~xM

+c13(�1)n

3~x1 + � � �+ cm3(�m)

n

3~xm + � � �+ cM3(�M)

n

3~xM

+etc., if there are more spurious roots (6.36)

Spurious roots arise if a method uses data from time level n � 1 or earlier toadvance the solution from time level n to n + 1. Such roots originate entirely fromthe numerical approximation of the time-marching method and have nothing to do

with the ODE being solved. However, generation of spurious roots does not, in itself,make a method inferior. In fact, many very accurate methods in practical use forintegrating some forms of ODE's have spurious roots.

It should be mentioned that methods with spurious roots are not self starting.

For example, if there is one spurious root to a method, all of the coe�cients (cm)2in Eq. 6.36 must be initialized by some starting procedure. The initial vector ~u0does not provide enough data to initialize all of the coe�cients. This results becausemethods which produce spurious roots require data from time level n � 1 or earlier.

For example, the leapfrog method requires ~un�1 and thus cannot be started using

only ~un.

Presumably (i.e., if one starts the method properly) the spurious coe�cients are

all initialized with very small magnitudes, and presumably the magnitudes of thespurious roots themselves are all less than one (see Chapter 7). Then the presence of

spurious roots does not contaminate the answer. That is, after some �nite time the

amplitude of the error associated with the spurious roots is even smaller then whenit was initialized. Thus while spurious roots must be considered in stability analysis,

they play virtually no role in accuracy analysis.

6.5.4 One-Root Time-Marching Methods

There are a number of time-marching methods that produce only one �-root for each

�-root. We refer to them as one-rootmethods. They are also called one-step methods.

6.6. ACCURACY MEASURES OF TIME-MARCHING METHODS 97

They have the signi�cant advantage of being self-starting which carries with it the

very useful property that the time-step interval can be changed at will throughout

the marching process. Three one-root methods were analyzed in Section 6.4.2. A

popular method having this property, the so-called �-method, is given by the formula

un+1 = un + hh(1� �)u0

n+ �u0

n+1

iThe �-method represents the explicit Euler (� = 0), the trapezoidal (� = 1

2), and the

implicit Euler methods (� = 1), respectively. Its �� relation is

� =1 + (1� �)�h

1� ��h

It is instructive to compare the exact solution to a set of ODE's (with a completeeigensystem) having time-invariant forcing terms with the exact solution to the O�E'sfor one-root methods. These are

~u(t) = c1�e�1h

�n~x1 + � � �+ cm

�e�mh

�n~xm + � � �+ cM

�e�Mh

�n~xM + A�1~f

~un = c1(�1)n ~x1 + � � �+ cm(�m)

n ~xm + � � �+ cM(�M )n ~xM + A�1~f (6.37)

respectively. Notice that when t and n = 0, these equations are identical, so that allthe constants, vectors, and matrices are identical except the ~u and the terms inside

the parentheses on the right hand sides. The only error made by introducing the time

marching is the error that � makes in approximating e�h.

6.6 Accuracy Measures of Time-Marching Meth-

ods

6.6.1 Local and Global Error Measures

There are two broad categories of errors that can be used to derive and evaluate time-marching methods. One is the error made in each time step. This is a local error such

as that found from a Taylor table analysis, see Section 3.4. It is usually used as the

basis for establishing the order of a method. The other is the error determined at theend of a given event which has covered a speci�c interval of time composed of many

time steps. This is a global error. It is useful for comparing methods, as we shall see

in Chapter 8.It is quite common to judge a time-marching method on the basis of results found

from a Taylor table. However, a Taylor series analysis is a very limited tool for �ndingthe more subtle properties of a numerical time-marching method. For example, it is

of no use in:


� �nding spurious roots.

� evaluating numerical stability and separating the errors in phase and amplitude.

� analyzing the particular solution of predictor-corrector combinations.

� �nding the global error.

The latter three of these are of concern to us here, and to study them we make use of

the material developed in the previous sections of this chapter. Our error measuresare based on the di�erence between the exact solution to the representative ODE,given by

u(t) = ce�t +ae�t

�� (6.38)

and the solution to the representative O�E's, including only the contribution from

the principal root, which can be written as

un = c1(�1)n+ ae�hn �

Q(e�h)

P (e�h)(6.39)

6.6.2 Local Accuracy of the Transient Solution (er�; j�j ; er!)

Transient error

The particular choice of an error measure, either local or global, is to some extent

arbitrary. However, a necessary condition for the choice should be that the measurecan be used consistently for all methods. In the discussion of the �-� relation we

saw that all time-marching methods produce a principal �-root for every �-root that

exists in a set of linear ODE's. Therefore, a very natural local error measure for the

transient solution is the value of the di�erence between solutions based on these two

roots. We designate this by er� and make the following de�nition

er� � e�h � �1

The leading error term can be found by expanding in a Taylor series and choosingthe �rst nonvanishing term. This is similar to the error found from a Taylor table.

The order of the method is the last power of �h matched exactly.


Amplitude and Phase Error

Suppose a � eigenvalue is imaginary. Such can indeed be the case when we study the

equations governing periodic convection which produces harmonic motion. For such

cases it is more meaningful to express the error in terms of amplitude and phase.

Let � = i! where ! is a real number representing a frequency. Then the numerical

method must produce a principal �-root that is complex and expressible in the form

�1 = �r + i�i � ei!h (6.40)

From this it follows that the local error in amplitude is measured by the deviation of

j�1j from unity, that is

era = 1� j�1j = 1�q(�1)2r + (�1)

2i

and the local error in phase can be de�ned as

er! � !h� tan�1 [(�1)i=(�1)r)] (6.41)

Amplitude and phase errors are important measures of the suitability of time-marching

methods for convection and wave propagation phenomena.The approach to error analysis described in Section 3.5 can be extended to the

combination of a spatial discretization and a time-marching method applied to the

linear convection equation. The principal root, �1(�h), is found using � = �ia��,where �� is the modi�ed wavenumber of the spatial discretization. Introducing the

Courant number, Cn = ah=�x, we have �h = �iCn��x. Thus one can obtain

values of the principal root over the range 0 � ��x � � for a given value of theCourant number. The above expression for er! can be normalized to give the error

in the phase speed, as follows

erp =er!

!h= 1 +

tan�1 [(�1)i=(�1)r)]

Cn��x(6.42)

where ! = �a�. A positive value of erp corresponds to phase lag (the numerical phase

speed is too small), while a negative value corresponds to phase lead (the numerical

phase speed is too large).

6.6.3 Local Accuracy of the Particular Solution (er�)

The numerical error in the particular solution is found by comparing the particularsolution of the ODE with that for the O�E. We have found these to be given by

P:S:(ODE) = ae�t �1

(�� )


and

P:S:(O�E) = ae�t �Q(e�h)

P (e�h)

respectively. For a measure of the local error in the particular solution we introducethe de�nition

er� � h

(P:S:(O�E)

P:S:(ODE)

� 1

)(6.43)

The multiplication by h converts the error from a global measure to a local one, so

that the order of er� and er� are consistent. In order to determine the leading errorterm, Eq. 6.43 can be written in terms of the characteristic and particular polynomials

as

er� =co

��

n(�� )Q

�e�h

�� P

�e�h

�o(6.44)

where

co = limh!0

h(�� )

P�e�h

�The value of co is a method-dependent constant that is often equal to one. If theforcing function is independent of time, � is equal to zero, and for this case, many

numerical methods generate an er� that is also zero.

The algebra involved in �nding the order of er� can be quite tedious. However,

this order is quite important in determining the true order of a time-marching method

by the process that has been outlined. An illustration of this is given in the section

on Runge-Kutta methods.

6.6.4 Time Accuracy For Nonlinear Applications

In practice, time-marching methods are usually applied to nonlinear ODE's, and itis necessary that the advertised order of accuracy be valid for the nonlinear cases as

well as for the linear ones. A necessary condition for this to occur is that the localaccuracies of both the transient and the particular solutions be of the same order.

More precisely, a time-marching method is said to be of order k if

er� = c1 � (�h)k1+1 (6.45)

er� = c2 � (�h)k2+1 (6.46)

where k = smallest of(k1; k2) (6.47)


The reader should be aware that this is not su�cient. For example, to derive all of

the necessary conditions for the fourth-order Runge-Kutta method presented later

in this chapter the derivation must be performed for a nonlinear ODE. However, the

analysis based on a linear nonhomogeneous ODE produces the appropriate conditions

for the majority of time-marching methods used in CFD.

6.6.5 Global Accuracy

In contrast to the local error measures which have just been discussed, we can also

de�ne global error measures. These are useful when we come to the evaluation of

time-marching methods for speci�c purposes. This subject is covered in Chapter 8

after our introduction to stability in Chapter 7.

Suppose we wish to compute some time-accurate phenomenon over a �xed intervalof time using a constant time step. We refer to such a computation as an \event".Let T be the �xed time of the event and h be the chosen step size. Then the required

number of time steps, is N, given by the relation

T = Nh

Global error in the transient

A natural extension of er� to cover the error in an entire event is given by

Er� � e�T � (�1(�h))N (6.48)

Global error in amplitude and phase

If the event is periodic, we are more concerned with the global error in amplitude andphase. These are given by

Era = 1�

�q(�1)2r + (�1)

2i

�N(6.49)

and

Er! � N

"!h� tan�1

(�1)i(�1)r

!#= !T �N tan�1 [(�1)i=(�1)r] (6.50)


Global error in the particular solution

Finally, the global error in the particular solution follows naturally by comparing the

solutions to the ODE and the O�E. It can be measured by

Er� � (�� )Q�e�h

�P�e�h

� � 1

6.7 Linear Multistep Methods

In the previous sections, we have developed the framework of error analysis for time

advance methods and have randomly introduced a few methods without addressingmotivational, developmental or design issues. In the subsequent sections, we introduce

classes of methods along with their associated error analysis. We shall not spend muchtime on development or design of these methods, since most of them have historicorigins from a wide variety of disciplines and applications. The Linear Multistep

Methods (LMM's) are probably the most natural extension to time marching of thespace di�erencing schemes introduced in Chapter 3 and can be analyzed for accuracyor designed using the Taylor table approach of Section 3.4.

6.7.1 The General Formulation

When applied to the nonlinear ODE

du

dt= u0 = F(u; t)

all linear multistep methods can be expressed in the general form

1Xk=1�K

�kun+k = h1X

k=1�K

�kFn+k (6.51)

where the notation for F is de�ned in Section 6.1. The methods are said to be linear

because the �'s and �'s are independent of u and n, and they are said to be K-step

because K time-levels of data are required to marching the solution one time-step, h.

They are explicit if �1 = 0 and implicit otherwise.

When Eq. 6.51 is applied to the representative equation, Eq. 6.8, and the result is

expressed in operational form, one �nds0@ 1Xk=1�K

�kEk

1Aun = h

0@ 1Xk=1�K

�kEk

1A(�un + ae�hn) (6.52)

6.7. LINEAR MULTISTEP METHODS 103

We recall from Section 6.5.2 that a time-marching method when applied to the repre-

sentative equation must provide a �-root, labeled �1, that approximates e�h through

the order of the method. The condition referred to as consistency simply means that

� ! 1 as h ! 0, and it is certainly a necessary condition for the accuracy of any

time marching method. We can also agree that, to be of any value in time accuracy,

a method should at least be �rst-order accurate, that is � ! (1 +�h) as h! 0. One

can show that these conditions are met by any method represented by Eq. 6.51 ifXk

�k = 0 andXk

�k =Xk

(K + k � 1)�k

Since both sides of Eq. 6.51 can be multiplied by an arbitrary constant, these methodsare often \normalized" by requiring X

k

�k = 1

Under this condition co = 1 in Eq. 6.44.

6.7.2 Examples

There are many special explicit and implicit forms of linear multistep methods. Two

well-known families of them, referred to as Adams-Bashforth (explicit) and Adams-Moulton (implicit), can be designed using the Taylor table approach of Section 3.4.

The Adams-Moulton family is obtained from Eq. 6.51 with

�1 = 1; �0 = �1; �k = 0; k = �1;�2; � � � (6.53)

The Adams-Bashforth family has the same �'s with the additional constraint that

�1 = 0. The three-step Adams-Moulton method can be written in the following form

un+1 = un + h(�1u0

n+1 + �0u0

n+ ��1u

0

n�1 + ��2u0

n�2) (6.54)

A Taylor table for Eq. 6.54 can be generated as

un h � u0n

h2 � u00n

h3 � u000n

h4 � u0000n

un+1 1 1 12

16

124

�un �1�h�1u

0

n+1 ��1 ��1 ��112 ��1

16

�h�0u0

n��0

�h��1u0

n�1 ��1 ��1 ��112 ��1

16

�h��2u0

n�2 �(�2)0��2 �(�2)1��2 �(�2)2��212 �(�2)3��2

16

Table 6.1. Taylor table for the Adams-Moulton three-step linear multistep method.


This leads to the linear system266641 1 1 1

2 0 �2 �4

3 0 3 12

4 0 �4 �32

3777526664�1�0��1��2

37775 =266641

1

1

1

37775 (6.55)

to solve for the �'s, resulting in

�1 = 9=24; �0 = 19=24; ��1 = �5=24; ��2 = 1=24 (6.56)

which produces a method which is fourth-order accurate.4 With �1 = 0 one obtains264 1 1 1

0 �2 �40 3 12

375264 �0��1��2

375 =264 1

11

375 (6.57)

giving

�0 = 23=12; ��1 = �16=12; ��2 = 5=12 (6.58)

This is the third-order Adams-Bashforth method.

A list of simple methods, some of which are very common in CFD applications,is given below together with identifying names that are sometimes associated withthem. In the following material AB(n) and AM(n) are used as abbreviations for the

(n)th order Adams-Bashforth and (n)th order Adams-Moulton methods. One canverify that the Adams type schemes given below satisfy Eqs. 6.55 and 6.57 up to theorder of the method.

Explicit Methods

un+1 = un + hu0n

Euler

un+1 = un�1 + 2hu0n

Leapfrog

un+1 = un +12hh3u0

n� u0

n�1

iAB2

un+1 = un +h12

h23u0

n� 16u0

n�1 + 5u0n�2

iAB3

Implicit Methods

un+1 = un + hu0n+1 Implicit Euler

un+1 = un +12hhu0n+ u0

n+1

iTrapezoidal (AM2)

un+1 = 13

h4un � un�1 + 2hu0

n+1

i2nd-order Backward

un+1 = un +h12

h5u0

n+1 + 8u0n� u0

n�1

iAM3

4Recall from Section 6.5.2 that a kth-order time-marching method has a leading truncation error

term which is O(hk+1).

6.7. LINEAR MULTISTEP METHODS 105

6.7.3 Two-Step Linear Multistep Methods

High resolution CFD problems usually require very large data sets to store the spatial

information from which the time derivative is calculated. This limits the interest

in multistep methods to about two time levels. The most general two-step linear

multistep method (i.e., K=2 in Eq. 6.51), that is at least �rst-order accurate, can be

written as

(1 + �)un+1 = [(1 + 2�)un � �un�1] + hh�u0

n+1 + (1� � + ')u0n� 'u0

n�1

i(6.59)

Clearly the methods are explicit if � = 0 and implicit otherwise. A list of methods

contained in Eq. 6.59 is given in Table 6.2. Notice that the Adams methods have� = 0, which corresponds to ��1 = 0 in Eq. 6.51. Methods with � = �1=2, which

corresponds to �0 = 0 in Eq. 6.51, are known as Milne methods.

� � ' Method Order

0 0 0 Euler 11 0 0 Implicit Euler 11=2 0 0 Trapezoidal or AM2 21 1=2 0 2nd Order Backward 23=4 0 �1=4 Adams type 21=3 �1=2 �1=3 Lees Type 21=2 �1=2 �1=2 Two{step trapezoidal 25=9 �1=6 �2=9 A{contractive 20 �1=2 0 Leapfrog 20 0 1=2 AB2 20 �5=6 �1=3 Most accurate explicit 31=3 �1=6 0 Third{order implicit 35=12 0 1=12 AM3 31=6 �1=2 �1=6 Milne 4

Table 6.2. Some linear one- and two-step methods, see Eq. 6.59.

One can show after a little algebra that both er� and er� are reduced to 0(h3)

(i.e., the methods are 2nd-order accurate) if

' = � � � +1

2

The class of all 3rd-order methods is determined by imposing the additional constraint

� = 2� �5

6

Finally a unique fourth-order method is found by setting � = �' = ��=3 = 16.


6.8 Predictor-Corrector Methods

There are a wide variety of predictor-corrector schemes created and used for a variety

of purposes. Their use in solving ODE's is relatively easy to illustrate and understand.

Their use in solving PDE's can be much more subtle and demands concepts5 which

have no counterpart in the analysis of ODE's.

Predictor-corrector methods constructed to time-march linear or nonlinear ODE's

are composed of sequences of linear multistep methods, each of which is referred to

as a family in the solution process. There may be many families in the sequence, andusually the �nal family has a higher Taylor-series order of accuracy than the inter-

mediate ones. Their use is motivated by ease of application and increased e�ciency,

where measures of e�ciency are discussed in the next two chapters.A simple one-predictor, one-corrector example is given by

~un+� = un + �hu0n

un+1 = un + hh�~u0

n+� + u0n

i(6.60)

where the parameters �; � and are arbitrary parameters to be determined. One

can analyze this sequence by applying it to the representative equation and using

the operational techniques outlined in Section 6.4. It is easy to show, following theexample leading to Eq. 6.26, that

P (E) = E��

hE � 1� ( + �)�h� ��2h2

i(6.61)

Q(E) = E�� h � [�E� + + ��h] (6.62)

Considering only local accuracy, one is led, by following the discussion in Section 6.6,to the following observations. For the method to be second-order accurate both er�and er� must be O(h3). For this to hold for er�, it is obvious from Eq. 6.61 that

+ � = 1 ; �� =1

2

which provides two equations for three unknowns. The situation for er� requires some

algebra, but it is not di�cult to show using Eq. 6.44 that the same conditions also

make it O(h3). One concludes, therefore, that the predictor-corrector sequence

~un+� = un + �hu0n

un+1 = un +1

2h

��1

�

�~u0n+� +

�2�� 1

�

�u0n

�(6.63)

is a second-order accurate method for any �.

5Such as alternating direction, fractional-step, and hybrid methods.

6.9. RUNGE-KUTTA METHODS 107

A classical predictor-corrector sequence is formed by following an Adams-Bashforth

predictor of any order with an Adams-Moulton corrector having an order one higher.

The order of the combination is then equal to the order of the corrector. If the order

of the corrector is (k), we refer to these as ABM(k) methods. The Adams-Bashforth-

Moulton sequence for k = 3 is

~un+1 = un +1

2hh3u0

n� u0

n�1

iun+1 = un +

h

12

h5~u0

n+1 + 8u0n� u0

n�1

i(6.64)

Some simple, speci�c, second-order accurate methods are given below. The Gazdag

method, which we discuss in Chapter 8, is

~un+1 = un +1

2hh3~u0

n� ~u0

n�1

iun+1 = un +

1

2hh~u0n+ ~u0

n+1

i(6.65)

The Burstein method, obtained from Eq. 6.63 with � = 1=2 is

~un+1=2 = un +1

2hu0

n

un+1 = un + h~u0n+1=2 (6.66)

and, �nally, MacCormack's method, presented earlier in this chapter, is

~un+1 = un + hu0n

un+1 =1

2[un + ~un+1 + h~u0

n+1] (6.67)

Note that MacCormack's method can also be written as

~un+1 = un + hu0n

un+1 = un +1

2h[u0

n+ ~u0

n+1] (6.68)

from which it is clear that it is obtained from Eq. 6.63 with � = 1.

6.9 Runge-Kutta Methods

There is a special subset of predictor-corrector methods, referred to as Runge-Kutta

methods,6 that produce just one �-root for each �-root such that �(�h) corresponds

6Although implicit and multi-step Runge-Kutta methods exist, we will consider only single-step,

explicit Runge-Kutta methods here.


to the Taylor series expansion of e�h out through the order of the method and then

truncates. Thus for a Runge-Kutta method of order k (up to 4th order), the principal

(and only) �-root is given by

� = 1 + �h +1

2�2h2 + � � �+

1

k!�khk (6.69)

It is not particularly di�cult to build this property into a method, but, as we pointed

out in Section 6.6.4, it is not su�cient to guarantee k'th order accuracy for the solution

of u0 = F (u; t) or for the representative equation. To ensure k'th order accuracy, the

method must further satisfy the constraint that

er� = O(hk+1) (6.70)

and this is much more di�cult.

The most widely publicized Runge-Kutta process is the one that leads to the

fourth-order method. We present it below in some detail. It is usually introduced inthe form

k1 = hF (un; tn)

k2 = hF (un + �k1; tn + �h)

k3 = hF (un + �1k1 + 1k2; tn + �1h)

k4 = hF (un + �2k1 + 2k2 + �2k3; tn + �2h)

followed by

u(tn + h)� u(tn) = �1k1 + �2k2 + �3k3 + �4k4 (6.71)

However, we prefer to present it using predictor-corrector notation. Thus, a scheme

entirely equivalent to 6.71 is

bun+� = un + �hu0n

~un+�1 = un + �1hu0

n+ 1hbu0n+�

un+�2 = un + �2hu0

n+ 2hbu0n+� + �2h~u

0

n+�1

un+1 = un + �1hu0

n+ �2hbu0n+� + �3h~u

0

n+�1+ �4hu

0

n+�2(6.72)

Appearing in Eqs. 6.71 and 6.72 are a total of 13 parameters which are to be

determined such that the method is fourth-order according to the requirements in

Eqs. 6.69 and 6.70. First of all, the choices for the time samplings, �, �1, and �2, are

not arbitrary. They must satisfy the relations

� = �

�1 = �1 + 1

�2 = �2 + 2 + �2 (6.73)

6.9. RUNGE-KUTTA METHODS 109

The algebra involved in �nding algebraic equations for the remaining 10 parameters

is not trivial, but the equations follow directly from �nding P (E) and Q(E) and then

satisfying the conditions in Eqs. 6.69 and 6.70. Using Eq. 6.73 to eliminate the �'s

we �nd from Eq. 6.69 the four conditions

�1 + �2 + �3 + �4 = 1 (1)

�2� + �3�1 + �4�2 = 1=2 (2)

�3� 1 + �4(� 2 + �1�2) = 1=6 (3)

�4� 1�2 = 1=24 (4)

(6.74)

These four relations guarantee that the �ve terms in � exactly match the �rst 5 terms

in the expansion of e�h. To satisfy the condition that er� = O(k5), we have to ful�llfour more conditions

�2�2 + �3�

21 + �4�

22 = 1=3 (3)

�2�3 + �3�

31 + �4�

32 = 1=4 (4)

�3�2 1 + �4(�

2 2 + �21�2) = 1=12 (4)

�3��1 1 + �4�2(� 2 + �1�2) = 1=8 (4)

(6.75)

The number in parentheses at the end of each equation indicates the order thatis the basis for the equation. Thus if the �rst 3 equations in 6.74 and the �rstequation in 6.75 are all satis�ed, the resulting method would be third-order accurate.

As discussed in Section 6.6.4, the fourth condition in Eq. 6.75 cannot be derivedusing the methodology presented here, which is based on a linear nonhomogenous

representative ODE. A more general derivation based on a nonlinear ODE can be

found in several books.7

There are eight equations in 6.74 and 6.75 which must be satis�ed by the 10

unknowns. Since the equations are overdetermined, two parameters can be set arbi-

trarily. Several choices for the parameters have been proposed, but the most popular

one is due to Runge. It results in the \standard" fourth-order Runge-Kutta method

expressed in predictor-corrector form as

bun+1=2 = un +1

2hu0

n

~un+1=2 = un +1

2hbu0

n+1=2

un+1 = un + h~u0n+1=2

un+1 = un +1

6hhu0n+ 2

�bu0n+1=2 + ~u0

n+1=2

�+ u0

n+1

i(6.76)

7The present approach based on a linear inhomogeneous equation provides all of the necessary

conditions for Runge-Kutta methods of up to third order.


Notice that this represents the simple sequence of conventional linear multistep meth-

ods referred to, respectively, as

Euler Predictor

Euler Corrector

Leapfrog Predictor

Milne Corrector

9>>>=>>>; � RK4

One can easily show that both the Burstein and the MacCormack methods given by

Eqs. 6.66 and 6.67 are second-order Runge-Kutta methods, and third-order methods

can be derived from Eqs. 6.72 by setting �4 = 0 and satisfying only Eqs. 6.74 and the

�rst equation in 6.75. It is clear that for orders one through four, RK methods of orderk require k evaluations of the derivative function to advance the solution one timestep. We shall discuss the consequences of this in Chapter 8. Higher-order Runge-

Kutta methods can be developed, but they require more derivative evaluations thantheir order. For example, a �fth-order method requires six evaluations to advance

the solution one step. In any event, storage requirements reduce the usefulness ofRunge-Kutta methods of order higher than four for CFD applications.

6.10 Implementation of Implicit Methods

We have presented a wide variety of time-marching methods and shown how to derive

their � � � relations. In the next chapter, we will see that these methods can have

widely di�erent properties with respect to stability. This leads to various trade-

o�s which must be considered in selecting a method for a speci�c application. Ourpresentation of the time-marching methods in the context of a linear scalar equation

obscures some of the issues involved in implementing an implicit method for systems

of equations and nonlinear equations. These are covered in this Section.

6.10.1 Application to Systems of Equations

Consider �rst the numerical solution of our representative ODE

u0 = �u+ ae�t (6.77)

using the implicit Euler method. Following the steps outlined in Section 6.2, we

obtained

(1� �h)un+1 � un = he�h � ae�hn (6.78)

6.10. IMPLEMENTATION OF IMPLICIT METHODS 111

Solving for un+1 gives

un+1 =1

1� �h(un + he�h � ae�hn) (6.79)

This calculation does not seem particularly onerous in comparison with the applica-

tion of an explicit method to this ODE, requiring only an additional division.

