lecture9 hyperbolic equation

8/22/2019 Lecture9 Hyperbolic Equation

1/31

AML 811Lecture 9

Approximate Factorization

Hyperbolic Equations: Leapfrog, FTBS


2/31

Recap : Implicit iterative method

Line Gauss-Seidel

Line Gauss-Seidel

Values at each line are updated simultaneously based on latest available values


3/31

Recap : Line Gauss-Seidel example

=

4

2,54,5

2,44,4

2,34,3

2,24,2

2

3,6

3,5

3,4

3,3

3,2

3,1

100000

141000

014100001410

000141

000001

u

uu

uu

uu

uu

u

u

u

u

u

u

u

Update

direction

System of tridiagonal

equations


4/31

Recap : Line Gauss-Seidel Method

Algorithm1. Initialize guess

2. Choose a direction for

sweeping (rows or columns)3. Sweep through lines

updating all points in eachline simultaneously to thek+1th level using the Gauss-Seidel update. This involvessolving a system oftridiagonal equations

4. After one sweep through the

domain check error.5. If error too high, repeat steps

3 and 4

Any efficient way of solving tridiagonal systems?


5/31

Recap : Solving Tridiagonal systems :

Thomas algorithm

Forward Sweep. Making the matrix upper-diagonal

Back-substitution to obtain x

iiiiiii dxcxbxa =++ + 11


6/31

Recap : Line SOR

Values at each line are updated simultaneously based on latest available values

( )SeidelGaussLine

k

ji

k

jiSORLine

k

ji uuu ++ += 1,,

1

, 1


7/31

Recap : Crank-Nicholson as a multi-step

methodStep 1: Explicit Update Step 2: Implicit Update

Total effect of explicit and implicit steps : Crank Nicholson


8/31

Recap : 2D parabolic equations

Use 1D parabolic equation methods as stepping stone for 2Dmethods

Explicit FTCS Implicit FTCS (same as BTCS)

Crank-Nicholson

As usual, we need to analyze consistency and stability for these

It turns out that all these schemes are consistent. Stability?

Example


9/31

Recap : Summary of Von-Neumann Analysis in

2D Step 1 : Fourier Decomposition:

Assume that solution is composed of a sum of waves of

the form

Step 2: Obtain evolution equation for the amplitude

Substitute Fourier decomposition in the original Finite

Difference equation and write it in the formnn GUU =+

1

Soln at grid

point i,j at

time step n

Gain oramplification

factor


10/31

Recap : Summary of Von-Neumann 2D Analysis

Step 3 : Find region of stabili ty

Find the conditions on x, y, t under which the amplitude of the

wave will be stable, i.e. not grow. This will happen if

Explicit FTCS for the diffusion equation

2sin4

2sin41 22

yx ddG =

2

122

+

=+

y

t

x

tdd yx

Stable only if Why not use FTCSimplicit instead?


11/31

Recap : Implicit formulation for 2D

parabolic equation

Banded, pentadiagonal system of equations to be solved at each time step


12/31


parabolic equation



13/31


parabolic equation for a 5x5 system


Very expensive. Is there a cheaper, equally stable alternative?


14/31

Recap : Alternate Direction Implicit (ADI)

Scheme

Update in two steps

Each step involves implicit space derivatives in only one direction This results in only a tridiagonal system of equations being solved in each step

Unconditionally stable. Second order in space and time

What is ADIs relationship to the 2D Crank-Nicholson scheme? Are theyidentical?

x sweep

y sweep


15/31

Summary of Lecture 8 Line Methods : Sweep through rows or columns updating all

values in a line simultaneously : Line Gauss Seidel, LSOR

Crank-Nicholson as a multi-step method involving separateexplicit and an implicit steps.

Von-Neumann stability analysis with 2 spatial dimensions Implicit schemes for 2D parabolic equations too expensive,

explicit schemes too restrictive on time step

Option :Alternate direction implicit scheme

Update in two steps with each step having an implicit update inone spatial direction

Unconditionally stable, second order accurate in time and space

Whats its relation to Crank-Nicholson?