Now let us apply the implicit Euler method to our generic system of equations

given by

~u0 = A~u� ~f(t) (6.80)

where ~u and ~f are vectors and we still assume that A is not a function of ~u or t. Now

the equivalent to Eq. 6.78 is

(I � hA)~un+1 � ~un = �h~f(t + h) (6.81)

or

~un+1 = (I � hA)�1[~un � h~f(t+ h)] (6.82)

The inverse is not actually performed, but rather we solve Eq. 6.81 as a linear system

of equations. For our one-dimensional examples, the system of equations which mustbe solved is tridiagonal (e.g., for biconvection, A = �aBp(�1; 0; 1)=2�x), and henceits solution is inexpensive, but in multidimensions the bandwidth can be very large. In

general, the cost per time step of an implicit method is larger than that of an explicitmethod. The primary area of application of implicit methods is in the solution of

sti� ODE's, as we shall see in Chapter 8.

6.10.2 Application to Nonlinear Equations

Now consider the general nonlinear scalar ODE given by

du

dt= F (u; t) (6.83)

Application of the implicit Euler method gives

un+1 = un + hF (un+1; tn+1) (6.84)

This is a nonlinear di�erence equation. As an example, consider the nonlinear ODE

du

dt+

1

2u2 = 0 (6.85)


solved using implicit Euler time marching, which gives

un+1 + h1

2u2n+1 = un (6.86)

which requires a nontrivial method to solve for un+1. There are several di�erent

approaches one can take to solving this nonlinear di�erence equation. An iterative

method, such as Newton's method (see below), can be used. In practice, the \initialguess" for this nonlinear problem can be quite close to the solution, since the \initial

guess" is simply the solution at the previous time step, which implies that a lineariza-

tion approach may be quite successful. Such an approach is described in the next

Section.

6.10.3 Local Linearization for Scalar Equations

General Development

Let us start the process of local linearization by considering Eq. 6.83. In order to

implement the linearization, we expand F (u; t) about some reference point in time.Designate the reference value by tn and the corresponding value of the dependent

variable by un. A Taylor series expansion about these reference quantities gives

F (u; t) = F (un; tn) +

@F

@u

!n

(u� un) +

@F

@t

!n

(t� tn)

+1

2

@2F

@u2

!n

(u� un)2 +

@2F

@u@t

!n

(u� un)(t� tn)

+1

2

@2F

@t2

!n

(t� tn)2 + � � � (6.87)

On the other hand, the expansion of u(t) in terms of the independent variable t is

u(t) = un + (t� tn)

@u

@t

!n

+1

2(t� tn)

2

@2u

@t2

!n

+ � � � (6.88)

If t is within h of tn, both (t � tn)k and (u � un)

k are O(hk), and Eq. 6.87 can be

written

F (u; t) = Fn +

@F

@u

!n

(u� un) +

@F

@t

!n

(t� tn) +O(h2) (6.89)


Notice that this is an expansion of the derivative of the function. Thus, relative to the

order of expansion of the function, it represents a second-order-accurate, locally-linear

approximation to F (u; t) that is valid in the vicinity of the reference station tn and

the corresponding un = u(tn). With this we obtain the locally (in the neighborhood

of tn) time-linear representation of Eq. 6.83, namely

du

dt=

@F

@u

!n

u+

Fn �

@F

@u

!n

un

!+

@F

@t

!n

(t� tn) +O(h2) (6.90)

Implementation of the Trapezoidal Method

As an example of how such an expansion can be used, consider the mechanics ofapplying the trapezoidal method for the time integration of Eq. 6.83. The trapezoidalmethod is given by

un+1 = un +1

2h[Fn+1 + Fn] + hO(h2) (6.91)

where we write hO(h2) to emphasize that the method is second order accurate. Using

Eq. 6.89 to evaluate Fn+1 = F (un+1; tn+1), one �nds

un+1 = un +1

2h

"Fn +

@F

@u

!n

(un+1 � un) + h

@F

@t

!n

+O(h2) + Fn

#+hO(h2) (6.92)

Note that the O(h2) term within the brackets (which is due to the local linearization)is multiplied by h and therefore is the same order as the hO(h2) error from the

Trapezoidal Method. The use of local time linearization updated at the end of each

time step, and the trapezoidal time march, combine to make a second-order-accurate

numerical integration process. There are, of course, other second-order implicit time-

marching methods that can be used. The important point to be made here is that

local linearization updated at each time step has not reduced the order of accuracy of

a second-order time-marching process.

A very useful reordering of the terms in Eq. 6.92 results in the expression

"1�

1

2h

@F

@u

!n

#�un = hFn +

1

2h2 @F

@t

!n

(6.93)

which is now in the delta form which will be formally introduced in Section 12.6. In

many uid mechanic applications the nonlinear function F is not an explicit function


of t. In such cases the partial derivative of F (u) with respect to t is zero and Eq. 6.93

simpli�es to the second-order accurate expression

"1�

1

2h

@F

@u

!n

#�un = hFn (6.94)

Notice that the RHS is extremely simple. It is the product of h and the RHS of

the basic equation evaluated at the previous time step. In this example, the basic

equation was the simple scalar equation 6.83, but for our applications, it is generally

the space-di�erenced form of the steady-state equation of some uid ow problem.

A numerical time-marching procedure using Eq. 6.94 is usually implemented asfollows:

1. Solve for the elements of h~Fn, store them in an array say ~R, and save ~un.

2. Solve for the elements of the matrix multiplying �~un and store in some appro-

priate manner making use of sparseness or bandedness of the matrix if possible.Let this storage area be referred to as B.

3. Solve the coupled set of linear equations

B�~un = ~R

for �~un. (Very seldom does one �nd B�1 in carrying out this step).

4. Find ~un+1 by adding �~un to ~un, thus

~un+1 = �~un + ~un

The solution for ~un+1 is generally stored such that it overwrites the value of ~unand the process is repeated.

Implementation of the Implicit Euler Method

We have seen that the �rst-order implicit Euler method can be written

un+1 = un + hFn+1 (6.95)

if we introduce Eq. 6.90 into this method, rearrange terms, and remove the explicitdependence on time, we arrive at the form"

1� h

@F

@u

!n

#�un = hFn (6.96)


We see that the only di�erence between the implementation of the trapezoidal method

and the implicit Euler method is the factor of 12 in the brackets of the left side of

Eqs. 6.94 and 6.96. Omission of this factor degrades the method in time accuracy by

one order of h. We shall see later that this method is an excellent choice for steady

problems.

Newton's Method

Consider the limit h ! 1 of Eq. 6.96 obtained by dividing both sides by h and

setting 1=h = 0. There results

�

@F

@u

!n

�un = Fn (6.97)

or

un+1 = un �

" @F

@u

!n

#�1

Fn (6.98)

This is the well-known Newton method for �nding the roots of a nonlinear equation

F (u) = 0. The fact that it has quadratic convergence is veri�ed by a glance at Eqs.6.87 and 6.88 (remember the dependence on t has been eliminated for this case). By

quadratic convergence, we mean that the error after a given iteration is proportional

to the square of the error at the previous iteration, where the error is the di�erencebetween the current solution and the converged solution. Quadratic convergence is

thus a very powerful property. Use of a �nite value of h in Eq. 6.96 leads to linearconvergence, i.e., the error at a given iteration is some multiple of the error at the

previous iteration. The reader should ponder the meaning of letting h ! 1 for the

trapezoidal method, given by Eq. 6.94.

6.10.4 Local Linearization for Coupled Sets of Nonlinear Equa-

tions

In order to present this concept, let us consider an example involving some sim-ple boundary-layer equations. We choose the Falkner-Skan equations from classical

boundary-layer theory. Our task is to apply the implicit trapezoidal method to theequations

d3f

dt3+ f

d2f

dt2+ �

0@1� df

dt

!21A = 0 (6.99)

Here f represents a dimensionless stream function, and � is a scaling factor.


First of all we reduce Eq. 6.99 to a set of �rst-order nonlinear equations by the

transformations

u1 =d2f

dt2; u2 =

df

dt; u3 = f (6.100)

This gives the coupled set of three nonlinear equations

u01 = F1 = �u1 u3 � ��1� u22

�u02 = F2 = u1

u03 = F3 = u2 (6.101)

and these can be represented in vector notation as

d~u

dt= ~F (~u) (6.102)

Now we seek to make the same local expansion that derived Eq. 6.90, except thatthis time we are faced with a nonlinear vector function, rather than a simple nonlinearscalar function. The required extension requires the evaluation of a matrix, called

the Jacobian matrix.8 Let us refer to this matrix as A. It is derived from Eq. 6.102

by the following process

A = (aij) = @Fi =@uj (6.103)

For the general case involving a third order matrix this is

A =

26666666664

@F1

@u1@F1

@u2@F1

@u3

@F2

@u1@F2

@u2@F2

@u3

@F3

@u1@F3

@u2@F3

@u3

37777777775(6.104)

The expansion of ~F (~u) about some reference state ~un can be expressed in a way

similar to the scalar expansion given by eq 6.87. Omitting the explicit dependency

on the independent variable t, and de�ning ~F n as ~F (~un), one has9

8Recall that we derived the Jacobian matrices for the two-dimensional Euler equations in Section

2.29The Taylor series expansion of a vector contains a vector for the �rst term, a matrix times a

vector for the second term, and tensor products for the terms of higher order.

6.11. PROBLEMS 117

~F (~u) = ~F n + An

�~u� ~un

�+O(h2) (6.105)

where t� tn and the argument for O(h2) is the same as in the derivation of Eq. 6.88.

Using this we can write the local linearization of Eq. 6.102 as

d~u

dt= An

~u+

�~F n � An

~un

�| {z }\constant"

+O(h2) (6.106)

which is a locally-linear, second-order-accurate approximation to a set of couplednonlinear ordinary di�erential equations that is valid for t � tn + h. Any �rst- or

second-order time-marching method, explicit or implicit, could be used to integratethe equations without loss in accuracy with respect to order. The number of times,

and the manner in which, the terms in the Jacobian matrix are updated as the solutionproceeds depends, of course, on the nature of the problem.

Returning to our simple boundary-layer example, which is given by Eq. 6.101, we

�nd the Jacobian matrix to be

A =

264 � u3 2�u2 �u11 0 00 1 0

375 (6.107)

The student should be able to derive results for this example that are equivalent to

those given for the scalar case in Eq. 6.93. Thus for the Falkner-Skan equations the

trapezoidal method results in266641 + h

2(u3)n ��h(u2)nh2 (u1)n

�h2 1 0

0 �h2 1

37775264 (�u1)n(�u2)n(�u3)n

375=h

264 � (u1u3)n � �(1� u22)n(u1)n(u2)n

375We �nd ~un+1 from �~un+~un, and the solution is now advanced one step. Re-evaluate

the elements using ~un+1 and continue. Without any iterating within a step advance,the solution will be second-order-accurate in time.

6.11 Problems

1. Find an expression for the nth term in the Fibonacci series, which is given

by 1; 1; 2; 3; 5; 8; : : : Note that the series can be expressed as the solution to a

di�erence equation of the form un+1 = un + un�1. What is u25? (Let the �rst

term given above be u0.)


2. The trapezoidal method un+1 = un +1

2h(u0

n+1 + u0n) is used to solve the repre-

sentative ODE.

(a) What is the resulting O�E?

(b) What is its exact solution?

(c) How does the exact steady-state solution of the O�E compare with the

exact steady-state solution of the ODE if � = 0?

3. The 2nd-order backward method is given by

un+1 =1

3

h4un � un�1 + 2hu0

n+1

i(a) Write the O�E for the representative equation. Identify the polynomials

P (E) and Q(E).

(b) Derive the �-� relation. Solve for the �-roots and identify them as principalor spurious.

(c) Find er� and the �rst two nonvanishing terms in a Taylor series expansionof the spurious root.

(d) Perform a �-root trace relative to the unit circle for both di�usion and

convection.

4. Consider the time-marching scheme given by

un+1 = un�1 +2h

3(u0

n+1 + u0n+ u0

n�1)

(a) Write the O�E for the representative equation. Identify the polynomialsP (E) and Q(E).

(b) Derive the �� relation.

(c) Find er�.

5. Find the di�erence equation which results from applying the Gazdag predictor-

corrector method (Eq. 6.65) to the representative equation. Find the �-� rela-tion.

6. Consider the following time-marching method:

~un+1=3 = un + hu0n=3

�un+1=2 = un + h~u0n+1=3=2

un+1 = un + h�u0n+1=2

6.11. PROBLEMS 119

Find the di�erence equation which results from applying this method to the

representative equation. Find the �-� relation. Find the solution to the di�er-

ence equation, including the homogeneous and particular solutions. Find er�and er�. What order is the homogeneous solution? What order is the particular

solution? Find the particular solution if the forcing term is �xed.

7. Write a computer program to solve the one-dimensional linear convection equa-tion with periodic boundary conditions and a = 1 on the domain 0 � x � 1.

Use 2nd-order centered di�erences in space and a grid of 50 points. For the

initial condition, use

u(x; 0) = e�0:5[(x�0:5)=�]2

with � = 0:08. Use the explicit Euler, 2nd-order Adams-Bashforth (AB2), im-

plicit Euler, trapezoidal, and 4th-order Runge-Kutta methods. For the explicitEuler and AB2 methods, use a Courant number, ah=�x, of 0.1; for the other

methods, use a Courant number of unity. Plot the solutions obtained at t = 1compared to the exact solution (which is identical to the initial condition).

8. Repeat problem 7 using 4th-order (noncompact) di�erences in space. Use only

4th-order Runge-Kutta time marching at a Courant number of unity. Show

solutions at t = 1 and t = 10 compared to the exact solution.

9. Using the computer program written for problem 7, compute the solution at

t = 1 using 2nd-order centered di�erences in space coupled with the 4th-order

Runge-Kutta method for grids of 100, 200, and 400 nodes. On a log-log scale,

plot the error given by vuuut MXj=1

(uj � uexactj )2

M

where M is the number of grid nodes and uexact is the exact solution. Find the

global order of accuracy from the plot.

10. Using the computer program written for problem 8, repeat problem 9 using4th-order (noncompact) di�erences in space.

11. Write a computer program to solve the one-dimensional linear convection equa-

tion with in ow-out ow boundary conditions and a = 1 on the domain 0 �x � 1. Let u(0; t) = sin!t with ! = 10�. Run until a periodic steady state

is reached which is independent of the initial condition and plot your solution


compared with the exact solution. Use 2nd-order centered di�erences in space

with a 1st-order backward di�erence at the out ow boundary (as in Eq. 3.69)

together with 4th-order Runge-Kutta time marching. Use grids with 100, 200,

and 400 nodes and plot the error vs. the number of grid nodes, as described in

problem 9. Find the global order of accuracy.

12. Repeat problem 11 using 4th-order (noncompact) centered di�erences. Use a

third-order forward-biased operator at the in ow boundary (as in Eq. 3.67). At

the last grid node, derive and use a 3rd-order backward operator (using nodes

j�3, j�2, j�1, and j) and at the second last node, use a 3rd-order backward-

biased operator (using nodes j�2, j�1, j, and j+1; see problem 1 in Chapter3).

13. Using the approach described in Section 6.6.2, �nd the phase speed error, erp,

and the amplitude error, era, for the combination of second-order centered dif-ferences and 1st, 2nd, 3rd, and 4th-order Runge-Kutta time-marching at a

Courant number of unity. Also plot the phase speed error obtained using exact

integration in time, i.e., that obtained using the spatial discretization alone.Note that the required �-roots for the various Runge-Kutta methods can be

deduced from Eq. 6.69, without actually deriving the methods. Explain yourresults.

� � ��

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h= οο

- oo,σ = e σ = ei h

oo,

a) Diffusion b) Convection

λ

ωhωhλ hλ

Ι(σ) Ι(σ)

(σ) (σ)RR

h= 0λ h= 0ω

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σ1

σ1

λ h=-2

σ1

σ1

σ2

σ2

λ h=-1

σ1σ2

σ2 σ1

a) Euler Explicit

b) Leapfrog

c) AB2

h = 1ω

ConvectionDiffusion

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σ1σ1

σ1

σ1

σ2

σ3

σ1σ1

σ1

σ1

σ2

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d) Trapezoidal

e) Gazdag

f) RK2

g) RK4

λ h=- οο h = 2/3ω

λ h=-2

λ h=-2.8 h = 2 2ω

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Chapter 8

CHOICE OF TIME-MARCHING

METHODS

In this chapter we discuss considerations involved in selecting a time-marching method

for a speci�c application. Examples are given showing how time-marching methods

can be compared in a given context. An important concept underlying much of this

discussion is sti�ness, which is de�ned in the next section.

8.1 Sti�ness De�nition for ODE's

8.1.1 Relation to �-Eigenvalues

The introduction of the concept referred to as \sti�ness" comes about from the nu-

merical analysis of mathematical models constructed to simulate dynamic phenom-

ena containing widely di�erent time scales. De�nitions given in the literature are

not unique, but fortunately we now have the background material to construct a

de�nition which is entirely su�cient for our purposes.

We start with the assumption that our CFD problem is modeled with su�cient

accuracy by a coupled set of ODE's producing an A matrix typi�ed by Eq. 7.1.

Any de�nition of sti�ness requires a coupled system with at least two eigenvalues,

and the decision to use some numerical time-marching or iterative method to solve

it. The di�erence between the dynamic scales in physical space is represented by

the di�erence in the magnitude of the eigenvalues in eigenspace. In the following

discussion we concentrate on the transient part of the solution. The forcing function

may also be time varying in which case it would also have a time scale. However,

we assume that this scale would be adequately resolved by the chosen time-marching

method, and, since this part of the ODE has no e�ect on the numerical stability of

149

150 CHAPTER 8. CHOICE OF TIME-MARCHING METHODS

StableRegion

AccurateRegion

1

I( h)

R( h)

λ

λ

λh

Figure 8.1: Stable and accurate regions for the explicit Euler method.

the homogeneous part, we exclude the forcing function from further discussion in this

section.

Consider now the form of the exact solution of a system of ODE's with a com-

plete eigensystem. This is given by Eq. 6.27 and its solution using a one-root, time-

marching method is represented by Eq. 6.28. For a given time step, the time integra-

tion is an approximation in eigenspace that is di�erent for every eigenvector ~xm. Inmany numerical applications the eigenvectors associated with the small j�mj are wellresolved and those associated with the large j�mj are resolved much less accurately,

if at all. The situation is represented in the complex �h plane in Fig. 8.1. In this

�gure the time step has been chosen so that time accuracy is given to the eigenvectors

associated with the eigenvalues lying in the small circle and stability without time

accuracy is given to those associated with the eigenvalues lying outside of the small

circle but still inside the large circle.

The whole concept of sti�ness in CFD arises from the fact that we often do

not need the time resolution of eigenvectors associated with the large j�mj inthe transient solution, although these eigenvectors must remain coupled into

the system to maintain a high accuracy of the spatial resolution.

8.1. STIFFNESS DEFINITION FOR ODE'S 151

8.1.2 Driving and Parasitic Eigenvalues

For the above reason it is convenient to subdivide the transient solution into two

parts. First we order the eigenvalues by their magnitudes, thus

j�1j � j�2j � � � � � j�M j (8.1)

Then we write

Transient

Solution=

pXm=1

cme�mt ~xm

| {z }Driving

+MX

m=p+1

cme�mt ~xm

| {z }Parasitic

(8.2)

This concept is crucial to our discussion. Rephrased, it states that we can separate

our eigenvalue spectrum into two groups; one [�1 ! �p] called the driving eigenvalues(our choice of a time-step and marching method must accurately approximate the time

variation of the eigenvectors associated with these), and the other, [�p+1 ! �M ], called

the parasitic eigenvalues (no time accuracy whatsoever is required for the eigenvectors

associated with these, but their presence must not contaminate the accuracy of the

complete solution). Unfortunately, we �nd that, although time accuracy requirements

are dictated by the driving eigenvalues, numerical stability requirements are dictated

by the parasitic ones.

8.1.3 Sti�ness Classi�cations

The following de�nitions are somewhat useful. An inherently stable set of ODE's is

sti� if

j�pj � j�M j

In particular we de�ne the ratio

Cr = j�M j = j�pj

and form the categories

Mildly-sti� Cr < 102

Strongly-sti� 103 < Cr < 105

Extremely-sti� 106 < Cr < 108

Pathologically-sti� 109 < Cr

It should be mentioned that the gaps in the sti� category de�nitions are intentional

because the bounds are arbitrary. It is important to notice that these de�nitions

make no distinction between real, complex, and imaginary eigenvalues.


8.2 Relation of Sti�ness to Space Mesh Size

Many ow �elds are characterized by a few regions having high spatial gradients of

the dependent variables and other domains having relatively low gradient phenomena.

As a result it is quite common to cluster mesh points in certain regions of space and

spread them out otherwise. Examples of where this clustering might occur are at a

shock wave, near an airfoil leading or trailing edge, and in a boundary layer.

One quickly �nds that this grid clustering can strongly a�ect the eigensystem of

the resulting A matrix. In order to demonstrate this, let us examine the eigensystems

of the model problems given in Section 4.3.2. The simplest example to discuss relates

to the model di�usion equation. In this case the eigenvalues are all real, negative

numbers that automatically obey the ordering given in Eq. 8.1. Consider the case

when all of the eigenvalues are parasitic, i.e., we are interested only in the converged

steady-state solution. Under these conditions, the sti�ness is determined by the ratio

�M=�1. A simple calculation shows that

�1 = �4�

�x2sin2

�

2(M + 1)

!� �

�4�

�x2

��x

2

�2= ��

�M � �4�

�x2sin2

��

2

�= �

4�

�x2

and the ratio is

�M=�1 �4

�x2= 4

�M + 1

�

�2

The most important information found from this example is the fact that the

sti�ness of the transient solution is directly related to the grid spacing. Furthermore,

in di�usion problems this sti�ness is proportional to the reciprocal of the space mesh

size squared. For a mesh size M = 40, this ratio is about 680. Even for a mesh of

this moderate size the problem is already approaching the category of strongly sti�.

For the biconvection model a similar analysis shows that

j�M j = j�1j �1

�x

Here the sti�ness parameter is still space-mesh dependent, but much less so than for

di�usion-dominated problems.

We see that in both cases we are faced with the rather annoying fact that the more

we try to increase the resolution of our spatial gradients, the sti�er our equations tend

to become. Typical CFD problems without chemistry vary between the mildly and

strongly sti� categories, and are greatly a�ected by the resolution of a boundary layer

since it is a di�usion process. Our brief analysis has been limited to equispaced

8.3. PRACTICAL CONSIDERATIONS FOR COMPARING METHODS 153

problems, but in general the sti�ness of CFD problems is proportional to the mesh

intervals in the manner shown above where the critical interval is the smallest one in

the physical domain.

8.3 Practical Considerations for Comparing Meth-

ods

We have presented relatively simple and reliable measures of stability and both the

local and global accuracy of time-marching methods. Since there are an endless

number of these methods to choose from, one can wonder how this information is

to be used to pick a \best" choice for a particular problem. There is no unique

answer to such a question. For example, it is, among other things, highly dependent

upon the speed, capacity, and architecture of the available computer, and technology

in uencing this is undergoing rapid and dramatic changes as this is being written.

Nevertheless, if certain ground rules are agreed upon, relevant conclusions can be

reached. Let us now examine some ground rules that might be appropriate. It should

then be clear how the analysis can be extended to other cases.

Let us consider the problem of measuring the e�ciency of a time{marching method

for computing, over a �xed interval of time, an accurate transient solution of a coupled

set of ODE's. The length of the time interval, T , and the accuracy required of the

solution are dictated by the physics of the particular problem involved. For example,

in calculating the amount of turbulence in a homogeneous ow, the time interval

would be that required to extract a reliable statistical sample, and the accuracy

would be related to how much the energy of certain harmonics would be permitted

to distort from a given level. Such a computation we refer to as an event.

The appropriate error measures to be used in comparing methods for calculating

an event are the global ones, Era, Er� and Er!, discussed in Section 6.6.5, rather

than the local ones er�, era, and erp discussed earlier.

The actual form of the coupled ODE's that are produced by the semi-discrete

approach is

d~u

dt= ~F (~u; t)

At every time step we must evaluate the function ~F (~u; t) at least once. This function

is usually nonlinear, and its computation usually consumes the major portion of the

computer time required to make the simulation. We refer to a single calculation of the

vector ~F (~u; t) as a function evaluation and denote the total number of such evaluations

by Fev.


8.4 Comparing the E�ciency of Explicit Methods

8.4.1 Imposed Constraints

As mentioned above, the e�ciency of methods can be compared only if one accepts a

set of limiting constraints within which the comparisons are carried out. The follow

assumptions bound the considerations made in this Section:

1. The time-march method is explicit.

2. Implications of computer storage capacity and access time are ignored. In some

contexts, this can be an important consideration.

3. The calculation is to be time-accurate, must simulate an entire event which

takes a total time T , and must use a constant time step size, h, so that

T = Nh

where N is the total number of time steps.

8.4.2 An Example Involving Di�usion

Let the event be the numerical solution of

du

dt= �u (8.3)

from t = 0 to T = � ln(0:25) with u(0) = 1. Eq. 8.3 is obtained from our representa-

tive ODE with � = �1, a = 0. Since the exact solution is u(t) = u(0)e�t, this makes

the exact value of u at the end of the event equal to 0.25, i.e., u(T ) = 0:25. To the

constraints imposed above, let us set the additional requirement

� The error in u at the end of the event, i.e., the global error, must be < 0:5%.

We judge the most e�cient method as the one that satis�es these conditions and

has the fewest number of evaluations, Fev. Three methods are compared | explicit

Euler, AB2, and RK4.

First of all, the allowable error constraint means that the global error in the am-

plitude, see Eq. 6.48, must have the property:

��Er�e�T

�� < 0:005

8.4. COMPARING THE EFFICIENCY OF EXPLICIT METHODS 155

Then, since h = T=N = � ln(0:25)=N , it follows that��1� (�1(ln(:25)=N))N=:25

�� < 0:005

where �1 is found from the characteristic polynomials given in Table 7.1. The results

shown in Table 8.1 were computed using a simple iterative procedure.

Method N h �1 Fev Er�Euler 193 :00718 :99282 193 :001248 worst

AB2 16 :0866 :9172 16 :001137RK4 2 :6931 :5012 8 :001195 best

Table 8.1: Comparison of time-marching methods for a simple dissipation problem.

In this example we see that, for a given global accuracy, the method with the

highest local accuracy is the most e�cient on the basis of the expense in evaluating

Fev. Thus the second-order Adams-Bashforth method is much better than the �rst-

order Euler method, and the fourth-order Runge-Kutta method is the best of all. The

main purpose of this exercise is to show the (usually) great superiority of second-order

over �rst-order time-marching methods.

8.4.3 An Example Involving Periodic Convection

Let us use as a basis for this example the study of homogeneous turbulence simulated

by the numerical solution of the incompressible Navier-Stokes equations inside a cube

with periodic boundary conditions on all sides. In this numerical experiment the

function evaluations contribute overwhelmingly to the CPU time, and the number of

these evaluations must be kept to an absolute minimum because of the magnitude of

the problem. On the other hand, a complete event must be established in order to

obtain meaningful statistical samples which are the essence of the solution. In this

case, in addition to the constraints given in Section 8.4.1, we add the following:

� The number of evaluations of ~F (~u; t) is �xed.

Under these conditions a method is judged as best when it has the highest global

accuracy for resolving eigenvectors with imaginary eigenvalues. The above constraint

has led to the invention of schemes that omit the function evaluation in the cor-

rector step of a predictor-corrector combination, leading to the so-called incomplete


predictor-corrector methods. The presumption is, of course, that more e�cient meth-

ods will result from the omission of the second function evaluation. An example is

the method of Gazdag, given in Section 6.8. Basically this is composed of an AB2

predictor and a trapezoidal corrector. However, the derivative of the fundamental

family is never found so there is only one evaluation required to complete each cycle.

The �-� relation for the method is shown as entry 10 in Table 7.1.

In order to discuss our comparisions we introduce the following de�nitions:

� Let a k-evaluation method be de�ned as one that requires k evaluations of~F (~u; t) to advance one step using that method's time interval, h.

� Let K represent the total number of allowable Fev.

� Let h1 be the time interval advanced in one step of a one-evaluation method.

The Gazdag, leapfrog, and AB2 schemes are all 1-evaluation methods. The second

and fourth order RK methods are 2- and 4-evaluation methods, respectively. For a 1-

evaluation method the total number of time steps, N , and the number of evaluations,

K, are the same, one evaluation being used for each step, so that for these methods

h = h1. For a 2-evaluation method N = K=2 since two evaluations are used for

each step. However, in this case, in order to arrive at the same time T after K

evaluations, the time step must be twice that of a one{evaluation method so h = 2h1.For a 4-evaluation method the time interval must be h = 4h1, etc. Notice that

as k increases, the time span required for one application of the method increases.

However, notice also that as k increases, the power to which �1 is raised to arrive

at the �nal destination decreases; see the Figure below. This is the key to the true

comparison of time-march methods for this type of problem.

0 T uNk = 1 j � � � � � � � j [�(�h1)]

8

k = 2 j 2h1 � � � j [�(2�h1)]4

k = 4 j 4h1 � j [�(4�h1)]2

Step sizes and powers of � for k-evaluation methods used to get to the same value

of T if 8 evaluations are allowed.

In general, after K evaluations, the global amplitude and phase error for k-

evaluation methods applied to systems with pure imaginary �-roots can be written1

Era = 1� j�1(ik!h1)jK=k

(8.4)

1See Eqs. 6.38 and 6.39.

8.4. COMPARING THE EFFICIENCY OF EXPLICIT METHODS 157

Er! = !T �K

ktan�1

"[�1(ik!h1)]imaginary

[�1(ik!h1)]real

#(8.5)

Consider a convection-dominated event for which the function evaluation is very

time consuming. We idealize to the case where � = i! and set ! equal to one. The

event must proceed to the time t = T = 10. We consider two maximum evaluation

limits K = 50 and K = 100 and choose from four possible methods, leapfrog, AB2,

Gazdag, and RK4. The �rst three of these are one-evaluation methods and the last

one is a four-evaluation method. It is not di�cult to show that on the basis of local

error (made in a single step) the Gazdag method is superior to the RK4 method in

both amplitude and phase. For example, for !h = 0:2 the Gazdag method produces

a j�1j = 0:9992276 whereas for !h = 0:8 (which must be used to keep the number of

evaluations the same) the RK4 method produces a j�1j = 0:998324. However, we are

making our comparisons on the basis of global error for a �xed number of evaluations.

First of all we see that for a one-evaluation method h1 = T=K. Using this, and the

fact that ! = 1, we �nd, by some rather simple calculations2 made using Eqs. 8.4 and

8.5, the results shown in Table 8.2. Notice that to �nd global error the Gazdag root

must be raised to the power of 50 while the RK4 root is raised only to the power of

50/4. On the basis of global error the Gazdag method is not superior to RK4 in either

amplitude or phase, although, in terms of phase error (for which it was designed) it

is superior to the other two methods shown.

K leapfrog AB2 Gazdag RK4

!h1 = :1 100 1:0 1:003 :995 :999

!h1 = :2 50 1:0 1:022 :962 :979

a. Amplitude, exact = 1.0.

K leapfrog AB2 Gazdag RK4

!h1 = :1 100 �:96 �2:4 :45 :12

!h1 = :2 50 �3:8 �9:8 1:5 1:5

b. Phase error in degrees.

Table 8.2: Comparison of global amplitude and phase errors for four methods.

2The �1 root for the Gazdag method can be found using a numerical root �nding routine to trace

the three roots in the �-plane, see Fig. 7.3e.


Using analysis such as this (and also considering the stability boundaries) the

RK4 method is recommended as a basic �rst choice for any explicit time-accurate

calculation of a convection-dominated problem.

8.5 Coping With Sti�ness

8.5.1 Explicit Methods

The ability of a numerical method to cope with sti�ness can be illustrated quite nicely

in the complex �h plane. A good example of the concept is produced by studying

the Euler method applied to the representative equation. The transient solution is

un = (1 + �h)n and the trace of the complex value of �h which makes j1 + �hj = 1

gives the whole story. In this case the trace forms a circle of unit radius centered at

(�1; 0) as shown in Fig. 8.1. If h is chosen so that all �h in the ODE eigensystem

fall inside this circle the integration will be numerically stable. Also shown by the

small circle centered at the origin is the region of Taylor series accuracy. If some �hfall outside the small circle but stay within the stable region, these �h are sti�, but

stable. We have de�ned these �h as parasitic eigenvalues. Stability boundaries for

some explicit methods are shown in Figs. 7.5 and 7.6.

For a speci�c example, consider the mildly sti� system composed of a coupled

two-equation set having the two eigenvalues �1 = �100 and �2 = �1. If uncoupledand evaluated in wave space, the time histories of the two solutions would appear as a

rapidly decaying function in one case, and a relatively slowly decaying function in the

other. Analytical evaluation of the time histories poses no problem since e�100t quickly

becomes very small and can be neglected in the expressions when time becomes large.

Numerical evaluation is altogether di�erent. Numerical solutions, of course, depend

upon [�(�mh)]n and no j�mj can exceed one for any �m in the coupled system or else

the process is numerically unstable.

Let us choose the simple explicit Euler method for the time march. The coupled

equations in real space are represented by

u1(n) = c1(1� 100h)nx11 + c2(1� h)nx12 + (PS)1

u2(n) = c1(1� 100h)nx21 + c2(1� h)nx22 + (PS)2 (8.6)

We will assume that our accuracy requirements are such that su�cient accuracy is

obtained as long as j�hj � 0:1. This de�nes a time step limit based on accuracy

considerations of h = 0:001 for �1 and h = 0:1 for �2. The time step limit based

on stability, which is determined from �1, is h = 0:02. We will also assume that

c1 = c2 = 1 and that an amplitude less than 0.001 is negligible. We �rst run 66

time steps with h = 0:001 in order to resolve the �1 term. With this time step the

8.5. COPING WITH STIFFNESS 159

�2 term is resolved exceedingly well. After 66 steps, the amplitude of the �1 term

(i.e., (1� 100h)n) is less than 0.001 and that of the �2 term (i.e., (1� h)n) is 0.9361.

Hence the �1 term can now be considered negligible. To drive the (1 � h)n term to

zero (i.e., below 0.001), we would like to change the step size to h = 0:1 and continue.

We would then have a well resolved answer to the problem throughout the entire

relevant time interval. However, this is not possible because of the coupled presence

of (1� 100h)n, which in just 10 steps at h = 0:1 ampli�es those terms by � 109, far

outweighing the initial decrease obtained with the smaller time step. In fact, with

h = 0:02, the maximum step size that can be taken in order to maintain stability,

about 339 time steps have to be computed in order to drive e�t to below 0.001. Thus

the total simulation requires 405 time steps.

8.5.2 Implicit Methods

Now let us re-examine the problem that produced Eq. 8.6 but this time using an

unconditionally stable implicit method for the time march. We choose the trapezoidal

method. Its behavior in the �h plane is shown in Fig. 7.4b. Since this is also a one-

root method, we simply replace the Euler � with the trapezoidal one and analyze the

result. It follows that the �nal numerical solution to the ODE is now represented in

real space by

u1(n) = c1

1� 50h

1 + 50h

!nx11 + c2

1� 0:5h

1 + 0:5h

!nx12 + (PS)1

u2(n) = c1

1� 50h

1 + 50h

!nx21 + c2

1� 0:5h

1 + 0:5h

!nx22 + (PS)2 (8.7)

In order to resolve the initial transient of the term e�100t, we need to use a step size

of about h = 0:001. This is the same step size used in applying the explicit Euler

method because here accuracy is the only consideration and a very small step size

must be chosen to get the desired resolution. (It is true that for the same accuracy

we could in this case use a larger step size because this is a second-order method,

but that is not the point of this exercise). After 70 time steps the �1 term has

amplitude less than 0.001 and can be neglected. Now with the implicit method we

can proceed to calculate the remaining part of the event using our desired step size

h = 0:1 without any problem of instability, with 69 steps required to reduce the

amplitude of the second term to below 0.001. In both intervals the desired solution

is second-order accurate and well resolved. It is true that in the �nal 69 steps one

�-root is [1 � 50(0:1)]=[1 + 50(0:1)] = 0:666 � � �, and this has no physical meaning

whatsoever. However, its in uence on the coupled solution is negligible at the end

of the �rst 70 steps, and, since (0:666 � � �)n < 1, its in uence in the remaining 70


steps is even less. Actually, although this root is one of the principal roots in the

system, its behavior for t > 0:07 is identical to that of a stable spurious root. The

total simulation requires 139 time steps.

8.5.3 A Perspective

It is important to retain a proper perspective on a problem represented by the above

example. It is clear that an unconditionally stable method can always be called upon

to solve sti� problems with a minimum number of time steps. In the example, the

conditionally stable Euler method required 405 time steps, as compared to about

139 for the trapezoidal method, about three times as many. However, the Euler

method is extremely easy to program and requires very little arithmetic per step. For

preliminary investigations it is often the best method to use for mildly-sti� di�usion

dominated problems. For re�ned investigations of such problems an explicit method

of second order or higher, such as Adams-Bashforth or Runge-Kutta methods, is

recommended. These explicit methods can be considered as e�ective mildly sti�-

stable methods. However, it should be clear that as the degree of sti�ness of the

problem increases, the advantage begins to tilt towards implicit methods, as the

reduced number of time steps begins to outweigh the increased cost per time step.

The reader can repeat the above example with �1 = �10; 000, �2 = �1, which is in

the strongly-sti� category.

There is yet another technique for coping with certain sti� systems in uid dynamic

applications. This is known as the multigrid method. It has enjoyed remarkable

success in many practical problems; however, we need an introduction to the theory

of relaxation before it can be presented.

8.6 Steady Problems

In Chapter 6 we wrote the O�E solution in terms of the principal and spurious roots

as follows:

un = c11(�1)n

1~x1 + � � �+ cm1(�m)

n

1~xm + � � �+ cM1(�M)

n

1~xM + P:S:

+c12(�1)n

2~x1 + � � �+ cm2(�m)

n

2~xm + � � �+ cM2(�M)

n

2~xM

+c13(�1)n

3~x1 + � � �+ cm3(�m)

n

3~xm + � � �+ cM3(�M)

n

3~xM

+etc., if there are more spurious roots (8.8)

When solving a steady problem, we have no interest whatsoever in the transient por-

tion of the solution. Our sole goal is to eliminate it as quickly as possible. Therefore,

8.7. PROBLEMS 161

the choice of a time-marching method for a steady problem is similar to that for a

sti� problem, the di�erence being that the order of accuracy is irrelevant. Hence the

explicit Euler method is a candidate for steady di�usion dominated problems, and

the fourth-order Runge-Kutta method is a candidate for steady convection dominated

problems, because of their stability properties. Among implicit methods, the implicit

Euler method is the obvious choice for steady problems.

When we seek only the steady solution, all of the eigenvalues can be considered to

be parasitic. Referring to Fig. 8.1, none of the eigenvalues are required to fall in the

accurate region of the time-marching method. Therefore the time step can be chosen

to eliminate the transient as quickly as possible with no regard for time accuracy.

For example, when using the implicit Euler method with local time linearization, Eq.

6.96, one would like to take the limit h !1, which leads to Newton's method, Eq.

6.98. However, a �nite time step may be required until the solution is somewhat close

to the steady solution.

8.7 Problems

1. Repeat the time-march comparisons for di�usion (Section 8.4.2) and periodic

convection (Section 8.4.3) using 2nd- and 3rd-order Runge-Kutta methods.

2. Repeat the time-march comparisons for di�usion (Section 8.4.2) and periodic

convection (Section 8.4.3) using the 3rd- and 4th-order Adams-Bashforth meth-

ods. Considering the stability bounds for these methods (see problem 4 in

Chapter 7) as well as their memory requirements, compare and contrast them

with the 3rd- and 4th-order Runge-Kutta methods.

3. Consider the di�usion equation (with � = 1) discretized using 2nd-order central

di�erences on a grid with 10 (interior) points. Find and plot the eigenvalues

and the corresponding modi�ed wavenumbers. If we use the explicit Euler time-

marching method what is the maximum allowable time step if all but the �rst

two eigenvectors are considered parasitic? Assume that su�cient accuracy is

obtained as long as j�hj � 0:1. What is the maximum allowable time step if all

but the �rst eigenvector are considered parasitic?

Chapter 9

RELAXATION METHODS

In the past three chapters, we developed a methodology for designing, analyzing,

and choosing time-marching methods. These methods can be used to compute the

time-accurate solution to linear and nonlinear systems of ODE's in the general form

d~u

dt= ~F (~u; t) (9.1)

which arise after spatial discretization of a PDE. Alternatively, they can be used to

solve for the steady solution of Eq. 9.1, which satis�es the following coupled system

of nonlinear algebraic equations:

~F (~u) = 0 (9.2)

In the latter case, the unsteady equations are integrated until the solution converges

to a steady solution. The same approach permits a time-marching method to be used

to solve a linear system of algebraic equations in the form

A~x = ~b (9.3)

To solve this system using a time-marching method, a time derivative is introduced

as follows

d~x

dt= A~x�~b (9.4)

and the system is integrated in time until the transient has decayed to a su�ciently

low level. Following a time-dependent path to steady state is possible only if all of

the eigenvalues of the matrix A (or �A) have real parts lying in the left half-plane.

Although the solution ~x = A�1~b exists as long as A is nonsingular, the ODE given by

Eq. 9.4 has a stable steady solution only if A meets the above condition.

163

164 CHAPTER 9. RELAXATION METHODS

The common feature of all time-marching methods is that they are at least �rst-

order accurate. In this chapter, we consider iterative methods which are not time

accurate at all. Such methods are known as relaxation methods. While they are

applicable to coupled systems of nonlinear algebraic equations in the form of Eq. 9.2,

our analysis will focus on their application to large sparse linear systems of equations

in the form

Ab~u� ~f b = 0 (9.5)

where Ab is nonsingular, and the use of the subscript b will become clear shortly. Such

systems of equations arise, for example, at each time step of an implicit time-marching

method or at each iteration of Newton's method. Using an iterative method, we seek

to obtain rapidly a solution which is arbitrarily close to the exact solution of Eq. 9.5,

which is given by

~u1 = A�1b~f b (9.6)

9.1 Formulation of the Model Problem

9.1.1 Preconditioning the Basic Matrix

It is standard practice in applying relaxation procedures to precondition the basic

equation. This preconditioning has the e�ect of multiplying Eq. 9.5 from the left

by some nonsingular matrix. In the simplest possible case the conditioning matrix

is a diagonal matrix composed of a constant D(b). If we designate the conditioning

matrix by C, the problem becomes one of solving for ~u in

CAb~u� C~f b = 0 (9.7)

Notice that the solution of Eq. 9.7 is

~u = [CAb]�1C~f b = A�1b C�1C~f b = A�1b

~f b (9.8)

which is identical to the solution of Eq. 9.5, provided C�1 exists.

In the following we will see that our approach to the iterative solution of Eq. 9.7

depends crucially on the eigenvalue and eigenvector structure of the matrix CAb, and,

equally important, does not depend at all on the eigensystem of the basic matrix Ab.

For example, there are well-known techniques for accelerating relaxation schemes if

the eigenvalues of CAb are all real and of the same sign. To use these schemes, the

9.1. FORMULATION OF THE MODEL PROBLEM 165

conditioning matrix C must be chosen such that this requirement is satis�ed. A

choice of C which ensures this condition is the negative transpose of Ab.

For example, consider a spatial discretization of the linear convection equation

using centered di�erences with a Dirichlet condition on the left side and no constraint

on the right side. Using a �rst-order backward di�erence on the right side (as in

Section 3.6), this leads to the approximation

�x~u =1

2�x

8>>>>>><>>>>>>:

26666664

0 1

�1 0 1

�1 0 1

�1 0 1

�2 2

37777775~u+

26666664

�ua0

0

0

0

37777775

9>>>>>>=>>>>>>;

(9.9)

The matrix in Eq. 9.9 has eigenvalues whose imaginary parts are much larger than

their real parts. It can �rst be conditioned so that the modulus of each element is 1.

This is accomplished using a diagonal preconditioning matrix

D = 2�x

26666664

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 12

37777775 (9.10)

which scales each row. We then further condition with multiplication by the negative

transpose. The result is

A2 = �AT1A1 =

26666664

0 1

�1 0 1

�1 0 1

�1 0 1

�1 �1

37777775

26666664

0 1

�1 0 1

�1 0 1

�1 0 1

�1 1

37777775

=

26666664

�1 0 1

0 �2 0 1

1 0 �2 0 1

1 0 �2 1

1 1 �2

37777775 (9.11)


If we de�ne a permutation matrix P 1 and carry out the process P T [�AT1A1]P (which

just reorders the elements of A1 and doesn't change the eigenvalues) we �nd

26666664

0 1 0 0 0

0 0 0 1 0

0 0 0 0 1

0 0 1 0 0

1 0 0 0 0

37777775

26666664

�1 0 1

0 �2 0 1

1 0 �2 0 1

1 0 �2 1

1 1 �2

37777775

26666664

0 0 0 0 1

1 0 0 0 0

0 0 0 1 0

0 1 0 0 0

0 0 1 0 0

37777775

=

26666664

�2 1

1 �2 1

1 �2 1

1 �2 1

1 �1

37777775 (9.12)

which has all negative real eigenvalues, as given in Appendix B. Thus even when

the basic matrix Ab has nearly imaginary eigenvalues, the conditioned matrix �ATb Ab

is nevertheless symmetric negative de�nite (i.e., symmetric with negative real eigen-

values), and the classical relaxation methods can be applied. We do not necessarily

recommend the use of �ATb as a preconditioner; we simply wish to show that a broad

range of matrices can be preconditioned into a form suitable for our analysis.

9.1.2 The Model Equations

Preconditioning processes such as those described in the last section allow us to

prepare our algebraic equations in advance so that certain eigenstructures are guar-

anteed. In the remainder of this chapter, we will thoroughly investigate some simple

equations which model these structures. We will consider the preconditioned system

of equations having the form

A~�� ~f = 0 (9.13)

where A is symmetric negative de�nite.2 The symbol for the dependent variable has

been changed to � as a reminder that the physics being modeled is no longer time

1A permutation matrix (de�ned as a matrix with exactly one 1 in each row and column and hasthe property that P T = P

�1) just rearranges the rows and columns of a matrix.2We use a symmetric negative de�nite matrix to simplify certain aspects of our analysis. Relax-

ation methods are applicable to more general matrices. The classical methods will usually convergeif Ab is diagonally dominant, as de�ned in Appendix A.

9.1. FORMULATION OF THE MODEL PROBLEM 167

accurate when we later deal with ODE formulations. Note that the solution of Eq.

9.13, ~� = A�1~f , is guaranteed to exist because A is nonsingular. In the notation of

Eqs. 9.5 and 9.7,

A = CAb and ~f = C~f b (9.14)

The above was written to treat the general case. It is instructive in formulating the

concepts to consider the special case given by the di�usion equation in one dimension

with unit di�usion coe�cient �:

@u

@t=

@2u

@x2� g(x) (9.15)

This has the steady-state solution

@2u

@x2= g(x) (9.16)

which is the one-dimensional form of the Poisson equation. Introducing the three-

point central di�erencing scheme for the second derivative with Dirichlet boundary

conditions, we �nd

d~u

dt=

1

�x2B(1;�2; 1)~u+ (~bc)� ~g (9.17)

where (~bc) contains the boundary conditions and ~g contains the values of the source

term at the grid nodes. In this case

Ab =1

�x2B(1;�2; 1)

~fb = ~g � (~bc) (9.18)

Choosing C = �x2I, we obtain

B(1;�2; 1)~� = ~f (9.19)

where ~f = �x2~fb. If we consider a Dirichlet boundary condition on the left side and

either a Dirichlet or a Neumann condition on the right side, then A has the form

A = B�1;~b; 1

�


~b = [�2;�2; � � � �;�2; s]T

s = �2 or � 1 (9.20)

Note that s = �1 is easily obtained from the matrix resulting from the Neumann

boundary condition given in Eq. 3.24 using a diagonal conditioning matrix. A tremen-

dous amount of insight to the basic features of relaxation is gained by an appropriate

study of the one-dimensional case, and much of the remaining material is devoted to

this case. We attempt to do this in such a way, however, that it is directly applicable

to two- and three-dimensional problems.

9.2 Classical Relaxation

9.2.1 The Delta Form of an Iterative Scheme

We will consider relaxation methods which can be expressed in the following delta

form:

Hh~�n+1 � ~�n

i= A~�n �

~f (9.21)

where H is some nonsingular matrix which depends upon the iterative method. The

matrix H is independent of n for stationary methods and is a function of n for

nonstationary ones. The iteration count is designated by the subscript n or the

superscript (n). The converged solution is designated ~�1so that

~�1= A�1~f (9.22)

9.2.2 The Converged Solution, the Residual, and the Error

Solving Eq. 9.21 for ~�n+1 gives

~�n+1 = [I +H�1A]~�n �H�1~f = G~�n �H�1~f (9.23)

where

G � I +H�1A (9.24)

Hence it is clear that H should lead to a system of equations which is easy to solve,

or at least easier to solve than the original system. The error at the nth iteration is

de�ned as

~en � ~�n � ~�1= ~�n � A�1~f (9.25)

9.2. CLASSICAL RELAXATION 169

where ~�1was de�ned in Eq. 9.22. The residual at the nth iteration is de�ned as

~rn � A~�n �~f (9.26)

Multiply Eq. 9.25 by A from the left, and use the de�nition in Eq. 9.26. There results

the relation between the error and the residual

A~en �~rn = 0 (9.27)

Finally, it is not di�cult to show that

~en+1 = G~en (9.28)

Consequently, G is referred to as the basic iteration matrix, and its eigenvalues, which

we designate as �m, determine the convergence rate of a method.

In all of the above, we have considered only what are usually referred to as sta-

tionary processes in which H is constant throughout the iterations. Nonstationary

processes in which H (and possibly C) is varied at each iteration are discussed in

Section 9.5.

9.2.3 The Classical Methods

Point Operator Schemes in One Dimension

Let us consider three classical relaxation procedures for our model equation

B(1;�2; 1)~� = ~f (9.29)

as given in Section 9.1.2. The Point-Jacobi method is expressed in point operator

form for the one-dimensional case as

�(n+1)j =

1

2

h�(n)j�1 + �

(n)j+1 � fj

i(9.30)

This operator comes about by choosing the value of �(n+1)j such that together with

the old values of �j�1 and �j+1, the jth row of Eq. 9.29 is satis�ed. The Gauss-Seidel

method is

�(n+1)j =

1

2

h�(n+1)j�1 + �

(n)j+1 � fj

i(9.31)

This operator is a simple extension of the point-Jacobi method which uses the most

recent update of �j�1. Hence the jth row of Eq. 9.29 is satis�ed using the new values of

�j and �j�1 and the old value of �j+1. The method of successive overrelaxation (SOR)


is based on the idea that if the correction produced by the Gauss-Seidel method tends

to move the solution toward ~�1, then perhaps it would be better to move further in

this direction. It is usually expressed in two steps as

~�j =1

2

h�(n+1)j�1 + �

(n)j+1 � fj

i�(n+1)j = �

(n)j + !

h~�j � �

(n)j

i(9.32)

where ! generally lies between 1 and 2, but it can also be written in the single line

�(n+1)j =

!

2�(n+1)j�1 + (1� !)�

(n)j +

!

2�(n)j+1 �

!

2fj (9.33)

The General Form

The general form of the classical methods is obtained by splitting the matrix A in

Eq. 9.13 into its diagonal, D, the portion of the matrix below the diagonal, L, and

the portion above the diagonal, U , such that

A = L+D + U (9.34)

Then the point-Jacobi method is obtained with H = �D, which certainly meets

the criterion that it is easy to solve. The Gauss-Seidel method is obtained with

H = �(L +D), which is also easy to solve, being lower triangular.