16/31

Recall : Alternate Direction Implicit (ADI)

Scheme

Update in two steps

Each step involves implicit space derivatives in only one direction This results in only a tridiagonal system of equations being solved in each step

Unconditionally stable. Second order in space and time

What is ADIs relationship to the 2D Crank-Nicholson scheme? Are theyidentical?

x sweep

y sweep


17/31

ADI and Crank-Nicholson methods in 2D

CN involves inverting a pentadiagonal matrix : P ADI involves inverting two tridiagonal matrices : T1 and T2

Whats the relation between P, T1 and T2?

Crank-Nicholson

ADI

x sweep

y sweep


18/31

ADI and Crank Nicholson: Approximate

Factorization Methods

Since ADI treats each direction successively at a time lag compared to the other directionit is still an approximation of the Crank-Nicholson scheme

However, since the difference between the two methods is less than the truncation error, it

is a valid approximation and retains consistency and the overall order of convergence Methods such as ADI which approximate a more computationally intensive matrix inversion

by a less intensive one (typically involving tridiagonal systems) are known asApproximateFactorization Methods.

Crank Nicholson

ADI

Difference between the two


19/31

Another approximate factorization :

Fractional Step Method

Treat each direction separately by Crank Nicholson.

Unlike ADI, which approximates the full 2D equation in each step, the fractionalstep method approximates only one 1D component in each step.

This method is also unconditionally stable and second order accurate in timeand space

Approximate factorization methods can be used for elliptic equations also

Step 1 : x update

Step 2 : y update


20/31

Hyperbolic Equations

Recall Have as many real characteristics as

order of PDE Information travels at a finite speed. The

effects of an action somewhere will befelt elsewhere after a time lag.Convection predominates in suchproblems

Marching Problems: Initial solution orcondition needs to be marched in timeto get solution at a later time

Domain of dependence : The region inspace and time on which the solution ata point depends on

Range of influence: The region in spaceand time which the value at a point willinfluence

The compressible, inviscid fluid dynamicequations are hyperbolic


21/31

Recall : A preliminary scheme to solve the

wave equation

Let us try and numerically solve the wave

equation with some given initial condition

( ) )][,( 2111 xtOuux

tauu

n

i

n

i

n

i

n

i +

= +

+

Use a forward difference scheme in time and

central difference scheme in space (FTCS

scheme)

0=

+

x

ua

t

u

Stencil


22/31

Numerically solving the wave equation.

Let the initial condition be u(x,0) = u0(x)


23/31

Exact Solution

Wave moving towards right at speed c


24/31

Recall : FTCS for wave equation. Using FTCS results in the following behavior irrespective

of time step and grid size

FTCS is unconditionally unstable.

What about FTFS,CTCS, BTCS, FTBS?


25/31

Leapfrog (CTCS) method

Some problems with Leapfrog

Need to give some initial condition for n =1 apartfrom the usual condition for n = 0

Odd and even time steps are decoupled. i.e. The

solution at the even time steps never depends onthe solution at odd time steps. Often, twoindependent solutions develop over time (Exercise)

Stability condition

1xta


26/31

FTBS for the wave equation

First order in space and time

Also known as the first order upwind

method

x

uua

t

uun

i

n

i

n

i

n

i

=

+

1

1

Stable for 1

=

x

tac Courant condition


27/31

FTBS for wave equation with CFL = 1

Gives the exact solution! Why?


28/31

FTBS for wave equation with CFL = 0.5

Why is the solution somewhat diffusive?


29/31

FTBS for wave equation with CFL = 0.9

Why is the solution somewhat diffusive but less diffusive than CFL = 0.5?

Is FTBS consistent?


30/31

Artificial viscosity

FTBS can be rewritten as

2

1111

1 2

x

uuu

x

uua

t

uun

i

n

i

n

i

n

i

n

i

n

i

n

i

+=

+

+++

Here the coefficient artificial viscosity goes

down as the grid size becomes smaller and

hence the scheme is consistent

However, the solution behaves as if it is the

solution to a convection-diffusion equation


31/31

Summary

ADI is an approximate factorization of the

Crank-Nicholson method Fractional Step Methods are another way of

reducing computational costs for parabolic

equations Hyperbolic equations

Stability analysis of Leapfrog method

Leapfrog can lead to odd-even decoupling FTBS is a stable method for hyperbolic equations

but it is first order accurate and can be diffusive

lecture9 hyperbolic equation

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