9.3 The ODE Approach to Classical Relaxation

9.3.1 The Ordinary Di�erential Equation Formulation

The particular type of delta form given by Eq. 9.21 leads to an interpretation of

relaxation methods in terms of solution techniques for coupled �rst-order ODE's,

about which we have already learned a great deal. One can easily see that Eq. 9.21

results from the application of the explicit Euler time-marching method (with h = 1)

to the following system of ODE's:

Hd~�

dt= A~�� ~f (9.35)

This is equivalent to

d~�

dt= H�1C

�Ab~�� ~f b

�= H�1[A~�� ~f ] (9.36)

9.3. THE ODE APPROACH TO CLASSICAL RELAXATION 171

In the special case where H�1A depends on neither ~u nor t, H�1~f is also independent

of t, and the eigenvectors of H�1A are linearly independent, the solution can be

written as

~� = c1e�1t~x1 + � � �+ cMe

�M t~xM| {z }error

+~�1

(9.37)

where what is referred to in time-accurate analysis as the transient solution, is now

referred to in relaxation analysis as the error. It is clear that, if all of the eigenvalues

of H�1A have negative real parts (which implies that H�1A is nonsingular), then the

system of ODE's has a steady-state solution which is approached as t!1, given by

~�1= A�1~f (9.38)

which is the solution of Eq. 9.13. We see that the goal of a relaxation method is to

remove the transient solution from the general solution in the most e�cient way pos-

sible. The � eigenvalues are �xed by the basic matrix in Eq. 9.36, the preconditioning

matrix in 9.7, and the secondary conditioning matrix in 9.35. The � eigenvalues are

�xed for a given �h by the choice of time-marching method. Throughout the remain-

ing discussion we will refer to the independent variable t as \time", even though no

true time accuracy is involved.

In a stationary method, H and C in Eq. 9.36 are independent of t, that is, they

are not changed throughout the iteration process. The generalization of this in our

approach is to make h, the \time" step, a constant for the entire iteration.

Suppose the explicit Euler method is used for the time integration. For this method

�m = 1 + �mh. Hence the numerical solution after n steps of a stationary relaxation

method can be expressed as (see Eq. 6.28)

~�n = c1~x1(1 + �1h)n + � � �+ cm~xm(1 + �mh)

n + � � �+ cM~xM(1 + �Mh)n| {z }

error

+~�1(9.39)

The initial amplitudes of the eigenvectors are given by the magnitudes of the cm.

These are �xed by the initial guess. In general it is assumed that any or all of the

eigenvectors could have been given an equally \bad" excitation by the initial guess,

so that we must devise a way to remove them all from the general solution on an

equal basis. Assuming that H�1A has been chosen (that is, an iteration process has

been decided upon), the only free choice remaining to accelerate the removal of the

error terms is the choice of h. As we shall see, the three classical methods have all

been conditioned by the choice of H to have an optimum h equal to 1 for a stationary

iteration process.


9.3.2 ODE Form of the Classical Methods

The three iterative procedures de�ned by Eqs. 9.30, 9.31 and 9.32 obey no apparent

pattern except that they are easy to implement in a computer code since all of the

data required to update the value of one point are explicitly available at the time

of the update. Now let us study these methods as subsets of ODE as formulated in

Section 9.3.1. Insert the model equation 9.29 into the ODE form 9.35. Then

Hd~�

dt= B(1;�2; 1)~�� ~f (9.40)

As a start, let us use for the numerical integration the explicit Euler method

�n+1 = �n + h�0n (9.41)

with a step size, h, equal to 1. We arrive at

H(~�n+1 � ~�n) = B(1;�2; 1)~�n �~f (9.42)

It is clear that the best choice of H from the point of view of matrix algebra is

�B(1;�2; 1) since then multiplication from the left by �B�1(1;�2; 1) gives the cor-rect answer in one step. However, this is not in the spirit of our study, since multi-

plication by the inverse amounts to solving the problem by a direct method without

iteration. The constraint on H that is in keeping with the formulation of the three

methods described in Section 9.2.3 is that all the elements above the diagonal (or

below the diagonal if the sweeps are from right to left) are zero. If we impose this

constraint and further restrict ourselves to banded tridiagonals with a single constant

in each band, we are led to

B(��;2

!; 0)(~�n+1 � ~�n) = B(1;�2; 1)~�n �

~f (9.43)

where � and ! are arbitrary. With this choice of notation the three methods presented

in Section 9.2.3 can be identi�ed using the entries in Table 9.1.

TABLE 9.1: VALUES OF � and ! IN EQ. 9.43 THAT

LEAD TO CLASSICAL RELAXATION METHODS

� ! Method Equation

0 1 Point-Jacobi 6:2:3

1 1 Gauss-Seidel 6:2:4

1 2=h1 + sin

��

M + 1

�iOptimum SOR 6:2:5

9.4. EIGENSYSTEMS OF THE CLASSICAL METHODS 173

The fact that the values in the tables lead to the methods indicated can be veri�ed

by simple algebraic manipulation. However, our purpose is to examine the whole

procedure as a special subset of the theory of ordinary di�erential equations. In this

light, the three methods are all contained in the following set of ODE's

d~�

dt= B�1(��;

2

!; 0)

�B(1;�2; 1)~�� ~f

�(9.44)

and appear from it in the special case when the explicit Euler method is used for its

numerical integration. The point operator that results from the use of the explicit

Euler scheme is

�(n+1)j =

!�

2�(n+1)j�1 +

!

2(h� �)�

(n)j�1

!��(!h� 1)�

(n)j

�+

!h

2�(n)j+1

!�!h

2fj(9.45)

This represents a generalization of the classical relaxation techniques.

9.4 Eigensystems of the Classical Methods

The ODE approach to relaxation can be summarized as follows. The basic equation

to be solved came from some time-accurate derivation

Ab~u� ~f b = 0 (9.46)

This equation is preconditioned in some manner which has the e�ect of multiplication

by a conditioning matrix C giving

A~�� ~f = 0 (9.47)

An iterative scheme is developed to �nd the converged, or steady-state, solution of

the set of ODE's

Hd~�

dt= A~�� ~f (9.48)

This solution has the analytical form

~�n =~en + ~�

1(9.49)

where ~en is the transient, or error, and ~�1� A�1~f is the steady-state solution. The

three classical methods, Point-Jacobi, Gauss-Seidel, and SOR, are identi�ed for the

one-dimensional case by Eq. 9.44 and Table 9.1.


Given our assumption that the component of the error associated with each eigen-

vector is equally likely to be excited, the asymptotic convergence rate is determined

by the eigenvalue �m of G (� I +H�1A) having maximum absolute value. Thus

Convergence rate � j�mjmax ; m = 1; 2; � � � ;M (9.50)

In this section, we use the ODE analysis to �nd the convergence rates of the three

classical methods represented by Eqs. 9.30, 9.31, and 9.32. It is also instructive to

inspect the eigenvectors and eigenvalues in the H�1A matrix for the three methods.

This amounts to solving the generalized eigenvalue problem

A~xm = �mH~xm (9.51)

for the special case

B(1;�2; 1)~xm = �mB(��;2

!; 0)~xm (9.52)

The generalized eigensystem for simple tridigonals is given in Appendix B.2. The

three special cases considered below are obtained with a = 1, b = �2, c = 1, d = ��,e = 2=!, and f = 0. To illustrate the behavior, we take M = 5 for the matrix order.

This special case makes the general result quite clear.

9.4.1 The Point-Jacobi System

If � = 0 and ! = 1 in Eq. 9.44, the ODE matrixH�1A reduces to simply B(12 ;�1;12).

The eigensystem can be determined from Appendix B.1 since both d and f are zero.

The eigenvalues are given by the equation

�m = �1 + cos

�m�

M + 1

�; m = 1; 2; : : : ;M (9.53)

The �-� relation for the explicit Euler method is �m = 1 + �mh. This relation can

be plotted for any h. The plot for h = 1, the optimum stationary case, is shown in

Fig. 9.1. For h < 1, the maximum j�mj is obtained with m = 1, Fig. 9.2 and for

h > 1, the maximum j�mj is obtained with m = M , Fig. 9.3. Note that for h > 1:0

(depending on M) there is the possibility of instability, i.e. j�mj > 1:0. To obtain the

optimal scheme we wish to minimize the maximum j�mj which occurs when h = 1,

j�1j = j�M j and the best possible convergence rate is achieved:

j�mjmax = cos

��

M + 1

�(9.54)


For M = 40, we obtain j�mjmax = 0:9971. Thus after 500 iterations the error content

associated with each eigenvector is reduced to no more than 0.23 times its initial level.

Again from Appendix B.1, the eigenvectors of H�1A are given by

~xj = (xj)m = sin

�j

�m�

M + 1

��; j = 1; 2; : : : ;M (9.55)

This is a very \well-behaved" eigensystem with linearly independent eigenvectors and

distinct eigenvalues. The �rst 5 eigenvectors are simple sine waves. For M = 5, the

eigenvectors can be written as

~x1 =

26666664

1=2p3=2

1p3=2

1=2

37777775 ;~x2 =

26666664

p3=2p3=2

0

�p3=2

�p3=2

37777775 ;~x3 =

26666664

1

0

�10

1

37777775 ;

~x4 =

26666664

p3=2

�p3=2

0p3=2

�p3=2

37777775 ;~x5 =

26666664

1=2

�p3=2

1

�p3=2

1=2

37777775 (9.56)

The corresponding eigenvalues are, from Eq. 9.53

�1 = �1 +p32 = �0:134 � � �

�2 = �1 + 12 = �0:5

�3 = �1 = �1:0

�4 = �1� 12 = �1:5

�5 = �1�p32 = �1:866 � � �

(9.57)

From Eq. 9.39, the numerical solution written in full is

~�n � ~�1

= c1[1� (1�

p3

2)h]n~x1

+ c2[1� (1�1

2)h]n~x2


σm

1.0

0.0

-2.0 -1.0 0.0

h = 1.0

Μ = 5

Values are equal

m=1

m=2

m=3

m=4

m=5

Figure 9.1: The �; � relation for Point-Jacobi, h = 1;M = 5.

+ c3[1� (1 )h]n~x3

+ c4[1� (1 +1

2)h]n~x4

+ c5[1� (1 +

p3

2)h]n~x5 (9.58)

9.4.2 The Gauss-Seidel System

If � and ! are equal to 1 in Eq. 9.44, the matrix eigensystem evolves from the relation

B(1;�2; 1)~xm = �mB(�1; 2; 0)~xm (9.59)

which can be studied using the results in Appendix B.2. One can show that the H�1A

matrix for the Gauss-Seidel method, AGS , is

AGS � B�1(�1; 2; 0)B(1;�2; 1) =


σm

λmh

1.0

0.0

-2.0 -1.0 0.0

h < 1.0

Decreasing h

Approaches 1.0

Decreasing h

Figure 9.2: The �; � relation for Point-Jacobi, h = 0:9;M = 5.

σm

Exceeds 1.0 at h 1.072 / Ustable h > 1.072∼∼

1.0

0.0

-2.0 -1.0 0.0

h > 1.0

Increasing h

λmh

M = 5

Increasing h

Figure 9.3: The �; � relation for Point-Jacobi, h = 1:1;M = 5.


2666666666664

�1 1=2

0 �3=4 1=2

0 1=8 �3=4 1=2

0 1=16 1=8 �3=4 1=2

0 1=32 1=16 1=8 �3=4 1=2...

.... . .

. . .. . .

. . .. . .

0 1=2M � � �

3777777777775

1

2

3

4

5...

M

(9.60)

The eigenvector structure of the Gauss-Seidel ODE matrix is quite interesting. If M

is odd there are (M + 1)=2 distinct eigenvalues with corresponding linearly indepen-

dent eigenvectors, and there are (M � 1)=2 defective eigenvalues with corresponding

principal vectors. The equation for the nondefective eigenvalues in the ODE matrix

is (for odd M)

�m = �1 + cos2(m�

M + 1) ; m = 1; 2; : : : ;

M + 1

2(9.61)

and the corresponding eigenvectors are given by

~xm =

�cos

�m�

M + 1

��j�1sin

�j

�m�

M + 1

��; m = 1; 2; : : : ;

M + 1

2(9.62)

The �-� relation for h = 1, the optimum stationary case, is shown in Fig. 9.4. The

�m with the largest amplitude is obtained with m = 1. Hence the convergence rate is

j�mjmax =�cos

��

M + 1

��2(9.63)

Since this is the square of that obtained for the Point-Jacobi method, the error associ-

ated with the \worst" eigenvector is removed in half as many iterations. ForM = 40,

j�mjmax = 0:9942. 250 iterations are required to reduce the error component of the

worst eigenvector by a factor of roughly 0.23.

The eigenvectors are quite unlike the Point-Jacobi set. They are no longer sym-

metrical, producing waves that are higher in amplitude on one side (the updated side)

than they are on the other. Furthermore, they do not represent a common family for

di�erent values of M .

The Jordan canonical form for M = 5 is

X�1AGSX = JGS =

2666666664

h~�1

ih~�2

i2664~�3 1

~�3 1~�3

3775

3777777775

(9.64)


σm

λ m

λm

1.0

0.0

-2.0 -1.0 0.0

h = 1.0

Μ = 5

-1.0-1.0

2 defective

h

Figure 9.4: The �; � relation for Gauss-Seidel, h = 1:0;M = 5.

The eigenvectors and principal vectors are all real. For M = 5 they can be written

~x1 =

26666664

1=2

3=4

3=4

9=16

9=32

37777775 ;~x2 =

26666664

p3=2p3=4

0

�p3=16

�p3=32

37777775 ;~x3 =

26666664

1

0

0

0

0

37777775 ;~x4 =

26666664

0

2

�10

0

37777775 ;~x5 =

26666664

0

0

4

�41

37777775 (9.65)

The corresponding eigenvalues are

�1 = �1=4�2 = �3=4�3 = �1

(4)

(5)

)Defective, linked to

�3 Jordan block(9.66)

The numerical solution written in full is thus

~�n � ~�1

= c1(1�h

4)n~x1

+ c2(1�3h

4)n~x2


+

"c3(1� h)n + c4h

n

1!(1� h)n�1 + c5h

2n(n� 1)

2!(1� h)n�2

#~x3

+

�c4(1� h)n + c5h

n

1!(1� h)n�1

�~x4

+ c5(1� h)n~x5 (9.67)

9.4.3 The SOR System

If � = 1 and 2=! = x in Eq. 9.44, the ODE matrix is B�1(�1; x; 0)B(1;�2; 1).One can show that this can be written in the form given below for M = 5. The

generalization to any M is fairly clear. The H�1A matrix for the SOR method,

ASOR � B�1(�1; x; 0)B(1;�2; 1), is

1

x5

26666664

�2x4 x4 0 0 0

�2x3 + x4 x3 � 2x4 x4 0 0

�2x2 + x3 x2 � 2x3 + x4 x3 � 2x4 x4 0

�2x + x2 x� 2x2 + x3 x2 � 2x3 + x4 x3 � 2x4 x4

�2 + x 1� 2x + x2 x� 2x2 + x3 x2 � 2x3 + x4 x3 � 2x4

37777775 (9.68)

Eigenvalues of the system are given by

�m = �1 +�!pm + zm

2

�2; m = 1; 2; : : :M (9.69)

where

zm = [4(1� !) + !2p2m]1=2

pm = cos[m�=(M + 1)]

If ! = 1, the system is Gauss-Seidel. If 4(1�!)+!2pm < 0; zm and �m are complex.

If ! is chosen such that 4(1 � !) + !2p21 = 0; ! is optimum for the stationary case,

and the following conditions hold:

1. Two eigenvalues are real, equal and defective.

2. If M is even, the remaining eigenvalues are complex and occur in conjugate

pairs.

3. If M is odd, one of the remaining eigenvalues is real and the others are complex

occurring in conjugate pairs.


One can easily show that the optimum ! for the stationary case is

!opt = 2=

�1 + sin

��

M + 1

��(9.70)

and for ! = !opt

�m = �2m � 1

~xm = �j�1m sin

�j

�m�

M + 1

��(9.71)

where

�m =!opt

2

�pm + i

qp21 � p2m

�

Using the explicit Euler method to integrate the ODE's, �m = 1 � h + h�2m, and if

h = 1, the optimum value for the stationary case, the �-� relation reduces to that

shown in Fig. 9.5. This illustrates the fact that for optimum stationary SOR all the

j�mj are identical and equal to !opt � 1. Hence the convergence rate is

j�mjmax = !opt � 1 (9.72)

!opt = 2=

�1 + sin

��

M + 1

��

For M = 40, j�mjmax = 0:8578. Hence the worst error component is reduced to less

than 0.23 times its initial value in only 10 iterations, much faster than both Gauss-

Seidel and Point-Jacobi. In practical applications, the optimum value of ! may have

to be determined by trial and error, and the bene�t may not be as great.

For odd M , there are two real eigenvectors and one real principal vector. The

remaining linearly independent eigenvectors are all complex. For M = 5 they can be

written

~x1 =

26666664

1=2

1=2

1=3

1=6

1=18

37777775 ;~x2 =

26666664

�69

16

13

6

37777775 ;~x3;4 =

26666664

p3(1 )=2p3(1� i

p2)=6

0p3(5� i

p2)=54p

3(7� 4ip2)=162

37777775 ;~x5 =

26666664

1

0

1=3

0

1=9

37777775 (9.73)

The corresponding eigenvalues are

�1 = �2=3(2) Defective linked to �1

�3 = �(10� 2p2i)=9

�4 = �(10 + 2p2i)=9

�5 = �4=3 (9.74)


σm

λ m

λ m

λ m

1.0

0.0

-2.0 -1.0 0.0

h = 1.0

Μ = 5

-1.0-1.0

2 real defective

1 real

2 complex

hλ m

Figure 9.5: The �; � relation for optimum stationary SOR, M = 5, h = 1.

The numerical solution written in full is

~�n � ~�1

= [c1(1� 2h=3)n + c2nh(1� 2h=3)n�1]~x1

+ c2(1� 2h=3)n~x2

+ c3[1� (10� 2p2i)h=9]n~x3

+ c4[1� (10 + 2p2i)h=9]n~x4

+ c5(1� 4h=3)n~x5 (9.75)

9.5 Nonstationary Processes

In classical terminology a method is said to be nonstationary if the conditioning

matrices, H and C, are varied at each time step. This does not change the steady-

state solution A�1b~f b, but it can greatly a�ect the convergence rate. In our ODE

approach this could also be considered and would lead to a study of equations with

nonconstant coe�cients. It is much simpler, however, to study the case of �xed

H and C but variable step size, h. This process changes the Point-Jacobi method

to Richardson's method in standard terminology. For the Gauss-Seidel and SOR

methods it leads to processes that can be superior to the stationary methods.

The nonstationary form of Eq. 9.39 is

9.5. NONSTATIONARY PROCESSES 183

~�N = c1~x1

NYn=1

(1 + �1hn) + � � �+ cm~xm

NYn=1

(1 + �mhn)

+ � � �+ cM~xM

NYn=1

(1 + �Mhn) + ~�1

(9.76)

where the symbol � stands for product. Since hn can now be changed at each step,

the error term can theoretically be completely eliminated in M steps by taking hm =

�1=�m, form = 1; 2; � � � ;M . However, the eigenvalues �m are generally unknown and

costly to compute. It is therefore unnecessary and impractical to set hm = �1=�mfor m = 1; 2; : : : ;M . We will see that a few well chosen h's can reduce whole clusters

of eigenvectors associated with nearby �'s in the �m spectrum. This leads to the

concept of selectively annihilating clusters of eigenvectors from the error terms as

part of a total iteration process. This is the basis for the multigrid methods discussed

in Chapter 10.

Let us consider the very important case when all of the �m are real and nega-

tive (remember that they arise from a conditioned matrix so this constraint is not

unrealistic for quite practical cases). Consider one of the error terms taken from

~eN � ~�N � ~�1=

MXm=1

cm~xm

NYn=1

(1 + �mhn) (9.77)

and write it in the form

cm~xmPe(�m) � cm~xm

NYn=1

(1 + �mhn) (9.78)

where Pe signi�es an \Euler" polynomial. Now focus attention on the polynomial

(Pe)N(�) = (1 + h1�)(1 + h2�) � � � (1 + hN�) (9.79)

treating it as a continuous function of the independent variable �. In the annihilation

process mentioned after Eq. 9.76, we considered making the error exactly zero by

taking advantage of some knowledge about the discrete values of �m for a particular

case. Now we pose a less demanding problem. Let us choose the hn so that the

maximum value of (Pe)N(�) is as small as possible for all � lying between �a and �bsuch that �b � � � �a � 0. Mathematically stated, we seek

max�b��a

j(Pe)N(�)j = minimum ; with(Pe)N(0) = 1 (9.80)


This problem has a well known solution due to Markov. It is

(Pe)N(�) =

TN

2�� a � �b

�a � �b

!

TN

��a � �b

�a � �b

! (9.81)

where

TN(y) = cos(N arccos y) (9.82)

are the Chebyshev polynomials along the interval �1 � y � 1 and

TN(y) =1

2

�y +

qy2 � 1

�N+1

2

�y �

qy2 � 1

�N(9.83)

are the Chebyshev polynomials for jyj > 1. In relaxation terminology this is generally

referred to as Richardson's method, and it leads to the nonstationary step size choice

given by

1

hn=

1

2

(��b � �a + (�b � �a) cos

"(2n� 1)�

2N

#); n = 1; 2; : : :N (9.84)

Remember that all � are negative real numbers representing the magnitudes of �m in

an eigenvalue spectrum.

The error in the relaxation process represented by Eq. 9.76 is expressed in terms

of a set of eigenvectors, ~xm, ampli�ed by the coe�cients cmQ(1 + �mhn). With each

eigenvector there is a corresponding eigenvalue. Eq. 9.84 gives us the best choice of a

series of hn that will minimize the amplitude of the error carried in the eigenvectors

associated with the eigenvalues between �b and �a.

As an example for the use of Eq. 9.84, let us consider the following problem:

Minimize the maximum error asso-

ciated with the � eigenvalues in the

interval �2 � � � �1 using only 3

iterations.

(9.85)

The three values of h which satisfy this problem are

hn = 2=

3� cos

"(2n� 1)�

6

#!(9.86)

9.5. NONSTATIONARY PROCESSES 185

and the amplitude of the eigenvector is reduced to

(Pe)3(�) = T3(2�+ 3)=T3(3) (9.87)

where

T3(3) =n[3 +

p8]3 + [3�

p8]3o=2 � 99 (9.88)

A plot of Eq. 9.87 is given in Fig. 9.6 and we see that the amplitudes of all the

eigenvectors associated with the eigenvalues in the range �2 � � � �1 have been

reduced to less than about 1% of their initial values. The values of h used in Fig. 9.6

are

h1 = 4=(6�p3)

h2 = 4=(6� 0)

h3 = 4=(6 +p3)

Return now to Eq. 9.76. This was derived from Eq. 9.37 on the condition that the

explicit Euler method, Eq. 9.41, was used to integrate the basic ODE's. If instead

the implicit trapezoidal rule

�n+1 = �n +1

2h(�0n+1 + �0n) (9.89)

is used, the nonstationary formula

~�N =MXm=1

cm~xm

NYn=1

0BB@1 +

1

2hn�m

1�1

2hn�m

1CCA+ ~�

1(9.90)

would result. This calls for a study of the rational \trapezoidal" polynomial, Pt:

(Pt)N(�) =NYn=1

0BB@1 +

1

2hn�

1�1

2hn�

1CCA (9.91)

under the same constraints as before, namely that

max�b��a

j(Pt)N(�)j = minimum , (9.92)

with (Pt)N(0) = 1


−2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

λ

(Pe )

3 (λ)

−2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

λ

(Pe )

3 (λ)

Figure 9.6: Richardson's method for 3 steps, minimization over �2 � � � �1.

9.6. PROBLEMS 187

The optimum values of h can also be found for this problem, but we settle here for

the approximation suggested by Wachspress

2

hn= ��b

�a

�b

!(n�1)=(N�1)

; n = 1; 2; � � � ; N (9.93)

This process is also applied to problem 9.85. The results for (Pt)3(�) are shown in

Fig. 9.7. The error amplitude is about 1/5 of that found for (Pe)3(�) in the same

interval of �. The values of h used in Fig. 9.7 are

h1 = 1

h2 =p2

h3 = 2

9.6 Problems

1. Given a relaxation method in the form

H�~�n = A~�n � ~f

show that

~�n = Gn~�0 + (I �Gn)A�1 ~f

where G = I +H�1A.

2. For a linear system of the form (A1 + A2)x = b, consider the iterative method

(I + �A1)~x = (I � �A2)xn + �b

(I + �A2)xn+1 = (I � �A1)~x+ �b

where � is a parameter. Show that this iterative method can be written in the

form

H(xk+1 � xk) = (A1 + A2)xk � b

Determine the iteration matrix G if � = �1=2.

3. Using Appendix B.2, �nd the eigenvalues of H�1A for the SOR method with

A = B(4 : 1;�2; 1) and ! = !opt. (You do not have to �nd H�1. Recall that

the eigenvalues of H�1A satisfy A~xm = �mH~xm.) Find the numerical values,

not just the expressions. Then �nd the corresponding j�mj values.


−2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

λ

(P

t )3 (

λ)

−2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

−3

λ

(P

t )3 (

λ)

Figure 9.7: Wachspress method for 3 steps, minimization over �2 � � � �1.

9.6. PROBLEMS 189

4. Solve the following equation on the domain 0 � x � 1 with boundary conditions

u(0) = 0, u(1) = 1:

@2u

@x2� 6x = 0

For the initial condition, use u(x) = 0. Use second-order centered di�erences

on a grid with 40 cells (M = 39). Iterate to steady state using

(a) the point-Jacobi method,

(b) the Gauss-Seidel method,

(c) the SOR method with the optimum value of !, and

(d) the 3-step Richardson method derived in Section 9.5.

Plot the solution after the residual is reduced by 2, 3, and 4 orders of mag-

nitude. Plot the logarithm of the L2-norm of the residual vs. the number of

iterations. Determine the asymptotic convergence rate. Compare with the the-

oretical asymptotic convergence rate.

Chapter 10

MULTIGRID

The idea of systematically using sets of coarser grids to accelerate the convergence of

iterative schemes that arise from the numerical solution to partial di�erential equa-

tions was made popular by the work of Brandt. There are many variations of the

process and many viewpoints of the underlying theory. The viewpoint presented here

is a natural extension of the concepts discussed in Chapter 9.

10.1 Motivation

10.1.1 Eigenvector and Eigenvalue Identi�cation with Space

Frequencies

Consider the eigensystem of the model matrix B(1;�2; 1). The eigenvalues and

eigenvectors are given in Sections 4.3.2 and 4.3.3, respectively. Notice that as the

magnitudes of the eigenvalues increase, the space-frequency (or wavenumber) of the

corresponding eigenvectors also increase. That is, if the eigenvalues are ordered such

that

j�1j � j�2j � � � � � j�M j (10.1)

then the corresponding eigenvectors are ordered from low to high space frequencies.

This has a rational explanation from the origin of the banded matrix. Note that

@2

@x2sin(mx) = �m2 sin(mx) (10.2)

and recall that

�xx~� =1

�x2B(1;�2; 1)~� = X

�1

�x2D(~�)

�X�1~� (10.3)

191

192 CHAPTER 10. MULTIGRID

where D(~�) is a diagonal matrix containing the eigenvalues. We have seen that

X�1~� represents a sine transform, and X~�, a sine synthesis. Therefore, the opera-

tion 1�x2

D(~�) represents the numerical approximation of the multiplication of the

appropriate sine wave by the negative square of its wavenumber, �m2. One �nds

that

1

�x2�m =

�M + 1

�

�2��2 + 2 cos

�m�

M + 1

�� m2 ; m << M (10.4)

Hence, the correlation of large magnitudes of �m with high space-frequencies is to be

expected for these particular matrix operators. This is consistent with the physics of

di�usion as well. However, this correlation is not necessary in general. In fact, the

complete counterexample of the above association is contained in the eigensystem

for B(12 ; 1;12). For this matrix one �nds, from Appendix B, exactly the opposite

behavior.

10.1.2 Properties of the Iterative Method

The second key motivation for multigrid is the following:

� Many iterative methods reduce error components corresponding to eigenvalues

of large amplitude more e�ectively than those corresponding to eigenvalues of

small amplitude.

This is to be expected of an iterative method which is time accurate. It is also

true, for example, of the Gauss-Seidel method and, by design, of the Richardson

method described in Section 9.5. The classical point-Jacobi method does not share

this property. As we saw in Section 9.4.1, this method produces the same value of j�j

for �min and �max. However, the property can be restored by using h < 1, as shown

in Fig. 9.2.

When an iterative method with this property is applied to a matrix with the

above correlation between the modulus of the eigenvalues and the space frequency of

the eigenvectors, error components corresponding to high space frequencies will be

reduced more quickly than those corresponding to low space frequencies. This is the

key concept underlying the multigrid process.

10.2 The Basic Process

First of all we assume that the di�erence equations representing the basic partial

di�erential equations are in a form that can be related to a matrix which has certain

10.2. THE BASIC PROCESS 193

basic properties. This form can be arrived at \naturally" by simply replacing the

derivatives in the PDE with di�erence schemes, as in the example given by Eq. 3.27,

or it can be \contrived" by further conditioning, as in the examples given by Eq. 9.11.

The basic assumptions required for our description of the multigrid process are:

1. The problem is linear.

2. The eigenvalues, �m, of the matrix are all real and negative.

3. The �m are fairly evenly distributed between their maximum and minimum

values.

4. The eigenvectors associated with the eigenvalues having largest magnitudes can

be correlated with high frequencies on the di�erencing mesh.

5. The iterative procedure used greatly reduces the amplitudes of the eigenvectors

associated with eigenvalues in the range between 12 j�jmax and j�jmax.

These conditions are su�cient to ensure the validity of the process described next.

Having preconditioned (if necessary) the basic �nite di�erencing scheme by a pro-

cedure equivalent to the multiplication by a matrix C, we are led to the starting

formulation

C[Ab~�1�~f b] = 0 (10.5)

where the matrix formed by the product CAb has the properties given above. In Eq.

10.5, the vector ~f b represents the boundary conditions and the forcing function, if

any, and ~�1

is a vector representing the desired exact solution. We start with some

initial guess for ~�1

and proceed through n iterations making use of some iterative

process that satis�es property 5 above. We do not attempt to develop an optimum

procedure here, but for clarity we suppose that the three-step Richardson method

illustrated in Fig. 9.6 is used. At the end of the three steps we �nd ~r, the residual,

where

~r = C[Ab~�� ~f b] (10.6)

Recall that the ~� used to compute ~r is composed of the exact solution ~�1

and the

error ~e in such a way that

A~e�~r = 0 (10.7)

where

A � CAb (10.8)


If one could solve Eq. 10.7 for ~e then

~�1= ~��~e (10.9)

Thus our goal now is to solve for ~e. We can write the exact solution for ~e in terms of

the eigenvectors of A, and the � eigenvalues of the Richardson process in the form:

~e =

M=2Xm=1

cm~xm

3Yn=1

[�(�mhn)] +MX

m=M=2+1

cm~xm

3Yn=1

[�(�mhn)]

| {z }very low amplitude

(10.10)

Combining our basic assumptions, we can be sure that the high frequency content of~e has been greatly reduced (about 1% or less of its original value in the initial guess).

In addition, assumption 4 ensures that the error has been smoothed.

Next we construct a permutation matrix which separates a vector into two parts,

one containing the odd entries, and the other the even entries of the original vector

(or any other appropriate sorting which is consistent with the interpolation approxi-

mation to be discussed below). For a 7-point example

2666666666664

e2e4e6e1e3e5e7

3777777777775=

2666666666664

0 1 0 0 0 0 0

0 0 0 1 0 0 0

0 0 0 0 0 1 0

1 0 0 0 0 0 0

0 0 1 0 0 0 0

0 0 0 0 1 0 0

0 0 0 0 0 0 1

3777777777775

2666666666664

e1e2e3e4e5e6e7

3777777777775

;

"~ee~eo

#= P~e (10.11)

Multiply Eq. 10.7 from the left by P and, since a permutation matrix has an inverse

which is its transpose, we can write

PA[P�1P ]~e = P~r (10.12)

The operation PAP�1 partitions the A matrix to form

264A1 A2

A3 A4

375"~ee~eo

#=

"~re~ro

#(10.13)

Notice that

A1~ee + A2

~eo = ~re (10.14)


is an exact expression.

At this point we make our one crucial assumption. It is that there is some connec-

tion between ~ee and ~eo brought about by the smoothing property of the Richardson

relaxation procedure. Since the top half of the frequency spectrum has been removed,

it is reasonable to suppose that the odd points are the average of the even points.

For example

e1 �

1

2(ea + e2)

e3 �

1

2(e2 + e4)

e5 �

1

2(e4 + e6) or ~eo = A02

~ee (10.15)

e7 �

1

2(e6 + eb)

It is important to notice that ea and eb represent errors on the boundaries where the

error is zero if the boundary conditions are given. It is also important to notice that we

are dealing with the relation between ~e and ~r so the original boundary conditions and

forcing function (which are contained in ~f in the basic formulation) no longer appear

in the problem. Hence, no aliasing of these functions can occur in subsequent steps.

Finally, notice that, in this formulation, the averaging of ~e is our only approximation,

no operations on ~r are required or justi�ed.

If the boundary conditions are Dirichlet, ea and eb are zero, and one can write for

the example case

A02 =1

2

266641 0 0

1 1 0

0 1 1

0 0 1

37775 (10.16)

With this approximation Eq. 10.14 reduces to

A1~ee + A2A

0

2~ee = ~re (10.17)

or

Ac~ee �~re = 0 (10.18)

where

Ac = [A1 + A2A0

2] (10.19)


The form of Ac, the matrix on the coarse mesh, is completely determined by the

choice of the permutation matrix and the interpolation approximation. If the original

A had been B(7 : 1;�2; 1), our 7-point example would produce

PAP�1 =

266666666666664

�2 1 1

�2 1 1

�2 1 1

1 �2

1 1 �2

1 1 �2

1 �2

377777777777775=

264 A1 A2

A3 A4

375 (10.20)

and Eq. 10.18 gives

A1z }| {264 �2

�2

�2

375+

A2z }| {264 1 1

1 1

1 1

375 �1

2

A0

2z }| {266641

1 1

1 1

1

37775 =

Acz }| {264 �1 1=2

1=2 �1 1=2

1=2 �1

375 (10.21)

If the boundary conditions are mixed Dirichlet-Neumann, A in the 1-D model

equation is B(1;~b; 1) where ~b = [�2;�2; :::;�2;�1]T . The eigensystem is given by

Eq. B.19. It is easy to show that the high space-frequencies still correspond to the

eigenvalues with high magnitudes, and, in fact, all of the properties given in Section

10.1 are met. However, the eigenvector structure is di�erent from that given in

Eq. 9.55 for Dirichlet conditions. In the present case they are given by

xjm = sin

"j

(2m� 1)�

2M + 1

!#; m = 1; 2; � � � ;M (10.22)

and are illustrated in Fig. 10.1. All of them go through zero on the left (Dirichlet)

side, and all of them re ect on the right (Neumann) side.

For Neumann conditions, the interpolation formula in Eq. 10.15 must be changed.

In the particular case illustrated in Fig. 10.1, eb is equal to eM . If Neumann conditions

are on the left, ea = e1. When eb = eM , the example in Eq. 10.16 changes to

A02 =1

2

266641 0 0

1 1 0

0 1 1

0 0 2

37775 (10.23)


X

Figure 10.1: Eigenvectors for the mixed Dirichlet{Neumann case.

The permutation matrix remains the same and both A1 and A2 in the partitioned

matrix PAP�1 are unchanged (only A4 is modi�ed by putting �1 in the lower right

element). Therefore, we can construct the coarse matrix from

A1z }| {264 �2

�2

�2

375+

A2z }| {264 1 1

1 1

1 1

375 �1

2

A0

2z }| {266641

1 1

1 1

2

37775 =

Acz }| {264 �1 1=2

1=2 �1 1=2

1=2 �1=2

375 (10.24)

which gives us what we might have \expected."

We will continue with Dirichlet boundary conditions for the remainder of this

Section. At this stage, we have reduced the problem from B(1;�2; 1)~e = ~r on the

�ne mesh to 12B(1;�2; 1)

~ee = ~re on the next coarser mesh. Recall that our goal is

to solve for ~e, which will provide us with the solution ~�1

using Eq. 10.9. Given ~eecomputed on the coarse grid (possibly using even coarser grids), we can compute ~eousing Eq. 10.15, and thus ~e. In order to complete the process, we must now determine

the relationship between ~ee and ~e.

In order to examine this relationship, we need to consider the eigensystems of A

and Ac:

A = X�X�1; Ac = Xc�cX�1c (10.25)

For A = B(M : 1;�2; 1) the eigenvalues and eigenvectors are

�m = �2

�1� cos

�m�

M + 1

��; ~xm = sin

�j

�m�

M + 1

��;

j = 1; 2; � � � ;M

m = 1; 2; � � � ;M(10.26)


Based on our assumptions, the most di�cult error mode to eliminate is that with

m = 1, corresponding to

�1 = �2

�1� cos

��

M + 1

��; ~x1 = sin

�j

��

M + 1

��; j = 1; 2; � � � ;M (10.27)

For example, with M = 51, �1 = �0:003649. If we restrict our attention to odd M ,

then Mc = (M �1)=2 is the size of Ac. The eigenvalue and eigenvector corresponding

to m = 1 for the matrix Ac =12B(Mc; 1;�2; 1) are

(�c)1 = �

�1� cos

�2�

M + 1

��; (~xc)1 = sin

�j

�2�

M + 1

��; j = 1; 2; � � � ;Mc (10.28)

For M = 51 (Mc = 25), we obtain (�c)1 = �0:007291 = 1:998�1. As M increases,

(�c)1 approaches 2�1. In addition, one can easily see that (~xc)1 coincides with ~x1 at

every second point of the latter vector, that is, it contains the even elements of ~x1.

Now let us consider the case in which all of the error consists of the eigenvector

component ~x1, i.e., ~e = ~x1. Then the residual is

~r = A~x1 = �1~x1 (10.29)

and the residual on the coarse grid is

~re = �1(~xc)1 (10.30)

since (~xc)1 contains the even elements of ~x1. The exact solution on the coarse grid

satis�es

~ee = A�1c~re = Xc�

�1c X�1

c �1(~xc)1

(10.31)

= �1Xc��1c

2666641

0...

0

377775

(10.32)

= �1Xc

2666641=(�c)1

0...

0

377775

(10.33)


=�1

(�c)1(~xc)1

(10.34)

�

1

2(~xc)1 (10.35)

Since our goal is to compute ~e = ~x1, in addition to interpolating ~ee to the �ne grid

(using Eq. 10.15), we must multiply the result by 2. This is equivalent to solving

1

2Ac~ee = ~re (10.36)

or

1

4B(Mc : 1;�2; 1)~ee = ~re (10.37)

In our case, the matrix A = B(M : 1;�2; 1) comes from a discretization of the

di�usion equation, which gives

Ab =�

�x2B(M : 1;�2; 1) (10.38)

and the preconditioning matrix C is simply

C =�x2

�I (10.39)

Applying the discretization on the coarse grid with the same preconditioning matrix

as used on the �ne grid gives, since �xc = 2�x,

C�

�x2cB(Mc : 1;�2; 1) =

�x2

�x2cB(Mc : 1;�2; 1) =

1

4B(Mc : 1;�2; 1) (10.40)

which is precisely the matrix appearing in Eq. 10.37. Thus we see that the process is

recursive. The problem to be solved on the coarse grid is the same as that solved on

the �ne grid.

The remaining steps required to complete an entire multigrid process are relatively

straightforward, but they vary depending on the problem and the user. The reduction

can be, and usually is, carried to even coarser grids before returning to the �nest level.

However, in each case the appropriate permutation matrix and the interpolation

approximation de�ne both the down- and up-going paths. The details of �nding

optimum techniques are, obviously, quite important but they are not discussed here.


10.3 A Two-Grid Process

We now describe a two-grid process for the linear problem A~� = ~f , which can be easily

generalized to a process with an arbitrary number of grids due to the recursive nature

of multigrid. Extension to nonlinear problems requires that both the solution and the

residual be transferred to the coarse grid in a process known as full approximation

storage multigrid.

1. Perform n1 iterations of the selected relaxation method on the �ne grid, starting

with ~� = ~�n. Call the result ~�(1). This gives1

~�(1) = Gn11~�n + (I �Gn1

1 )A�1 ~f (10.41)

where

G1 = I +H�11 A1 (10.42)

and H1 is de�ned as in Chapter 9 (e.g., Eq. 9.21). Next compute the residual based

on ~�(1):

~r(1) = A~�(1) � ~f = AGn11~�n + A (I �Gn1

1 )A�1 ~f � ~f

= AGn11~�n � AGn1

1 A�1 ~f (10.43)

2. Transfer (or restrict) ~r(1) to the coarse grid:

~r(2) = R21~r

(1) (10.44)

In our example in the preceding section, the restriction matrix is

R21 =

264 0 1 0 0 0 0 0

0 0 0 1 0 0 0

0 0 0 0 0 1 0

375 (10.45)

that is, the �rst three rows of the permutation matrix P in Eq. 10.11. This type of

restriction is known as \simple injection." Some form of weighted restriction can also

be used.

3. Solve the problem A2~e(2) = ~r(2) on the coarse grid exactly:2

~e(2) = A�12 ~r(2) (10.46)

1See problem 1 of Chapter 9.2Note that the coarse grid matrix denoted A2 here was denoted Ac in the preceding section.

10.3. A TWO-GRID PROCESS 201

Here A2 can be formed by applying the discretization on the coarse grid. In the

preceding example (eq. 10.40), A2 =14B(Mc : 1;�2; 1). It is at this stage that the

generalization to a multigrid procedure with more than two grids occurs. If this is the

coarsest grid in the sequence, solve exactly. Otherwise, apply the two-grid process

recursively.

4. Transfer (or prolong) the error back to the �ne grid and update the solution:

~�n+1 = ~�(1) � I12~e(2) (10.47)

In our example, the prolongation matrix is

I12 =

2666666666664

1=2 0 0

1 0 0

1=2 1=2 0

0 1 0

0 1=2 1=2

0 0 1

0 0 1=2

3777777777775

(10.48)

which follows from Eq. 10.15.

Combining these steps, one obtains

~�n+1 = [I � I12A�12 R2

1A]Gn11~�n � [I � I12A

�12 R2

1A]Gn11 A

�1 ~f + A�1 ~f (10.49)

Thus the basic iteration matrix is

[I � I12A�12 R2

1A]Gn11 (10.50)

The eigenvalues of this matrix determine the convergence rate of the two-grid process.

The basic iteration matrix for a three-grid process is found from Eq. 10.50 by

replacing A�12 with (I �G23)A

�12 , where

G23 = [I � I23A

�13 R3

2A2]Gn22 (10.51)

In this expression n2 is the number of relaxation steps on grid 2, I23 and R32 are the

transfer operators between grids 2 and 3, and A3 is obtained by discretizing on grid

3. Extension to four or more grids proceeds in similar fashion.


10.4 Problems

1. Derive Eq. 10.51.

2. Repeat problem 4 of Chapter 9 using a four-grid multigrid method together

with

(a) the Gauss-Seidel method,

(b) the 3-step Richardson method derived in Section 9.5.

Solve exactly on the coarsest grid. Plot the solution after the residual is reduced

by 2, 3, and 4 orders of magnitude. Plot the logarithm of the L2-norm of the

residual vs. the number of iterations. Determine the asymptotic convergence

rate. Calculate the theoretical asymptotic convergence rate and compare.

Chapter 11

NUMERICAL DISSIPATION

Up to this point, we have emphasized the second-order centered-di�erence approxima-

tions to the spatial derivatives in our model equations. We have seen that a centered

approximation to a �rst derivative is nondissipative, i.e., the eigenvalues of the as-

sociated circulant matrix (with periodic boundary conditions) are pure imaginary.

In processes governed by nonlinear equations, such as the Euler and Navier-Stokes

equations, there can be a continual production of high-frequency components of the

solution, leading, for example, to the production of shock waves. In a real phys-

ical problem, the production of high frequencies is eventually limited by viscosity.

However, when we solve the Euler equations numerically, we have neglected viscous

e�ects. Thus the numerical approximation must contain some inherent dissipation to

limit the production of high-frequency modes. Although numerical approximations

to the Navier-Stokes equations contain dissipation through the viscous terms, this

can be insu�cient, especially at high Reynolds numbers, due to the limited grid res-

olution which is practical. Therefore, unless the relevant length scales are resolved,

some form of added numerical dissipation is required in the numerical solution of

the Navier-Stokes equations as well. Since the addition of numerical dissipation is

tantamount to intentionally introducing nonphysical behavior, it must be carefully

controlled such that the error introduced is not excessive. In this Chapter, we discuss

some di�erent ways of adding numerical dissipation to the spatial derivatives in the

linear convection equation and hyperbolic systems of PDE's.

203

204 CHAPTER 11. NUMERICAL DISSIPATION

11.1 One-Sided First-Derivative Space Di�erenc-

ing

We investigate the properties of one-sided spatial di�erence operators in the context

of the biconvection model equation given by

@u

@t= �a

@u

@x(11.1)

with periodic boundary conditions. Consider the following point operator for the

spatial derivative term

�a(�xu)j =�a2�x

[�(1 + �)uj�1 + 2�uj + (1� �)uj+1]

=�a2�x

[(�uj�1 + uj+1) + �(�uj�1 + 2uj � uj+1)] (11.2)

The second form shown divides the operator into an antisymmetric component (�uj�1+

uj+1)=2�x and a symmetric component �(�uj�1 + 2uj � uj+1)=2�x. The antisym-

metric component is the second-order centered di�erence operator. With � 6= 0, the

operator is only �rst-order accurate. A backward di�erence operator is given by � = 1

and a forward di�erence operator is given by � = �1.For periodic boundary conditions the corresponding matrix operator is

�a�x =�a2�x

Bp(�1� �; 2�; 1� �)

The eigenvalues of this matrix are

�m =�a�x

��

�1� cos

�2�m

M

��+ i sin

�2�m

M

��for m = 0; 1; : : : ;M� 1

If a is positive, the forward di�erence operator (� = �1) produces Re(�m) > 0,

the centered di�erence operator (� = 0) produces Re(�m) = 0, and the backward

di�erence operator produces Re(�m) < 0. Hence the forward di�erence operator is

inherently unstable while the centered and backward operators are inherently stable.

If a is negative, the roles are reversed. When Re(�m) = 0, the solution will either

grow or decay with time. In either case, our choice of di�erencing scheme produces

nonphysical behavior. We proceed next to show why this occurs.

11.2. THE MODIFIED PARTIAL DIFFERENTIAL EQUATION 205

11.2 The Modi�ed Partial Di�erential Equation

First carry out a Taylor series expansion of the terms in Eq. 11.2. We are lead to the

expression

(�xu)j =1

2�x

242�x

@u

@x

!j

� ��x2 @2u

@x2

!j

+�x3

3

@3u

@x3

!j

��x4

12

@4u

@x4

!j

+ : : :

35

We see that the antisymmetric portion of the operator introduces odd derivative

terms in the truncation error while the symmetric portion introduces even derivatives.

Substituting this into Eq. 11.1 gives

@u

@t= �a

@u

@x+a��x

2

@2u

@x2�a�x2

6

@3u

@x3+a��x3

24

@4u

@x4+ : : : (11.3)

This is the partial di�erential equation we are really solving when we apply the

approximation given by Eq. 11.2 to Eq. 11.1. Notice that Eq. 11.3 is consistent with

Eq. 11.1, since the two equations are identical when �x! 0. However, when we use

a computer to �nd a numerical solution of the problem, �x can be small but it is

not zero. This means that each term in the expansion given by Eq. 11.3 is excited to

some degree. We refer to Eq. 11.3 as the modi�ed partial di�erential equation. We

proceed next to investigate the implications of this concept.

Consider the simple linear partial di�erential equation

@u

@t= �a

@u

@x+ �

@2u

@x2+

@3u

@x3+ �

@4u

@x4(11.4)

Choose periodic boundary conditions and impose an initial condition u = ei�x. Underthese conditions there is a wave-like solution to Eq. 11.4 of the form

u(x; t) = ei�xe(r+is)t

provided r and s satisfy the condition

r + is = �ia�� 2 � i �3 + ��4

or

r = ��2(� � ��2); s = ��(a + �2)

The solution is composed of both amplitude and phase terms. Thus

u = e��2(��2)| {z }

amplitude

ei�[x�(a+ �2)t]| {z }phase

(11.5)


It is important to notice that the amplitude of the solution depends only upon � and

� , the coe�cients of the even derivatives in Eq. 11.4, and the phase depends only on

a and , the coe�cients of the odd derivatives.

If the wave speed a is positive, the choice of a backward di�erence scheme (� = 1)

produces a modi�ed PDE with � � ��2 > 0 and hence the amplitude of the solution

decays. This is tantamount to deliberately adding dissipation to the PDE. Under the

same condition, the choice of a forward di�erence scheme (� = �1) is equivalent todeliberately adding a destabilizing term to the PDE.

By examining the term governing the phase of the solution in Eq. 11.5, we see

that the speed of propagation is a + �2. Referring to the modi�ed PDE, Eq. 11.3

we have = �a�x2=6. Therefore, the phase speed of the numerical solution is less

than the actual phase speed. Furthermore, the numerical phase speed is dependent

upon the wavenumber �. This we refer to as dispersion.

Our purpose here is to investigate the properties of one-sided spatial di�erencing

operators relative to centered di�erence operators. We have seen that the three-

point centered di�erence approximation of the spatial derivative produces a modi�ed

PDE that has no dissipation (or ampli�cation). One can easily show, by using the

antisymmetry of the matrix di�erence operators, that the same is true for any cen-

tered di�erence approximation of a �rst derivative. As a corollary, any departure

from antisymmetry in the matrix di�erence operator must introduce dissipation (or

ampli�cation) into the modi�ed PDE.

Note that the use of one-sided di�erencing schemes is not the only way to in-

troduce dissipation. Any symmetric component in the spatial operator introduces

dissipation (or ampli�cation). Therefore, one could choose � = 1=2 in Eq. 11.2. The

resulting spatial operator is not one-sided but it is dissipative. Biased schemes use

more information on one side of the node than the other. For example, a third-order

backward-biased scheme is given by

(�xu)j =1

6�x(uj�2 � 6uj�1 + 3uj + 2uj+1)

=1

12�x[(uj�2 � 8uj�1 + 8uj+1 � uj+2)

+ (uj�2 � 4uj�1 + 6uj � 4uj+1 + uj+2)] (11.6)

The antisymmetric component of this operator is the fourth-order centered di�erence

operator. The symmetric component approximates �x3uxxxx=12. Therefore, this

operator produces fourth-order accuracy in phase with a third-order dissipative term.

11.3. THE LAX-WENDROFF METHOD 207

11.3 The Lax-Wendro� Method

In order to introduce numerical dissipation using one-sided di�erencing, backward

di�erencing must be used if the wave speed is positive, and forward di�erencing must

be used if the wave speed is negative. Next we consider a method which introduces

dissipation independent of the sign of the wave speed, known as the Lax-Wendro�

method. This explicit method di�ers conceptually from the methods considered pre-

viously in which spatial di�erencing and time-marching are treated separately.

Consider the following Taylor-series expansion in time:

u(x; t+ h) = u+ h@u

@t+

1

2h2@2u

@t2+O(h3) (11.7)

First replace the time derivatives with space derivatives according to the PDE (in

this case, the linear convection equation @u

@t+ a@u

@x= 0). Thus

@u

@t= �a

@u

@x;

@2u

@t2= a2

@2u

@x2(11.8)

Now replace the space derivatives with three-point centered di�erence operators, giv-

ing

u(n+1)j = u

(n)j �

1

2

ah

�x(u

(n)j+1 � u

(n)j�1) +

1

2

ah

�x

!2

(u(n)j+1 � 2u

(n)j + u

(n)j�1) (11.9)

This is the Lax-Wendro� method applied to the linear convection equation. It is a

fully-discrete �nite-di�erence scheme. There is no intermediate semi-discrete stage.

For periodic boundary conditions, the corresponding fully-discrete matrix operator

is

~un+1 = Bp

0@12

24 ah�x

+

ah

�x

!235 ; 1�

ah

�x

!2

;1

2

24� ah

�x+

ah

�x

!2351A ~un

The eigenvalues of this matrix are

�m = 1� ah

�x

!2 �1� cos

�2�m

M

�� i

ah

�xsin

�2�m

M

�for m = 0; 1; : : : ;M � 1

For j ah�xj � 1 all of the eigenvalues have modulus less than or equal to unity and hence

the method is stable independent of the sign of a. The quantity j ah�xj is known as the

Courant (or CFL) number. It is equal to the ratio of the distance travelled by a wave

in one time step to the mesh spacing.


The nature of the dissipative properties of the Lax-Wendro� scheme can be seen

by examining the modi�ed partial di�erential equation, which is given by

@u

@t+ a

@u

@x= �

a

6(�x2 � a2h2)

@3u

@x3�a2h

8(�x2 � a2h2)

@4u

@x4+ : : :

This is derived by substituting Taylor series expansions for all terms in Eq. 11.9 and

converting the time derivatives to space derivatives using Eq. 11.8. The two leading

error terms appear on the right side of the equation. Recall that the odd derivatives on

the right side lead to unwanted dispersion and the even derivatives lead to dissipation

(or ampli�cation, depending on the sign). Therefore, the leading error term in the

Lax-Wendro� method is dispersive and proportional to

�a

6(�x2 � a2h2)

@3u

@x3= �

a�x2

6(1� C2

n)@3u

@x3

The dissipative term is proportional to

�a2h

8(�x2 � a2h2)

@4u

@x4= �

a2h�x2

8(1� C2

n)@4u

@x4

This term has the appropriate sign and hence the scheme is truly dissipative as long

as Cn � 1.

A closely related method is that of MacCormack. Recall MacCormack's time-

marching method, presented in Chapter 6:

~un+1 = un + hu0n

un+1 =1

2[un + ~un+1 + h~u0

n+1] (11.10)

If we use �rst-order backward di�erencing in the �rst stage and �rst-order forward

di�erencing in the second stage,1 a dissipative second-order method is obtained. For

the linear convection equation, this approach leads to

~u(n+1)j = u

(n)j �

ah

�x(u

(n)j � u

(n)j�1)

u(n+1)j =

1

2[u

(n)j + ~u

(n+1)j �

ah

�x(~u

(n+1)j+1 � ~u

(n+1)j )] (11.11)

which can be shown to be identical to the Lax-Wendro� method. Hence MacCor-

mack's method has the same dissipative and dispersive properties as the Lax-Wendro�

method. The two methods di�er when applied to nonlinear hyperbolic systems, how-

ever.

1Or vice-versa; for nonlinear problems, these should be applied alternately.

11.4. UPWIND SCHEMES 209

11.4 Upwind Schemes

In Section 11.1, we saw that numerical dissipation can be introduced in the spatial

di�erence operator using one-sided di�erence schemes or, more generally, by adding

a symmetric component to the spatial operator. With this approach, the direction

of the one-sided operator (i.e., whether it is a forward or a backward di�erence)

or the sign of the symmetric component depends on the sign of the wave speed.

When a hyperbolic system of equations is being solved, the wave speeds can be both

positive and negative. For example, the eigenvalues of the ux Jacobian for the one-

dimensional Euler equations are u; u+ a; u� a. When the ow is subsonic, these are

of mixed sign. In order to apply one-sided di�erencing schemes to such systems, some

form of splitting is required. This is avoided in the Lax-Wendro� scheme. However,

as a result of their superior exibility, schemes in which the numerical dissipation

is introduced in the spatial operator are generally preferred over the Lax-Wendro�

approach.

Consider again the linear convection equation:

@u

@t+ a

@u

@x= 0 (11.12)

where we do not make any assumptions as to the sign of a. We can rewrite Eq. 11.12

as@u

@t+ (a+ + a�)

@u

@x= 0 ; a� =

a� jaj2

If a � 0, then a+ = a � 0 and a� = 0. Alternatively, if a � 0, then a+ = 0 and

a� = a � 0. Now for the a+ (� 0) term we can safely backward di�erence and for the

a� (� 0) term forward di�erence. This is the basic concept behind upwind methods,

that is, some decomposition or splitting of the uxes into terms which have positive

and negative characteristic speeds so that appropriate di�erencing schemes can be

chosen. In the next two sections, we present two splitting techniques commonly used

with upwind methods. These are by no means unique.

The above approach to obtaining a stable discretization independent of the sign

of a can be written in a di�erent, but entirely equivalent, manner. From Eq. 11.2, we

see that a stable discretization is obtained with � = 1 if a � 0 and with � = �1 if

a � 0. This is achieved by the following point operator:

�a(�xu)j =�12�x

[a(�uj�1 + uj+1) + jaj(�uj�1 + 2uj � uj+1)] (11.13)

This approach is extended to systems of equations in Section 11.5.

In this section, we present the basic ideas of ux-vector and ux-di�erence splitting.

For more subtle aspects of implementation and application of such techniques to


nonlinear hyperbolic systems such as the Euler equations, the reader is referred to

the literature on this subject.

11.4.1 Flux-Vector Splitting

Recall from Section 2.5 that a linear, constant-coe�cient, hyperbolic system of partial

di�erential equations given by

@u

@t+@f

@x=

@u

@t+ A

@u

@x= 0 (11.14)

can be decoupled into characteristic equations of the form

@wi

@t+ �i

@wi

@x= 0 (11.15)

where the wave speeds, �i, are the eigenvalues of the Jacobian matrix, A, and the

wi's are the characteristic variables. In order to apply a one-sided (or biased) spatial

di�erencing scheme, we need to apply a backward di�erence if the wave speed, �i, is

positive, and a forward di�erence if the wave speed is negative. To accomplish this,

let us split the matrix of eigenvalues, �, into two components such that

� = �+ + �� (11.16)

where

�+ =�+ j�j

2; �� =

�� j�j2

(11.17)

With these de�nitions, �+ contains the positive eigenvalues and �� contains the neg-

ative eigenvalues. We can now rewrite the system in terms of characteristic variables

as

@w

@t+ �

@w

@x=

@w

@t+ �+@w

@x+ ��

@w

@x= 0 (11.18)

The spatial terms have been split into two components according to the sign of the

wave speeds. We can use backward di�erencing for the �+ @w

@xterm and forward

di�erencing for the �� @w

@xterm. Premultiplying by X and inserting the product

X�1X in the spatial terms gives

@Xw

@t+@X�+X�1Xw

@x+@X��X�1Xw

@x= 0 (11.19)

11.4. UPWIND SCHEMES 211

With the de�nitions2

A+ = X�+X�1; A� = X��X�1 (11.20)

and recalling that u = Xw, we obtain

@u

@t+@A+u

@x+@A�u

@x= 0 (11.21)

Finally the split ux vectors are de�ned as

f+ = A+u; f� = A�u (11.22)

and we can write

@u

@t+@f+

@x+@f�

@x= 0 (11.23)

In the linear case, the de�nition of the split uxes follows directly from the de�ni-

tion of the ux, f = Au. For the Euler equations, f is also equal to Au as a result of

their homogeneous property, as discussed in Appendix C. Note that

f = f+ + f� (11.24)

Thus by applying backward di�erences to the f+ term and forward di�erences to the

f� term, we are in e�ect solving the characteristic equations in the desired manner.

This approach is known as ux-vector splitting.

When an implicit time-marching method is used, the Jacobians of the split ux

vectors are required. In the nonlinear case,

@f+

@u6= A+;

@f�

@u6= A� (11.25)

Therefore, one must �nd and use the new Jacobians given by

A++ =@f+

@u; A�� =

@f�

@u(11.26)

For the Euler equations, A++ has eigenvalues which are all positive, and A�� has all

negative eigenvalues.

2With these de�nitions A+ has all positive eigenvalues, and A� has all negative eigenvalues.


11.4.2 Flux-Di�erence Splitting

Another approach, more suited to �nite-volume methods, is known as ux-di�erence

splitting. In a �nite-volume method, the uxes must be evaluated at cell bound-

aries. We again begin with the diagonalized form of the linear, constant-coe�cient,

hyperbolic system of equations

@w

@t+ �

@w

@x= 0 (11.27)

The ux vector associated with this form is g = �w. Now, as in Chapter 5, we

consider the numerical ux at the interface between nodes j and j + 1, gj+1=2, as a

function of the states to the left and right of the interface, wL and wR, respectively.

The centered approximation to gj+1=2, which is nondissipative, is given by

gj+1=2 =1

2(g(wL) + g(wR)) (11.28)

In order to obtain a one-sided upwind approximation, we require

(gi)j+1=2 =

(�i(wi)L if �i > 0

�i(wi)R if �i < 0(11.29)

where the subscript i indicates individual components of w and g. This is achieved

with

(gi)j+1=2 =1

2�i [(wi)L + (wi)R] +

1

2j�ij [(wi)L � (wi)R] (11.30)

or

gj+1=2 =1

2� (wL + wR) +

1

2j�j (wL � wR) (11.31)

Now, as in Eq. 11.19, we premultiply by X to return to the original variables and

insert the product X�1X after � and j�j to obtain

Xgj+1=2 =1

2X�X�1X (wL + wR) +

1

2Xj�jX�1X (wL � wR) (11.32)

and thus

fj+1=2 =1

2(fL + fR) +

1

2jAj (uL � uR) (11.33)

where

jAj = Xj�jX�1 (11.34)

11.5. ARTIFICIAL DISSIPATION 213

and we have also used the relations f = Xg, u = Xw, and A = X�X�1.

In the linear, constant-coe�cient case, this leads to an upwind operator which is

identical to that obtained using ux-vector splitting. However, in the nonlinear case,

there is some ambiguity regarding the de�nition of jAj at the cell interface j + 1=2.

In order to resolve this, consider a situation in which the eigenvalues of A are all of

the same sign. In this case, we would like our de�nition of fj+1=2 to satisfy

fj+1=2 =

(fL if all �i

0s > 0

fR if all �i0s < 0

(11.35)

giving pure upwinding. If the eigenvalues of A are all positive, jAj = A; if they are

all negative, jAj = �A. Hence satisfaction of Eq. 11.35 is obtained by the de�nition

fj+1=2 =1

2(fL + fR) +

1

2jAj+1=2j (uL � uR) (11.36)

if Aj+1=2 satis�es

fL � fR = Aj+1=2 (uL � uR) (11.37)

For the Euler equations for a perfect gas, Eq. 11.37 is satis�ed by the ux Jacobian

evaluated at the Roe-average state given by

uj+1=2 =

p�LuL +

p�RuR

p�L +

p�R

(11.38)

Hj+1=2 =

p�LHL +

p�RHR

p�L +

p�R

(11.39)

where u and H = (e + p)=� are the velocity and the total enthalpy per unit mass,

respectively.3

11.5 Arti�cial Dissipation

We have seen that numerical dissipation can be introduced by using one-sided dif-

ferencing schemes together with some form of ux splitting. We have also seen that

such dissipation can be introduced by adding a symmetric component to an antisym-

metric (dissipation-free) operator. Thus we can generalize the concept of upwinding

to include any scheme in which the symmetric portion of the operator is treated in

such a manner as to be truly dissipative.

3Note that the ux Jacobian can be written in terms of u and H only; see problem 6 at the end

of this chapter.


For example, let

(�axu)

j=

uj+1 � uj�1

2�x; (�s

xu)

j=�uj+1 + 2uj � uj�1

2�x(11.40)

Applying �x = �ax+ �s

xto the spatial derivative in Eq. 11.15 is stable if �i � 0 and

unstable if �i < 0. Similarly, applying �x = �ax� �s

xis stable if �i � 0 and unstable if

�i > 0. The appropriate implementation is thus

�i�x = �i�a

x+ j�ij�sx (11.41)

Extension to a hyperbolic system by applying the above approach to the characteristic

variables, as in the previous two sections, gives

�x(Au) = �ax(Au) + �s

x(jAju) (11.42)

or

�xf = �axf + �s

x(jAju) (11.43)

where jAj is de�ned in Eq. 11.34. The second spatial term is known as arti�cial

dissipation. It is also sometimes referred to as arti�cial di�usion or arti�cial viscosity.

With appropriate choices of �axand �s

x, this approach can be related to the upwind

approach. This is particularly evident from a comparison of Eqs. 11.36 and 11.43.

It is common to use the following operator for �sx

(�sxu)

j=

�

�x(uj�2 � 4uj�1 + 6uj � 4uj+1 + uj+2) (11.44)

where � is a problem-dependent coe�cient. This symmetric operator approximates

��x3uxxxx and thus introduces a third-order dissipative term. With an appropriate

value of �, this often provides su�cient damping of high frequency modes without

greatly a�ecting the low frequency modes. For details of how this can be implemented

for nonlinear hyperbolic systems, the reader should consult the literature. A more

complicated treatment of the numerical dissipation is also required near shock waves

and other discontinuities, but is beyond the scope of this book.

11.6 Problems

1. A second-order backward di�erence approximation to a 1st derivative is given

as a point operator by

(�xu)j =1

2�x(uj�2 � 4uj�1 + 3uj)

11.6. PROBLEMS 215

(a) Express this operator in banded matrix form (for periodic boundary condi-

tions), then derive the symmetric and skew-symmetric matrices that have

the matrix operator as their sum. (See Appendix A.3 to see how to con-

struct the symmetric and skew-symmetric components of a matrix.)

(b) Using a Taylor table, �nd the derivative which is approximated by the

corresponding symmetric and skew-symmetric operators and the leading

error term for each.

2. Find the modi�ed wavenumber for the �rst-order backward di�erence operator.

Plot the real and imaginary parts of ��x vs. ��x for 0 � ��x � �. Using

Fourier analysis as in Section 6.6.2, �nd j�j for the combination of this spatial

operator with 4th-order Runge-Kutta time marching at a Courant number of

unity and plot vs. ��x for 0 � ��x � �.

3. Find the modi�ed wavenumber for the operator given in Eq. 11.6. Plot the real

and imaginary parts of ��x vs. ��x for 0 � ��x � �. Using Fourier analysisas in Section 6.6.2, �nd j�j for the combination of this spatial operator with

4th-order Runge-Kutta time marching at a Courant number of unity and plot

vs. ��x for 0 � ��x � �.

4. Consider the spatial operator obtained by combining second-order centered dif-

ferences with the symmetric operator given in Eq. 11.44. Find the modi�ed

wavenumber for this operator with � = 0; 1=12; 1=24, and 1=48. Plot the real

and imaginary parts of ��x vs. ��x for 0 � ��x � �. Using Fourier analysis

as in Section 6.6.2, �nd j�j for the combination of this spatial operator with

4th-order Runge-Kutta time marching at a Courant number of unity and plot

vs. ��x for 0 � ��x � �.

5. Consider the hyperbolic system derived in problem 8 of Chapter 2. Find the

matrix jAj. Form the plus-minus split ux vectors as in Section 11.4.1.

6. Show that the ux Jacobian for the 1-D Euler equations can be written in terms

of u and H. Show that the use of the Roe average state given in Eqs. 11.38 and

11.39 leads to satisfaction of Eq. 11.37.

Chapter 12

SPLIT AND FACTORED FORMS

In the next two chapters, we present and analyze split and factored algorithms. This

gives the reader a feel for some of the modi�cations which can be made to the basic

algorithms in order to obtain e�cient solvers for practical multidimensional applica-

tions, and a means for analyzing such modi�ed forms.

12.1 The Concept

Factored forms of numerical operators are used extensively in constructing and ap-

plying numerical methods to problems in uid mechanics. They are the basis for a

wide variety of methods variously known by the labels \hybrid", \time split", and

\fractional step". Factored forms are especially useful for the derivation of practical

algorithms that use implicit methods. When we approach numerical analysis in the

light of matrix derivative operators, the concept of factoring is quite simple to present

and grasp. Let us start with the following observations:

1. Matrices can be split in quite arbitrary ways.

2. Advancing to the next time level always requires some reference to a previous

one.

3. Time marching methods are valid only to some order of accuracy in the step

size, h.

Now recall the generic ODE's produced by the semi-discrete approach

d~u

dt= A~u� ~f (12.1)

217

218 CHAPTER 12. SPLIT AND FACTORED FORMS

and consider the above observations. From observation 1 (arbitrary splitting of A) :

d~u

dt= [A1 +A2]~u�

~f (12.2)

where A = [A1 +A2] but A1 and A2 are not unique. For the time march let us choose

the simple, �rst-order,1 explicit Euler method. Then, from observation 2 (new data~un+1 in terms of old ~un):

~un+1 = [ I + hA1 + hA2]~un � h~f +O(h2) (12.3)

or its equivalent

~un+1 =h[ I + hA1][ I + hA2]� h2A1A2

i~un � h~f +O(h2)

Finally, from observation 3 (allowing us to drop higher order terms �h2A1A2~un):

~un+1 = [ I + hA1][ I + hA2]~un � h~f +O(h2) (12.4)

Notice that Eqs. 12.3 and 12.4 have the same formal order of accuracy and, in

this sense, neither one is to be preferred over the other. However, their numerical

stability can be quite di�erent, and techniques to carry out their numerical evaluation

can have arithmetic operation counts that vary by orders of magnitude. Both of these

considerations are investigated later. Here we seek only to apply to some simple cases

the concept of factoring.

12.2 Factoring Physical Representations | Time

Splitting

Suppose we have a PDE that represents both the processes of convection and dissi-

pation. The semi-discrete approach to its solution might be put in the form

d~u

dt= Ac

~u+Ad~u+ ~(bc) (12.5)

where Ac and Ad are matrices representing the convection and dissipation terms,

respectively; and their sum forms the A matrix we have considered in the previous

sections. Choose again the explicit Euler time march so that

~un+1 = [ I + hAd + hAc]~un + h ~(bc) +O(h2) (12.6)

1Second-order time-marching methods are considered later.

12.2. FACTORING PHYSICAL REPRESENTATIONS | TIME SPLITTING 219

Now consider the factored form

~un+1 = [ I + hAd]�[ I + hAc]~un + h ~(bc)

�

= [ I + hAd + hAc]~un + h ~(bc)| {z }Original Unfactored Terms

+ h2Ad

�Ac~un +

~(bc)�

| {z }Higher Order Terms

+O(h2) (12.7)

and we see that Eq. 12.7 and the original unfactored form Eq. 12.6 have identical

orders of accuracy in the time approximation. Therefore, on this basis, their selection

is arbitrary. In practical applications2 equations such as 12.7 are often applied in a

predictor-corrector sequence. In this case one could write

~un+1 = [ I + hAc]~un + h ~(bc)

~un+1 = [ I + hAd]~un+1 (12.8)

Factoring can also be useful to form split combinations of implicit and explicit

techniques. For example, another way to approximate Eq. 12.6 with the same order

of accuracy is given by the expression

~un+1 = [ I � hAd]�1

�[ I + hAc]~un + h ~(bc)

�

= [ I + hAd + hAc]~un + h ~(bc)| {z }Original Unfactored Terms

+O(h2) (12.9)

where in this approximation we have used the fact that

[ I � hAd]�1

= I + hAd + h2A2d+ � � �

if h � jjAdjj < 1, where jjAdjj is some norm of [Ad]. This time a predictor-corrector

interpretation leads to the sequence

~un+1 = [ I + hAc]~un + h ~(bc)

[ I � hAd]~un+1 = ~un+1 (12.10)

The convection operator is applied explicitly, as before, but the di�usion operator is

now implicit, requiring a tridiagonal solver if the di�usion term is central di�erenced.

Since numerical sti�ness is generally much more severe for the di�usion process, this

factored form would appear to be superior to that provided by Eq. 12.8. However,

the important aspect of stability has yet to be discussed.

2We do not suggest that this particular method is suitable for use. We have yet to determine its

stability, and a �rst-order time-march method is usually unsatisfactory.


We should mention here that Eq. 12.9 can be derived for a di�erent point of view

by writing Eq. 12.6 in the form

un+1 � un

h= Acun +Adun+1 +

~(bc) +O(h2)

Then

[ I � hAd]un+1 = [ I + hAc]un + h ~(bc)

which is identical to Eq. 12.10.

12.3 Factoring Space Matrix Operators in 2{D

12.3.1 Mesh Indexing Convention

Factoring is widely used in codes designed for the numerical solution of equations

governing unsteady two- and three-dimensional ows. Let us study the basic concept

of factoring by inspecting its use on the linear 2-D scalar PDE that models di�usion:

@u

@t=@2u

@x2+@2u

@y2(12.11)

We begin by reducing this PDE to a coupled set of ODE's by di�erencing the space

derivatives and inspecting the resulting matrix operator.

A clear description of a matrix �nite-di�erence operator in 2- and 3-D requires some

reference to a mesh. We choose the 3 � 4 point mesh3 shown in the Sketch 12.12.

In this example Mx, the number of (interior) x points, is 4 and My, the number of

(interior) y points is 3. The numbers 11 ; 12 ; � � � ; 43 represent the location in the

mesh of the dependent variable bearing that index. Thus u32 represents the value of

u at j = 3 and k = 2.

� � � �

My � 13 23 33 43 �

k � 12 22 32 42 �

1 � 11 21 31 41 �

� � � �

1 j � � � Mx

Mesh indexing in 2-D.

(12.12)

3This could also be called a 5 � 6 point mesh if the boundary points (labeled � in the sketch)

were included, but in these notes we describe the size of a mesh by the number of interior points.

12.3. FACTORING SPACE MATRIX OPERATORS IN 2{D 221

12.3.2 Data Bases and Space Vectors

The dimensioned array in a computer code that allots the storage locations of the

dependent variable(s) is referred to as a data-base. There are many ways to lay out

a data-base. Of these, we consider only two: (1), consecutively along rows that are

themselves consecutive from k = 1 to My, and (2), consecutively along columns that

are consecutive from j = 1 to Mx. We refer to each row or column group as a

space vector (they represent data along lines that are continuous in space) and label

their sum with the symbol U . In particular, (1) and (2) above are referred to as x-

vectors and y-vectors, respectively. The symbol U by itself is not enough to identify

the structure of the data-base and is used only when the structure is immaterial or

understood.

To be speci�c about the structure, we label a data{base composed of x-vectors with

U (x) , and one composed of y-vectors with U (y). Examples of the order of indexing

for these space vectors are given in Eq. 12.16 part a and b.

12.3.3 Data Base Permutations

The two vectors (arrays) are related by a permutation matrix P such that

U (x) = PxyU(y) and U (y) = PyxU

(x) (12.13)

where

Pyx = PT

xy= P�1

xy

Now consider the structure of a matrix �nite-di�erence operator representing 3-

point central-di�erencing schemes for both space derivatives in two dimensions. When

the matrix is multiplying a space vector U , the usual (but ambiguous) representation

is given by Ax+y. In this notation the ODE form of Eq. 12.11 can be written 4

dU

dt= Ax+yU + ~(bc) (12.14)

If it is important to be speci�c about the data-base structure, we use the notation

A(x)x+y or A

(y)x+y, depending on the data{base chosen for the U it multiplies. Examples

are in Eq. 12.16 part a and b. Notice that the matrices are not the same although

they represent the same derivative operation. Their structures are similar, however,

and they are related by the same permutation matrix that relates U (x) to U (y). Thus

A(x)x+y = Pxy � A

(y)x+y � Pyx (12.15)

4Notice that Ax+y and U , which are notations used in the special case of space vectors, are

subsets of A and ~u, used in the previous sections.


A(x)x+y � U

(x) =

26666666666666666666666666664

� x j o j

x � x j o j

x � x j o j

x � j o j

o j � x j o

o j x � x j o

o j x � x j o

o j x � j o

j o j � x

j o j x � x

j o j x � x

j o j x �

37777777777777777777777777775

�

11

21

31

41

��

12

22

32

42

��

13

23

33

43

a:Elements in 2-dimensional, central-di�erence, matrix

operator, Ax+y, for 3�4 mesh shown in Sketch 12.12.

Data base composed of My x{vectors stored in U (x).

Entries for x! x, for y ! o, for both ! �.

A(y)x+y � U

(y) =

2666666666666666666666666666664

� o j x j j

o � o j x j j

o � j x j j

x j � o j x j

x j o � o j x j

x j o � j x j

j x j � o j x

j x j o � o j x

j x j o � j x

j j x j � o

j j x j o � o

j j x j o �

3777777777777777777777777777775

�

11

12

13

��

21

22

23

��

31

32

33

��

41

42

43

b: Elements in 2-dimensional, central-di�erence, matrix

operator, Ax+y, for 3�4 mesh shown in Sketch 12.12.

Data base composed of Mx y{vectors stored in U (y).

Entries for x! x, for y ! o, for both ! �.

(12.16)


12.3.4 Space Splitting and Factoring

We are now prepared to discuss splitting in two dimensions. It should be clear that

the matrix A(x)x+y can be split into two matrices such that

A(x)x+y = A(x)

x+ A(x)

y(12.17)

where A(x)x

and A(x)y

are shown in Eq. 12.22. Similarily

A(y)x+y = A(y)

x+ A(y)

y(12.18)

where the split matrices are shown in Eq. 12.23.

The permutation relation also holds for the split matrices so

A(x)y

= PxyA(y)y

Pyx

and

A(x)x

= PxyA(y)x

Pyx

The splittings in Eqs. 12.17 and 12.18 can be combined with factoring in the

manner described in Section 12.2. As an example (�rst-order in time), applying the

implicit Euler method to Eq. 12.14 gives

U(x)n+1 = U (x)

n+ h

hA(x)

x+ A(x)

y

iU

(x)n+1 + h ~(bc)

or

hI � hA(x)

x� hA(x)

y

iU

(x)n+1 = U (x)

n+ h ~(bc) +O(h2) (12.19)

As in Section 12.2, we retain the same �rst order accuracy with the alternative

hI � hA(x)

x

ihI � hA(x)

y

iU

(x)n+1 = U (x)

n+ h ~(bc) +O(h2) (12.20)

Write this in predictor-corrector form and permute the data base of the second row.

There results

hI � hA(x)

x

i~U (x) = U (x)

n+ h ~(bc)h

I � hA(y)y

iU

(y)n+1 = ~U (y) (12.21)


A(x)x� U (x) =

26666666666666666666666666664

x x j j

x x x j j

x x x j j

x x j j

j x x j

j x x x j

j x x x j

j x x j

j j x x

j j x x x

j j x x x

j j x x

37777777777777777777777777775

� U (x)

A(x)y� U (x) =

26666666666666666666666666664

o j o j

o j o j

o j o j

o j o j

o j o j o

o j o j o

o j o j o

o j o j o

j o j o

j o j o

j o j o

j o j o

37777777777777777777777777775

� U (x)

The splitting of A(x)x+y.

(12.22)


A(y)x� U (y) =

2666666666666666666666666666664

x j x j j

x j x j j

x j x j j

x j x j x j

x j x j x j

x j x j x j

j x j x j x

j x j x j x

j x j x j x

j j x j x

j j x j x

j j x j x

3777777777777777777777777777775

� U (y)

A(y)y� U (y) =

2666666666666666666666666666664

o o j j j

o o o j j j

o o j j j

j o o j j

j o o o j j

j o o j j

j j o o j

j j o o o j

j j o o j

j j j o o

j j j o o o

j j j o o

3777777777777777777777777777775

� U (y)

The splitting of A(y)x+y.

(12.23)


12.4 Second-Order Factored Implicit Methods

Second-order accuracy in time can be maintained in a certain factored implicit meth-

ods. For example, apply the trapezoidal method to Eq. 12.14 where the derivative

operators have been split as in Eq. 12.17 or 12.18. Let the data base be immaterial

and the ~(bc) be time invariant. There results

�I �

1

2hAx �

1

2hAy

�Un+1 =

�I +

1

2hAx +

1

2hAy

�Un + h ~(bc) +O(h3) (12.24)

Factor both sides giving� �

I �1

2hAx

��I �

1

2hAy

��

1

4h2AxAy

�Un+1

=

� �I +

1

2hAx

��I +

1

2hAy

��

1

4h2AxAy

�Un + h ~(bc) +O(h3) (12.25)

Then notice that the combination 14h

2[AxAy](Un+1 � Un) is proportional to h3 since

the leading term in the expansion of (Un+1 � Un) is proportional to h. Therefore, we

can write�I �

1

2hAx

��I �

1

2hAy

�Un+1 =

�I +

1

2hAx

��I +

1

2hAy

�Un + h ~(bc) +O(h3)(12.26)

and both the factored and unfactored form of the trapezoidal method are second-order

accurate in the time march.

An alternative form of this kind of factorization is the classical ADI (alternating

direction implicit) method5 usually written

�I �

1

2hAx

�~U =

�I +

1

2hAy

�Un +

1

2hFn

�I �

1

2hAy

�Un+1 =

�I +

1

2hAx

�~U +

1

2hFn+1 +O(h3) (12.27)

For idealized commuting systems the methods given by Eqs. 12.26 and 12.27 di�er

only in their evaluation of a time-dependent forcing term.

12.5 Importance of Factored Forms in 2 and 3 Di-

mensions

When the time-march equations are sti� and implicit methods are required to permit

reasonably large time steps, the use of factored forms becomes a very valuable tool

5A form of the Douglas or Peaceman-Rachford methods.

12.5. IMPORTANCE OF FACTORED FORMS IN 2 AND 3 DIMENSIONS 227

for realistic problems. Consider, for example, the problem of computing the time

advance in the unfactored form of the trapezoidal method given by Eq. 12.24�I �

1

2hAx+y

�Un+1 =

�I +

1

2hAx+y

�Un + h ~(bc)

Forming the right hand side poses no problem, but �nding Un+1 requires the solution

of a sparse, but very large, set of coupled simultaneous equations having the matrix

form shown in Eq. 12.16 part a and b. Furthermore, in real cases involving the Euler

or Navier-Stokes equations, each symbol (o; x; �) represents a 4 � 4 block matrix with

entries that depend on the pressure, density and velocity �eld. Suppose we were to

solve the equations directly. The forward sweep of a simple Gaussian elimination �lls6

all of the 4 � 4 blocks between the main and outermost diagonal7 (e.g. between �

and o in Eq. 12.16 part b.). This must be stored in computer memory to be used to

�nd the �nal solution in the backward sweep. If Ne represents the order of the small

block matrix (4 in the 2-D Euler case), the approximate memory requirement is

(Ne �My) � (Ne �My) �Mx

oating point words. Here it is assumed that My < Mx. If My > Mx, My and Mx

would be interchanged. A moderate mesh of 60 � 200 points would require over 11

million words to �nd the solution. Actually current computer power is able to cope

rather easily with storage requirements of this order of magnitude. With computing

speeds of over one giga op,8 direct solvers may become useful for �nding steady-state

solutions of practical problems in two dimensions. However, a three-dimensional

solver would require a memory of approximatly

N2e�M2

y�M2

z�Mx

words and, for well resolved ow �elds, this probably exceeds memory availability for

some time to come.

On the other hand, consider computing a solution using the factored implicit equa-

tion 12.25. Again computing the right hand side poses no problem. Accumulate the

result of such a computation in the array (RHS). One can then write the remaining

terms in the two-step predictor-corrector form�I �

1

2hA(x)

x

�~U (x) = (RHS)(x)

�I �

1

2hA(y)

y

�U

(y)n+1 = ~U (y) (12.28)

6For matrices as small as those shown there are many gaps in this \�ll", but for meshes of

practical size the �ll is mostly dense.7The lower band is also computed but does not have to be saved unless the solution is to be

repeated for another vector.8One billion oating-point operations per second.


which has the same appearance as Eq. 12.21 but is second-order time accurate. The

�rst step would be solved using My uncoupled block tridiagonal solvers9. Inspecting

the top of Eq. 12.22, we see that this is equivalent to solving My one-dimensional

problems, each with Mx blocks of order Ne. The temporary solution ~U (x) would then

be permuted to ~U (y) and an inspection of the bottom of Eq. 12.23 shows that the �nal

step consists of solving Mx one-dimensional implicit problems each with dimension

My.

12.6 The Delta Form

Clearly many ways can be devised to split the matrices and generate factored forms.

One way that is especially useful, for ensuring a correct steady-state solution in a

converged time-march, is referred to as the \delta form" and we develop it next.

Consider the unfactored form of the trapezoidal method given by Eq. 12.24, and

let the ~(bc) be time invariant:

�I �

1

2hAx �

1

2hAy

�Un+1 =

�I +

1

2hAx +

1

2hAy

�Un + h ~(bc) +O(h3)

From both sides subtract �I �

1

2hAx �

1

2hAy

�Un

leaving the equality unchanged. Then, using the standard de�nition of the di�erence

operator �,

�Un = Un+1 � Un

one �nds

�I �

1

2hAx �

1

2hAy

��Un = h

hAx+yUn +

~(bc)i+O(h3) (12.29)

Notice that the right side of this equation is the product of h and a term that is

identical to the right side of Eq. 12.14, our original ODE. Thus, if Eq. 12.29 converges,

it is guaranteed to converge to the correct steady-state solution of the ODE. Now we

can factor Eq. 12.29 and maintain O(h2) accuracy. We arrive at the expression

�I �

1

2hAx

��I �

1

2hAy

��Un = h

hAx+yUn +

~(bc)i+O(h3) (12.30)

This is the delta form of a factored, 2nd-order, 2-D equation.

9A block tridiagonal solver is similar to a scalar solver except that small block matrix operations

replace the scalar ones, and matrix multiplications do not commute.

12.7. PROBLEMS 229

The point at which the factoring is made may not a�ect the order of time-accuracy,

but it can have a profound e�ect on the stability and convergence properties of a

method. For example, the unfactored form of a �rst-order method derived from the

implicit Euler time march is given by Eq. 12.19, and if it is immediately factored,

the factored form is presented in Eq. 12.20. On the other hand, the delta form of the

unfactored Eq. 12.19 is

[I � hAx � hAy]�Un = hhAx+yUn +

~(bc)i

and its factored form becomes10

[I � hAx][I � hAy]�Un = hhAx+yUn + ~(bc)

i(12.31)

In spite of the similarities in derivation, we will see in the next chapter that the

convergence properties of Eq. 12.20 and Eq. 12.31 are vastly di�erent.

12.7 Problems

1. Consider the 1-D heat equation:

@u

@t= �

@2u

@x20 � x � 9

Let u(0; t) = 0 and u(9; t) = 0, so that we can simplify the boundary conditions.

Assume that second order central di�erencing is used, i.e.,

(�xxu)j =1

�x2(uj�1 � 2uj + uj+1)

The uniform grid has �x = 1 and 8 interior points.

(a) Space vector de�nition

i. What is the space vector for the natural ordering (monotonically in-

creasing in index), u(1) ? Only include the interior points.

ii. If we reorder the points with the odd points �rst and then the even

points, write the space vector, u(2) ?

iii. Write down the permutation matrices,(P12,P21).

10Notice that the only di�erence between the O(h2) method given by Eq. 12.30 and the O(h)

method given by Eq. 12.31 is the appearance of the factor 12 on the left side of the O(h2) method.


iv. The generic ODE representing the discrete form of the heat equation

is

du(1)

dt= A1u

(1) + f

Write down the matrix A1. (Note f = 0, due to the boundary condi-

tions) Next �nd the matrix A2 such that

du(2)

dt= A2u

(2)

Note that A2 can be written as

A2 =

264D UT

U D

375

De�ne D and U .

v. Applying implicit Euler time marching, write the delta form of the

implicit algorithm. Comment on the form of the resulting implicit

matrix operator.

(b) System de�nition

In problem 1a, we de�ned u(1); u(2); A1; A2; P12, and P21 which partition

the odd points from the even points. We can put such a partitioning to

use. First de�ne extraction operators

I(o) =

26666666666664

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

37777777777775

=

264I4 04

04 04

375

I(e) =

26666666666664

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

37777777777775

=

26404 04

04 I4

375

12.7. PROBLEMS 231

which extract the odd even points from u(2) as follows: u(o) = I(o)u(2) and

u(e) = I(e)u(2).

i. Beginning with the ODE written in terms of u(2), de�ne a splitting

A2 = Ao + Ae, such that Ao operates only on the odd terms, and Ae

operates only on the even terms. Write out the matrices Ao and Ae.

Also, write them in terms of D and U de�ned above.

ii. Apply implicit Euler time marching to the split ODE. Write down the

delta form of the algorithm and the factored delta form. Comment on

the order of the error terms.

iii. Examine the implicit operators for the factored delta form. Comment

on their form. You should be able to argue that these are now trangu-

lar matrices (a lower and an upper). Comment on the solution process

this gives us relative to the direct inversion of the original system.

Chapter 13

LINEAR ANALYSIS OF SPLIT

AND FACTORED FORMS

In Section 4.4 we introduced the concept of the representative equation, and used

it in Chapter 7 to study the stability, accuracy, and convergence properties of time-

marching schemes. The question is: Can we �nd a similar equation that will allow

us to evaluate the stability and convergence properties of split and factored schemes?

The answer is yes | for certain forms of linear model equations.

The analysis in this chapter is useful for estimating the stability and steady-state

properties of a wide variety of time-marching schemes that are variously referred

to as time-split, fractional-step, hybrid, and (approximately) factored. When these

methods are applied to practical problems, the results found from this analysis are

neither necessary nor su�cient to guarantee stability. However, if the results indicate

that a method has an instability, the method is probably not suitable for practical

use.

13.1 The Representative Equation for Circulant

Operators

Consider linear PDE's with coe�cients that are �xed in both space and time and with

boundary conditions that are periodic. We have seen that under these conditions

a semi-discrete approach can lead to circulant matrix di�erence operators, and we

discussed circulant eigensystems1 in Section 4.3. In this and the following section

we assume circulant systems and our analysis depends critically on the fact that all

circulant matrices commute and have a common set of eigenvectors.

1See also the discussion on Fourier stability analysis in Section 7.7.

233

234 CHAPTER 13. LINEAR ANALYSIS OF SPLIT AND FACTORED FORMS

Suppose, as a result of space di�erencing the PDE, we arrive at a set of ODE's

that can be written

d~u

dt= A

ap~u+A

bp~u� ~f(t) (13.1)

where the subscript p denotes a circulant matrix. Since both matrices have the same

set of eigenvectors, we can use the arguments made in Section 4.2.3 to uncouple the

set and form the M set of independent equations

w0

1 = (�a + �b)1w1 � g1(t)

...

w0

m= (�a + �b)mwm � gm(t)

...

w0

M= (�a + �b)MwM � gM(t) (13.2)

The analytic solution of the m'th line is

wm(t) = cme(�a+�b)mt + P:S:

Note that each �a pairs with one, and only one2, �b since they must share a common

eigenvector. This suggests (see Section 4.4:

The representative equation for split, circulant systems is

du

dt= [�a + �b + �c + � � �]u+ ae�t (13.3)

where �a + �b + �c + � � � are the sum of the eigenvalues in Aa; A

b; A

c; � � � that

share the same eigenvector.

13.2 Example Analysis of Circulant Systems

13.2.1 Stability Comparisons of Time-Split Methods

Consider as an example the linear convection-di�usion equation:

@u

@t+ a

@u

@x= �

@2u

@x2(13.4)

2This is to be contrasted to the developments found later in the analysis of 2-D equations.

13.2. EXAMPLE ANALYSIS OF CIRCULANT SYSTEMS 235

If the space di�erencing takes the form

d~u

dt= �

a

2�xBp(�1; 0; 1)~u+

�

�x2Bp(1;�2; 1)~u (13.5)

the convection matrix operator and the di�usion matrix operator, can be represented

by the eigenvalues �c and �d, respectively, where (see Section 4.3.2):

(�c)m =ia

�xsin �m

(�d)m = �

4�

�x2sin2

�m

2(13.6)

In these equations �m = 2m�=M , m = 0 ; 1 ; � � � ; M � 1 , so that 0 � �m � 2�.

Using these values and the representative equation 13.4, we can analyze the stability

of the two forms of simple time-splitting discussed in Section 12.2. In this section we

refer to these as

1. the explicit-implicit Euler method, Eq. 12.10.

2. the explicit-explicit Euler method, Eq. 12.8.

1. The Explicit-Implicit Method

When applied to Eq. 13.4, the characteristic polynomial of this method is

P (E) = (1� h�d)E � (1 + h�c)

This leads to the principal � root

� =1 + i

ah

�xsin �m

1 + 4h�

�x2sin2

�m

2

where we have made use of Eq. 13.6 to quantify the eigenvalues. Now introduce the

dimensionless numbers

Cn =ah

�x; Courant number

R� =a�x

�; mesh Reynolds number

and we can write for the absolute value of �

j�j =

q1 + C2

nsin2 �m

1 + 4Cn

R�

sin2�m

2

; 0 � �m � 2� (13.7)


0 4 8

R∆

C n

0

0.4

0.8

1.2

C n R∆= 2/

Explicit-Implicit

0 4 8

R∆

C n

0

0.4

0.8

1.2

Explicit-Explicit

C n

R∆= 2/C n

R∆= /2

Figure 13.1: _Stability regions for two simple time-split methods.

A simple numerical parametric study of Eq. 13.7 shows that the critical range of

�m for any combination of Cn and R� occurs when �m is near 0 (or 2�). From this

we �nd that the condition on Cn and R� that make j�j � 1 is

h1 + C2

nsin2 �

i=

�1 + 4

Cn

R�

sin2�

2

�2

As �! 0 this gives the stability region

Cn <2

R�

which is bounded by a hyperbola and shown in Fig. 13.1.

2. The Explicit-Explicit Method

An analysis similar to the one given above shows that this method produces

j�j =q1 + C2

nsin2 �m

"1� 4

Cn

R�

sin2�m

2

#; 0 � �m � 2�

Again a simple numerical parametric study shows that this has two critical ranges

of �m, one near 0, which yields the same result as in the previous example, and the

13.2. EXAMPLE ANALYSIS OF CIRCULANT SYSTEMS 237

other near 180o, which produces the constraint that

Cn <1

2R� for R� � 2

The resulting stability boundary is also shown in Fig. 13.1. The totaly explicit,

factored method has a much smaller region of stability when R� is small, as we

should have expected.

13.2.2 Analysis of a Second-Order Time-Split Method

Next let us analyze a more practical method that has been used in serious compu-

tational analysis of turbulent ows. This method applies to a ow in which there is

a combination of di�usion and periodic convection. The convection term is treated

explicitly using the second-order Adams-Bashforth method. The di�usion term is

integrated implicitly using the trapezoidal method. Our model equation is again the

linear convection-di�usion equation 13.4 which we split in the fashion of Eq. 13.5. In

order to evaluate the accuracy, as well as the stability, we include the forcing func-

tion in the representative equation and study the e�ect of our hybrid, time-marching

method on the equation

u0 = �cu+ �du+ ae�t

First let us �nd expressions for the two polynomials, P (E) and Q(E). The char-

acteristic polynomial follows from the application of the method to the homogeneous

equation, thus

un+1 = un +1

2h�c(3un � un�1) +

1

2h�d(un+1 + un)

This produces

P (E) = (1�1

2h�d)E

2� (1 +

3

2h�c +

1

2h�d)E +

1

2h�c

The form of the particular polynomial depends upon whether the forcing function is

carried by the AB2 method or by the trapezoidal method. In the former case it is

Q(E) =1

2h(3E � 1) (13.8)

and in the latter

Q(E) =1

2h(E2 + E) (13.9)


Accuracy

From the characteristic polynomial we see that there are two �{roots and they are

given by the equation

� =1 +

3

2h�c +

1

2h�d �

s�1 +

3

2h�c +

1

2h�d

�2

� 2h�c

�1�

1

2h�d

�

2

�1�

1

2h�d

� (13.10)

The principal �-root follows from the plus sign and one can show

�1 = 1 + (�c + �d)h+1

2(�c + �d)

2h2 +1

4

��3d+ �c�

2d� �2

c�d � �3

c

�h3

From this equation it is clear that 16�

3 = 16(�c + �d)

3 does not match the coe�cient

of h3 in �1, so

er� = O(h3)

Using P (e�h) and Q(e�h) to evaluate er� in Section 6.6.3, one can show

er� = O(h3)

using either Eq. 13.8 or Eq. 13.9. These results show that, for the model equation,

the hybrid method retains the second-order accuracy of its individual components.

Stability

The stability of the method can be found from Eq. 13.10 by a parametric study of cnand R� de�ned in Eq. 13.7. This was carried out in a manner similar to that used

to �nd the stability boundary of the �rst-order explicit-implicit method in Section

13.2.1. The results are plotted in Fig. 13.2. For values of R� � 2 this second-order

method has a much greater region of stability than the �rst-order explicit-implicit

method given by Eq. 12.10 and shown in Fig. 13.1.

13.3 The Representative Equation for Space-Split

Operators

Consider the 2-D model3 equations

@u

@t=

@2u

@x2+@2u

@y2(13.11)

3The extension of the following to 3-D is simple and straightforward.

13.3. THE REPRESENTATIVE EQUATION FOR SPACE-SPLIT OPERATORS239

R∆

0 4 8

0.4

0.8

1.2

Cn

Stable

Figure 13.2: _Stability regions for the second-order time-split method.

and

@u

@t+ ax

@u

@x+ ay

@u

@y= 0 (13.12)

Reduce either of these, by means of spatial di�erencing approximations, to the coupled

set of ODE's:

dU

dt= [Ax + Ay]U + ~(bc) (13.13)

for the space vector U . The form of the Ax and Ay matrices for three-point central

di�erencing schemes are shown in Eqs. 12.22 and 12.23 for the 3 � 4 mesh shown

in Sketch 12.12. Let us inspect the structure of these matrices closely to see how we

can diagonalize [Ax + Ay] in terms of the individual eigenvalues of the two matrices

considered separately.

First we write these matrices in the form

A(x)x

=

264B B

B

375 A(x)

y=

264~b0 � I ~b1 � I~b�1 � I ~b0 � I ~b1 � I

~b�1 � I ~b0 � I

375

where B is a banded matrix of the form B(b�1; b0; b1). Now �nd the block eigenvector


matrix that diagonalizes B and use it to diagonalize A(x)x. Thus2

64X�1

X�1

X�1

375264B B

B

375264X X

X

375 =

264� �

�

375

where

� =

26664�1

�2�3

�4

37775

Notice that the matrix A(x)y

is transparent to this transformation. That is, if we

set X � diag(X)

X�1

264~b0 � I ~b1 � I~b�1 � I ~b0 � I ~b1 � I

~b�1 � I ~b0 � I

375X =

264~b0 � I ~b1 � I~b�1 � I ~b0 � I ~b1 � I

~b�1 � I ~b0 � I

375

One now permutes the transformed system to the y-vector data-base using the per-

mutation matrix de�ned by Eq. 12.13. There results

Pyx �X�1hA(x)x

+ A(x)y

iX � Pxy =2

6664�1 � I

�2 � I

�3 � I

�4 � I

37775+

266664

~B~B

~B~B

377775 (13.14)

where ~B is the banded tridiagonal matrix B(~b�1;~b0;~b1), see the bottom of Eq. 12.23.

Next �nd the eigenvectors ~X that diagonalize the ~B blocks. Let ~B � diag( ~B) and~X � diag( ~X) and form the second transformation

~X�1 ~B ~X =

26664~�

~�~�

~�

37775 ; ~� =

264~�1

~�2~�3

375

This time, by the same argument as before, the �rst matrix on the right side of

Eq. 13.14 is transparent to the transformation, so the �nal result is the complete

diagonalization of the matrix Ax+y:h~X�1

� Pyx �X�1ihA(x)x+y

ihX � Pxy � ~X

i=2

66664�1I + ~�

�2I + ~�

�3I + ~�

�4I + ~�

377775 (13.15)

13.3. THE REPRESENTATIVE EQUATION FOR SPACE-SPLIT OPERATORS241

It is important to notice that:

� The diagonal matrix on the right side of Eq. 13.15 contains every possible com-

bination of the individual eigenvalues of B and ~B.

Now we are ready to present the representative equation for two dimensional sys-

tems. First reduce the PDE to ODE by some choice4 of space di�erencing. This

results in a spatially split A matrix formed from the subsets

A(x)x

= diag(B) ; A(y)y

= diag( ~B) (13.16)

where B and ~B are any two matrices that have linearly independent eigenvectors (this

puts some constraints on the choice of di�erencing schemes).

Although Ax and Ay do commute, this fact, by itself, does not ensure the prop-

erty of \all possible combinations". To obtain the latter property the structure of

the matrices is important. The block matrices B and ~B can be either circulant or

noncirculant; in both cases we are led to the �nal result:

The 2{D representative equation for model linear systems is

du

dt= [�x + �y]u+ ae�t

where �x and �y are any combination of eigenvalues from Ax and Ay, a and � are

(possibly complex) constants, and where Ax and Ay satisfy the conditions in 13.16.

Often we are interested in �nding the value of, and the convergence rate to, the

steady-state solution of the representative equation. In that case we set � = 0 and

use the simpler form

du

dt= [�x + �y]u+ a (13.17)

which has the exact solution

u(t) = ce(�x+�y)t �a

�x + �y(13.18)

4We have used 3-point central di�erencing in our example, but this choice was for convenience

only, and its use is not necessary to arrive at Eq. 13.15.


13.4 Example Analysis of 2-D Model Equations

In the following we analyze four di�erent methods for �nding a �xed, steady-state

solution to the 2-D representative equation 13.16. In each case we examine

1. The stability.

2. The accuracy of the �xed, steady-state solution.

3. The convergence rate to reach the steady-state.

13.4.1 The Unfactored Implicit Euler Method

Consider �rst this unfactored, �rst-order scheme which can then be used as a reference

case for comparison with the various factored ones. The form of the method is given

by Eq. 12.19, and when it is applied to the representative equation, we �nd

(1� h �x � h �y)un+1 = un + ha

from which

P (E) = (1� h �x � h �y)E � 1

Q(E) = h (13.19)

giving the solution

un = c

"1

1� h �x � h �y

#n

�

a

�x + �y

Like its counterpart in the 1-D case, this method:

1. Is unconditionally stable.

2. Produces the exact (see Eq. 13.18) steady-state solution (of the ODE) for any

h.

3. Converges very rapidly to the steady-state when h is large.

Unfortunately, however, use of this method for 2-D problems is generally impractical

for reasons discussed in Section 12.5.

13.4. EXAMPLE ANALYSIS OF 2-D MODEL EQUATIONS 243

13.4.2 The Factored Nondelta Form of the Implicit Euler

Method

Now apply the factored Euler method given by Eq. 12.20 to the 2-D representative

equation. There results

(1� h �x)(1� h �y)un+1 = un + ha

from which

P (E) = (1� h �x)(1� h �y)E � 1

Q(E) = h (13.20)

giving the solution

un = c

"1

(1� h �x)(1� h �y)

#n

�

a

�x + �y � h�x�y

We see that this method:


2. Produces a steady state solution that depends on the choice of h.

3. Converges rapidly to a steady-state for large h, but the converged solution is

completely wrong.

The method requires far less storage then the unfactored form. However, it is not

very useful since its transient solution is only �rst-order accurate and, if one tries to

take advantage of its rapid convergence rate, the converged value is meaningless.

13.4.3 The Factored Delta Form of the Implicit Euler Method

Next apply Eq. 12.31 to the 2-D representative equation. One �nds

(1� h �x)(1� h �y)(un+1 � un) = h(�xun + �yun + a)

which reduces to

(1� h �x)(1� h �y)un+1 =�1 + h2�x�y

�un + ha

and this has the solution

un = c

"1 + h2�x�y

(1� h �x)(1� h �y)

#n�

a

�x + �y

This method:



2. Produces the exact steady-state solution for any choice of h.

3. Converges very slowly to the steady{state solution for large values of h, since

j�j ! 1 as h!1.

Like the factored nondelta form, this method demands far less storage than the un-

factored form, as discussed in Section 12.5. The correct steady solution is obtained,

but convergence is not nearly as rapid as that of the unfactored form.

13.4.4 The Factored Delta Form of the Trapezoidal Method

Finally consider the delta form of a second-order time-accurate method. Apply Eq.

12.30 to the representative equation and one �nds�1�

1

2h�x

��1�

1

2h�y

�(un+1 � un) = h(�xun + �yun + a)

which reduces to�1�

1

2h�x

��1�

1

2h�y

�un+1 =

�1 +

1

2h�x

��1 +

1

2h�y

�un + ha

and this has the solution

un = c

2664�1 +

1

2h�x

��1 +

1

2h�y

��1�

1

2h�x

��1�

1

2h�y

�3775n

�

a

�x + �y

This method:


2. Produces the exact steady{state solution for any choice of h.

3. Converges very slowly to the steady{state solution for large values of h, since

j�j ! 1 as h!1.

All of these properties are identical to those found for the factored delta form of the

implicit Euler method. Since it is second order in time, it can be used when time

accuracy is desired, and the factored delta form of the implicit Euler method can be

used when a converged steady-state is all that is required.5 A brief inspection of eqs.

12.26 and 12.27 should be enough to convince the reader that the �'s produced by

those methods are identical to the � produced by this method.

5In practical codes, the value of h on the left side of the implicit equation is literally switched

from h to 12h.

13.5. EXAMPLE ANALYSIS OF THE 3-D MODEL EQUATION 245

13.5 Example Analysis of the 3-D Model Equation

The arguments in Section 13.3 generalize to three dimensions and, under the condi-

tions given in 13.16 with an A(z)z

included, the model 3-D cases6 have the following

representative equation (with � = 0):

du

dt= [�x + �y + �z]u+ a (13.21)

Let us analyze a 2nd-order accurate, factored, delta form using this equation. First

apply the trapezoidal method:

un+1 = un +1

2h[(�x + �y + �z)un+1 + (�x + �y + �z)un + 2a]

Rearrange terms:

�1�

1

2h(�x + �y + �z)

�un+1 =

�1 +

1

2h(�x + �y + �z)

�un + ha

Put this in delta form:�1�

1

2h(�x + �y + �z)

��un = h[(�x + �y + �z)un + a]

Now factor the left side:�1�

1

2h�x

��1�

1

2h�y

��1�

1

2h�z

��un = h[(�x + �y + �z)un + a] (13.22)

This preserves second order accuracy since the error terms

1

4h2(�x�y + �x�z + �y�z)�un and

1

8h3�x�y�z

are both O(h3). One can derive the characteristic polynomial for Eq. 13.22, �nd the

� root, and write the solution either in the form

un = c

26641 +

1

2h(�x + �y + �z) +

1

4h2(�x�y + �x�z + �y�z)�

1

8h3�x�y�z

1�1

2h(�x + �y + �z) +

1

4h2(�x�y + �x�z + �y�z)�

1

8h3�x�y�z

3775n

�

a

�x + �y + �z(13.23)

6Eqs. 13.11 and 13.12, each with an additional term.


or in the form

un = c

2664�1 +

1

2h�x

��1 +

1

2h�y

��1 +

1

2h�z

��

1

4h3�x�y�z�

1�1

2h�x

��1�

1

2h�y

��1�

1

2h�z

�3775n

�

a

�x + �y + �z(13.24)

It is interesting to notice that a Taylor series expansion of Eq. 13.24 results in

� = 1 + h(�x + �y + �z) +1

2h2(�x + �y + �z)

2(13.25)

+1

4h3h�3z+ (2�y + 2�x) +

�2�2

y+ 3�x�y + 2�2

y

�+ �3

y+ 2�x�

2y+ 2�2

x�y + �3

x

i+ � � �

which veri�es the second order accuracy of the factored form. Furthermore, clearly,

if the method converges, it converges to the proper steady-state.7

With regards to stability, it follows from Eq. 13.23 that, if all the �'s are real and

negative, the method is stable for all h. This makes the method unconditionally stable

for the 3-D di�usion model when it is centrally di�erenced in space.

Now consider what happens when we apply this method to the biconvection model,

the 3-D form of Eq. 13.12 with periodic boundary conditions. In this case, central

di�erencing causes all of the �'s to be imaginary with spectrums that include both

positive and negative values. Remember that in our analysis we must consider every

possible combination of these eigenvalues. First write the � root in Eq. 13.23 in the

form

� =1 + i�� + i

1� i�� + i

where �, � and are real numbers that can have any sign. Now we can always �nd

one combination of the �'s for which �, and are both positive. In that case since

the absolute value of the product is the product of the absolute values

j�j2=

(1� �)2 + (� + )2

(1� �)2 + (�� )2> 1

and the method is unconditionally unstable for the model convection problem.

From the above analysis one would come to the conclusion that the method rep-

resented by Eq. 13.22 should not be used for the 3-D Euler equations. In practical

cases, however, some form of dissipation is almost always added to methods that are

used to solve the Euler equations and our experience to date is that, in the presence

of this dissipation, the instability disclosed above is too weak to cause trouble.

7However, we already knew this because we chose the delta form.

13.6. PROBLEMS 247

13.6 Problems

1. Starting with the generic ODE,

du

dt= Au+ f

we can split A as follows: A = A1+A2+A3+A4. Applying implicit Euler time

marching gives

un+1 � un

h= A1un+1 + A2un+1 + A3un+1 + A4un+1 + f

(a) Write the factored delta form. What is the error term?

(b) Instead of making all of the split terms implicit, leave two explicit:

un+1 � un

h= A1un+1 + A2un + A3un+1 + A4un + f

Write the resulting factored delta form and de�ne the error terms.

(c) The scalar representative equation is

du

dt= (�1 + �2 + �3 + �4)u+ a

For the fully implicit scheme of problem 1a, �nd the exact solution to the

resulting scalar di�erence equation and comment on the stability, conver-

gence, and accuracy of the converged steady-state solution.

(d) Repeat 1c for the explicit-implicit scheme of problem 1b.

Appendix A

USEFUL RELATIONS AND

DEFINITIONS FROM LINEAR

ALGEBRA

A basic understanding of the fundamentals of linear algebra is crucial to our develop-

ment of numerical methods and it is assumed that the reader is at least familar withthis subject area. Given below is some notation and some of the important relations

between matrices and vectors.

A.1 Notation

1. In the present context a vector is a vertical column or string. Thus

~v =

26664

v1v2...vm

37775

and its transpose ~vTis the horizontal row

~vT= [v1; v2; v3; : : : ; vm] ; ~v = [v1; v2; v3; : : : ; vm]

T

2. A general m�m matrix A can be written

A = (aij) =

26664

a11 a12 � � � a1ma21 a22 � � � a2m

. . .

am1 am2 � � � amm

37775

249

250APPENDIX A. USEFUL RELATIONS ANDDEFINITIONS FROM LINEAR ALGEBRA

3. An alternative notation for A is

A =h~a1;~a2; : : : ;~am

i

and its transpose AT is

AT =

2666664

~aT

1

~aT

2

...~aT

m

3777775

4. The inverse of a matrix (if it exists) is written A�1 and has the property that

A�1A = AA�1 = I, where I is the identity matrix.

A.2 De�nitions

1. A is symmetric if AT = A.

2. A is skew-symmetric or antisymmetric if AT = �A.

3. A is diagonally dominant if aii �P

j 6=i jaijj ; i = 1; 2; : : : ; m and aii >P

j 6=i jaijj

for at least one i.

4. A is orthogonal if aij are real and ATA = AAT = I

5. �A is the complex conjugate of A.

6. P is a permutation matrix if P~v is a simple reordering of ~v.

7. The trace of a matrix isP

i aii.

8. A is normal if ATA = AAT .

9. det[A] is the determinant of A.

10. AH is the conjugate transpose of A, (Hermitian).

11. If

A =

�a b

c d

�

thendet[A] = ad� bc

and

A�1 =1

det[A]

�d �b

�c a

�

A.3. ALGEBRA 251

A.3 Algebra

We consider only square matrices of the same dimension.

1. A and B are equal if aij = bij for all i; j = 1; 2; : : : ; m.

2. A+ (B + C) = (C + A) +B, etc.

3. sA = (saij) where s is a scalar.

4. In general AB 6= BA.

5. Transpose equalities:

(A+B)T = AT +BT

(AT )T = A

(AB)T = BTAT

6. Inverse equalities (if the inverse exists):

(A�1)�1 = A

(AB)�1 = B�1A�1

(AT )�1 = (A�1)T

7. Any matrix A can be expressed as the sum of a symmetric and a skew-symmetric

matrix. Thus:

A =1

2

�A+ AT

�+

1

2

�A� AT

�

A.4 Eigensystems

1. The eigenvalue problem for a matrix A is de�ned as

A~x = �~x or [A� �I]~x = 0

and the generalized eigenvalue problem, including the matrix B, as

A~x = �B~x or [A� �B]~x = 0

2. If a square matrix with real elements is symmetric, its eigenvalues are all real.

If it is asymmetric, they are all imaginary.


3. Gershgorin's theorem: The eigenvalues of a matrix lie in the complex plane in

the union of circles having centers located by the diagonals with radii equal to

the sum of the absolute values of the corresponding o�-diagonal row elements.

4. In general, an m �m matrix A has n~x linearly independent eigenvectors with

n~x � m and n� distinct eigenvalues (�i) with n� � n~x � m.

5. A set of eigenvectors is said to be linearly independent if

a � ~xm + b � ~xn 6= ~xk ; m 6= n 6= k

for any complex a and b and for all combinations of vectors in the set.

6. If A posseses m linearly independent eigenvectors then A is diagonalizable, i.e.,

X�1AX = �

where X is a matrix whose columns are the eigenvectors,

X =h~x1; ~x2; : : : ; ~xm

i

and � is the diagonal matrix

� =

266664

�1 0 � � � 0

0 �2. . .

......

. . .. . . 0

0 � � � 0 �m

377775

If A can be diagonalized, its eigenvectors completely span the space, and A issaid to have a complete eigensystem.

7. If A has m distinct eigenvalues, then A is always diagonalizable, and witheach distinct eigenvalue there is one associated eigenvector, and this eigenvector

cannot be formed from a linear combination of any of the other eigenvectors.

8. In general, the eigenvalues of a matrix may not be distinct, in which case the

possibility exists that it cannot be diagonalized. If the eigenvalues of a matrix

are not distinct, but all of the eigenvectors are linearly independent, the matrixis said to be derogatory, but it can still be diagonalized.

9. If a matrix does not have a complete set of linearly independent eigenvectors,

it cannot be diagonalized. The eigenvectors of such a matrix cannot span the

space and the matrix is said to have a defective eigensystem.

A.4. EIGENSYSTEMS 253

10. Defective matrices cannot be diagonalized but they can still be put into a com-

pact form by a similarity transform, S, such that

J = S�1AS =

266664

J1 0 � � � 0

0 J2. . .

......

. . .. . . 0

0 � � � 0 Jk

377775

where there are k linearly independent eigenvectors and Ji is either a Jordan

subblock or �i.

11. A Jordan submatrix has the form

Ji =

266666664

�i 1 0 � � � 0

0 �i 1. . .

...

0 0 �i. . . 0

.... . .

. . . 1

0 � � � 0 0 �i

377777775

12. Use of the transform S is known as putting A into its Jordan Canonical form.A repeated root in a Jordan block is referred to as a defective eigenvalue. For

each Jordan submatrix with an eigenvalue �i of multiplicity r, there exists oneeigenvector. The other r�1 vectors associated with this eigenvalue are referred

to as principal vectors. The complete set of principal vectors and eigenvectors

are all linearly independent.

13. Note that if P is the permutation matrix

P =

2640 0 10 1 01 0 0

375 ; P T = P�1 = P

then

P�1

264� 1 0

0 � 10 0 �

375P =

264� 0 0

1 � 00 1 �

375

14. Some of the Jordan subblocks may have the same eigenvalue. For example, the


matrix 266666666666664

264�1 1

�1 1

�1

375

�1 ��1 1

�1

��2 1

�2

�

�3

377777777777775

is both defective and derogatory, having:

� 9 eigenvalues

� 3 distinct eigenvalues

� 3 Jordan blocks

� 5 linearly independent eigenvectors

� 3 principal vectors with �1

� 1 principal vector with �2

A.5 Vector and Matrix Norms

1. The spectral radius of a matrix A is symbolized by �(A) such that

�(A) = j�mjmax

where �m are the eigenvalues of the matrix A.

2. A p-norm of the vector ~v is de�ned as

jjvjjp =

0@ MXj=1

jvjjp

1A1=p

3. A p-norm of a matrix A is de�ned as

jjAjjp = maxx6=0

jjAvjjp

jjvjjp

A.5. VECTOR AND MATRIX NORMS 255

4. Let A and B be square matrices of the same order. All matrix norms must have

the properties

jjAjj � 0; jjAjj = 0 implies A = 0

jjc � Ajj = jcj � jjAjj

jjA+Bjj � jjAjj+ jjBjj

jjA �Bjj � jjAjj � jjBjj

5. Special p-norms are

jjAjj1 = maxj=1;��;MPM

i=1 jaijj maximum column sum

jjAjj2 =q�(A

T� A)

jjAjj1 = maxi=1;2;��;MPM

j=1 jaijj maximum row sum

where jjAjjp is referred to as the Lp norm of A.

6. In general �(A) does not satisfy the conditions in 4, so in general �(A) is not atrue norm.

7. When A is normal, �(A) is a true norm, in fact, in this case it is the L2 norm.

8. The spectral radius of A, �(A), is the lower bound of all the norms of A.

Appendix B

SOME PROPERTIES OF

TRIDIAGONAL MATRICES

B.1 Standard Eigensystem for Simple Tridiagonals

In this work tridiagonal banded matrices are prevalent. It is useful to list some of

their properties. Many of these can be derived by solving the simple linear di�erence

equations that arise in deriving recursion relations.

Let us consider a simple tridiagonal matrix, i.e., a tridiagonal with constant scalar

elements a,b, and c, see Section 3.4. If we examine the conditions under which the

determinant of this matrix is zero, we �nd (by a recursion exercise)

det[B(M : a; b; c)] = 0

if

b + 2pac cos

�m�

M + 1

�= 0 ; m = 1; 2; � � � ;M

From this it follows at once that the eigenvalues of B(a; b; c) are

�m = b + 2pac cos

�m�

M + 1

�; m = 1; 2; � � � ;M (B.1)

The right-hand eigenvector of B(a; b; c) that is associated with the eigenvalue �msatis�es the equation

B(a; b; c)~xm = �m~xm (B.2)

and is given by

~xm = (xj)m =

�a

c

�j � 12 sin

�j

�m�

M + 1

��; m = 1; 2; � � � ;M (B.3)

257

258 APPENDIX B. SOME PROPERTIES OF TRIDIAGONAL MATRICES

These vectors are the columns of the right-hand eigenvector matrix, the elements of

which are

X = (xjm) =

�a

c

�j � 12 sin

�jm�

M + 1

�;

j = 1; 2; � � � ;Mm = 1; 2; � � � ;M (B.4)

Notice that if a = �1 and c = 1,

�a

c

�j � 12 = e

i(j�1)�2 (B.5)

The left-hand eigenvector matrix of B(a; b; c) can be written

X�1 =2

M + 1

�c

a

�m� 12 sin

�mj�

M + 1

�;

m = 1; 2; � � � ;Mj = 1; 2; � � � ;M

In this case notice that if a = �1 and c = 1

�c

a

�m� 12 = e

�i(m�1)�2 (B.6)

B.2 Generalized Eigensystem for Simple Tridiag-

onals

This system is de�ned as follows26666664

b c

a b c

a b. . . ca b

37777775

26666664

x1x2x3...

xM

37777775= �

26666664

e f

d e f

d e. . . fd e

37777775

26666664

x1x2x3...

xM

37777775

In this case one can show after some algebra that

det[B(a� �d; b� �e; c� �f ] = 0 (B.7)

if

b� �me+ 2q(a� �md)(c� �mf) cos

�m�

M + 1

�= 0 ; m = 1; 2; � � � ;M (B.8)

If we de�ne

�m =m�

M + 1; pm = cos �m

B.3. THE INVERSE OF A SIMPLE TRIDIAGONAL 259

�m =eb� 2(cd+ af)p2

m+ 2pm

q(ec� fb)(ea� bd) + [(cd� af)pm]

2

e2 � 4fdp2m

The right-hand eigenvectors are

~xm =

"a� �md

c� �mf

#j � 12

sin [j�m] ;m = 1; 2; � � � ;Mj = 1; 2; � � � ;M

These relations are useful in studying relaxation methods.

B.3 The Inverse of a Simple Tridiagonal

The inverse of B(a; b; c) can also be written in analytic form. Let DM represent the

determinant of B(M : a; b; c)

DM � det[B(M : a; b; c)]

De�ning D0 to be 1, it is simple to derive the �rst few determinants, thus

D0 = 1

D1 = b

D2 = b2 � ac

D3 = b3 � 2abc (B.9)

One can also �nd the recursion relation

DM = bDM�1 � acDM�2 (B.10)

Eq. B.10 is a linear O�E the solution of which was discussed in Section 4.2. Its

characteristic polynomial P (E) is P (E2 � bE + ac) and the two roots to P (�) = 0

result in the solution

DM =1p

b2 � 4ac

8<:"b +

pb2 � 4ac

2

#M+1

�"b�

pb2 � 4ac

2

#M+19=;

M = 0; 1; 2; � � � (B.11)

where we have made use of the initial conditions D0 = 1 and D1 = b. In the limiting

case when b2 � 4ac = 0, one can show that

DM = (M + 1)

b

2

!M; b2 = 4ac


Then for M = 4

B�1 =1

D4

26664

D3 �cD2 c2D1 �c3D0

�aD2 D1D2 �cD1D1 c2D1

a2D1 �aD1D1 D2D1 �cD2

�a3D0 a2D1 �aD2 D3

37775

and for M = 5

B�1 =1

D5

2666664

D4 �cD3 c2D2 �c3D1 c4D0

�aD3 D1D3 �cD1D2 c2D1D1 �c3D1

a2D2 �aD1D2 D2D2 �cD2D1 c2D2

�a3D1 a2D1D1 �aD2D1 D3D1 �cD3

a4D0 �a3D1 a2D2 �aD3 D4

3777775

The general element dmn is

Upper triangle:

m = 1; 2; � � � ;M � 1 ; n = m + 1; m+ 2; � � � ;M

dmn = Dm�1DM�n(�c)n�m=DM

Diagonal:

n = m = 1; 2; � � � ;M

dmm = DM�1DM�m=DM

Lower triangle:

m = n + 1; n+ 2; � � � ;M ; n = 1; 2; � � � ;M � 1

dmn = DM�mDn�1(�a)m�n=DM

B.4 Eigensystems of Circulant Matrices

B.4.1 Standard Tridiagonals

Consider the circulant (see Section 3.4.4) tridiagonal matrix

Bp(M : a; b; c; ) (B.12)

B.4. EIGENSYSTEMS OF CIRCULANT MATRICES 261

The eigenvalues are

�m = b + (a+ c) cos

�2�m

M

�� i(a� c) sin

�2�m

M

�; m = 0; 1; 2; � � � ;M � 1

(B.13)

The right-hand eigenvector that satis�es Bp(a; b; c)~xm = �m~xm is

~xm = (xj)m = ei j (2�m=M) ; j = 0; 1; � � � ;M � 1 (B.14)

where i �p�1, and the right-hand eigenvector matrix has the form

X = (xjm) = eij

�2�mM

�;

j = 0; 1; � � � ;M � 1

m = 0; 1; � � � ;M � 1

The left-hand eigenvector matrix with elements x0 is

X�1 = (x0mj) =

1

Me�im

�2�jM

�;

m = 0; 1; � � � ;M � 1

j = 0; 1; � � � ;M � 1

Note that both X and X�1 are symmetric and that X�1 = 1MX�, where X� is the

conjugate transpose of X.

B.4.2 General Circulant Systems

Notice the remarkable fact that the elements of the eigenvector matrices X and X�1

for the tridiagonal circulant matrix given by eq. B.12 do not depend on the elements

a; b; c in the matrix. In fact, all circulant matrices of order M have the same set of

linearly independent eigenvectors, even if they are completely dense. An example of

a dense circulant matrix of order M = 4 is26664b0 b1 b2 b3b3 b0 b1 b2b2 b3 b0 b1b1 b2 b3 b0

37775 (B.15)

The eigenvectors are always given by eq. B.14, and further examination shows that

the elements in these eigenvectors correspond to the elements in a complex harmonic

analysis or complex discrete Fourier series.

Although the eigenvectors of a circulant matrix are independent of its elements,

the eigenvalues are not. For the element indexing shown in eq. B.15 they have the

general form

�m =M�1Xj=0

bjei(2�jm=M)

of which eq. B.13 is a special case.


B.5 Special Cases Found From Symmetries

Consider a mesh with an even number of interior points such as that shown in Fig.

B.1. One can seek from the tridiagonal matrix B(2M : a; b; a; ) the eigenvector subset

that has even symmetry when spanning the interval 0 � x � �. For example, we seek

the set of eigenvectors ~xm for which

2666666664

b aa b a

a. . .

. . . a

a b a

a b

3777777775

26666666664

x1x2......

x2x1

37777777775= �m

26666666664

x1x2......

x2x1

37777777775

This leads to the subsystem of order M which has the form

B(M : a;~b; a)~xm =

266666666664

b a

a b a

a. . .

. . . aa b a

a b + a

377777777775~xm = �m~xm (B.16)

By folding the known eigenvectors of B(2M : a; b; a) about the center, one can show

from previous results that the eigenvalues of eq. B.16 are

�m = b + 2a cos

(2m� 1)�

2M + 1

!; m = 1; 2; � � � ;M (B.17)

B.6. SPECIAL CASES INVOLVING BOUNDARY CONDITIONS 263

and the corresponding eigenvectors are

~xm = sin

�j(2m� 1)�2M + 1

�;

j = 1; 2; � � � ;M

Imposing symmetry about the same interval

but for a mesh with an odd number of points,

see Fig. B.1, leads to the matrix

B(M : ~a; b; a) =

2666666664

b a

a b a

a. . .

. . . a

a b a2a b

3777777775

By folding the known eigenvalues of B(2M �1 : a; b; a) about the center, one can show

from previous results that the eigenvalues of

eq. B.17 are

Line of Symmetry

x = 0 x = ��

j 0 = 1 2 3 4 5 6

M 0

j = 1 2 3

M

a. An even-numbered mesh

Line of Symmetry

x = 0 x = ��

j 0 = 1 2 3 4 5

M 0

j = 1 2 3

M

b. An odd{numbered mesh

Figure B.1 { Symmetrical folds for

special cases

�m = b + 2a cos

(2m� 1)�

2M

!; m = 1; 2; � � � ;M

and the corresponding eigenvectors are

~xm = sin

j(2m� 1)�

2M

!; j = 1; 2; � � � ;M

B.6 Special Cases Involving Boundary Conditions

We consider two special cases for the matrix operator representing the 3-point central

di�erence approximation for the second derivative @2=@x2 at all points away from the

boundaries, combined with special conditions imposed at the boundaries.


Note: In both cases

m = 1; 2; � � � ;Mj = 1; 2; � � � ;M

�2 + 2 cos(�) = �4 sin2(�=2)

When the boundary conditions are Dirichlet on both sides,

26666664

�2 1

1 �2 1

1 �2 1

1 �2 1

1 �2

37777775

�m = �2 + 2 cos�

m�M + 1

�~xm = sin

hj�

m�M + 1

�i (B.18)

When one boundary condition is Dirichlet and the other is Neumann (and a diagonal

preconditioner is applied to scale the last equation),

26666664

�2 1

1 �2 1

1 �2 1

1 �2 1

1 �1

37777775

�m = �2 + 2 cos

�(2m� 1)�2M + 1

�~xm = sin

�j

�(2m� 1)�2M + 1

�� (B.19)

Appendix C

THE HOMOGENEOUS

PROPERTY OF THE EULER

EQUATIONS

The Euler equations have a special property that is sometimes useful in constructing

numerical methods. In order to examine this property, let us �rst inspect Euler's

theorem on homogeneous functions. Consider �rst the scalar case. If F (u; v) satis�es

the identity

F (�u; �v) = �nF (u; v) (C.1)

for a �xed n, F is called homogeneous of degree n. Di�erentiating both sides with

respect to � and setting � = 1 (since the identity holds for all �), we �nd

u@F

@u+ v

@F

@v= nF (u; v) (C.2)

Consider next the theorem as it applies to systems of equations. If the vector

F (Q) satis�es the identity

F (�Q) = �nF (Q) (C.3)

for a �xed n, F is said to be homogeneous of degree n and we �nd

"@F

@q

#Q = nF (Q) (C.4)

265

266APPENDIX C. THE HOMOGENEOUS PROPERTY OF THE EULER EQUATIONS

Now it is easy to show, by direct use of eq. C.3, that both E and F in eqs. 2.11 and

2.12 are homogeneous of degree 1, and their Jacobians, A and B, are homogeneous

of degree 0 (actually the latter is a direct consequence of the former).1 This being

the case, we notice that the expansion of the ux vector in the vicinity of tn which,

according to eq. 6.105 can be written in general as,

E = En + An(Q�Qn) +O(h2)

F = Fn +Bn(Q�Qn) +O(h2) (C.5)

can be written

E = AnQ+O(h2)

F = BnQ+O(h2) (C.6)

since the terms En � AnQn and Fn � BnQn are identically zero for homogeneous

vectors of degree 1, see eq. C.4. Notice also that, under this condition, the constant

term drops out of eq. 6.106.

As a �nal remark, we notice from the chain rule that for any vectors F and Q

@F (Q)

@x=

"@F

@Q

#@Q

@x= A

@Q

@x(C.7)

We notice also that for a homogeneous F of degree 1, F = AQ and

@F

@x= A

@Q

@x+

"@A

@x

#Q (C.8)

Therefore, if F is homogeneous of degree 1,

"@A

@x

#Q = 0 (C.9)

in spite of the fact that individually [@A=@x] and Q are not equal to zero.

1Note that this depends on the form of the equation of state. The Euler equations are homoge-

neous if the equation of state can be written in the form p = �f(�), where � is the internal energy

per unit mass.

Bibliography

[1] D. Anderson, J. Tannehill, and R. Pletcher. Computational Fluid Mechanics and

Heat Transfer. McGraw-Hill Book Company, New York, 1984.

[2] C. Hirsch. Numerical Computation of Internal and External Flows, volume 1,2.

John Wiley & Sons, New York, 1988.

267

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