hyperbolic conservation laws - eth z · to the conservation law. this unique solution is called the...

Research Collection

Doctoral Thesis

High-order accurate entropy stable numercial schemes forhyperbolic conservation laws

Author(s): Fjordholm, Ulrik Skre

Publication Date: 2013

Permanent Link: https://doi.org/10.3929/ethz-a-007622508

Rights / License: In Copyright - Non-Commercial Use Permitted

This page was generated automatically upon download from the ETH Zurich Research Collection. For moreinformation please consult the Terms of use.

ETH Library

https://doi.org/10.3929/ethz-a-007622508

http://rightsstatements.org/page/InC-NC/1.0/

https://www.research-collection.ethz.ch

https://www.research-collection.ethz.ch/terms-of-use

Diss. ETH No. 21025

HIGH-ORDER ACCURATE ENTROPY STABLE NUMERICALSCHEMES FOR HYPERBOLIC CONSERVATION LAWS

A dissertation submitted toETH ZURICH

for the degree ofDoctor of Sciences

presented by

ULRIK SKRE FJORDHOLM

M.Sc. Computational Science and Engineering, University of Osloborn 1 December 1985

citizen of Norway

accepted on the recommendation of

Prof. Dr. Siddhartha Mishra, ETH Zurich, examinerProf. Dr. Eitan Tadmor, University of Maryland, co-examiner

Prof. Dr. Chi-Wang Shu, Brown University, co-examiner

2013

ISBN 978-3-906031-29-3

DOI 10.3929/ethz-a-007622508

Abstract

The role of entropy for the stability of hyperbolic conservation laws is well-understood,and to some extent, also for first- and second-order accurate numerical schemes. The stabilityof higher-order accurate schemes, however, is to a large degree an open problem. In thisthesis we adopt the framework of entropy stability as a design principle. We combine entropyconservative fluxes with appropriate diffusion operators, and to obtain high order accuracy,we consider diffusion operators using the ENO reconstruction procedure. We show that theENO procedure satisfies the so-called sign property, which ensures entropy stability of our(arbitrarily) high-order accurate finite difference scheme. Moreover, we pose and argue for aconjecture on the total variation of the ENO reconstruction, which would imply a weak totalvariation bound for our scheme. For hyperbolic systems, the reconstruction is performedusing a novel, computationally efficient characteristic-wise decomposition. The scheme iseasily generalized to multi-dimensional systems on Cartesian meshes.

To study the convergence properties of our scheme for scalar equations, we consider theframework of compensated compactness. Under the assumption that our conjecture on thetotal variation bound of ENO holds, we prove convergence to the entropy solution. For multi-dimensional hyperbolic systems we are able to prove convergence to an entropy measure-valued solution.

The robustness, accuracy and computational efficiency of the scheme are demonstratedin a series of numerical experiments.

iii

Zusammenfassung

Die Rolle der Entropie fur die Stabilitat hyperbolischer Differentialgleichungen ist gutverstanden. Dies gilt teilweise auch fur numerische Verfahren erster und zweiter Ordnung.Die Stabilitat von Verfahren hoherer Ordnung ist jedoch ein offenes Problem. In dieser Dis-sertation wird Entropiestabilitat als Gestaltungsprinzip verwendet. Wir kombinieren entropie-erhaltende Flusse mit geeigneten Diffusionsoperatoren. Um eine hohe Ordnung zu erhalten,benutzen wir Diffusionsoperatoren basierend auf dem ENO-Rekonstruktionsverfahren. Wirzeigen, dass das ENO-Verfahren vorzeichenerhaltend ist und somit Entropiestabilitat unseresVerfahrens bei (beliebig) hoher Ordnung gilt. Zudem stellen wir eine Vermutung uber die to-tale Variation der ENO-Verfahren auf, die eine schwache obere Schranke der totalen Variationunserer Methode darstellen wurde. Fur hyperbolische Systeme rekonstruieren wir die En-tropievariablen mittels einer neuen, effizienten Zerlegung, basierend auf den Eigenvektorendes Diffusionsoperators. Das Verfahren lasst sich leicht auf mehrdimensionale, kartesischeGitter verallgemeinern.

Der Zugang uber kompensierte Kompaktheit wird benutzt, um die Konvergenzeigen-schaften unserer Methode fur skalare Gleichungen zu untersuchen. Unter der Annahme derobigen Vermutung zeigen wir Konvergenz des Verfahrens zur Entropielosung. Fur mehr-dimensionale hyperbolische Systeme beweisen wir Konvergenz zu einer masswertigen Entropie-losung.

Die Robustheit, Genauigkeit und Effizienz der Verfahren werden in verschiedenen nu-merischen Experimenten demonstriert.

v

Acknowledgements

Of all the people who have helped me through the past 3.5 years, I wish to mention three inparticular. First, I am very grateful to Sidharth Mishra for all that he has taught me and foralways having time for a discussion, math-related or not.Second, I thank Eitan Tadmor for generously sharing from his immense experience, for hiskeen interest and enthusiasm, and for his warm hospitality in Maryland.Finally, thank you, Ingeborg, for your immense patience with me. Your support has meantmore to me than I am able to convey here.

vii

Contents

Abstract iii

Zusammenfassung v

Acknowledgements vii

Introduction xiNotation xiv

Chapter 1. Hyperbolic conservation laws 11.1. Basic notions and examples 11.2. Entropy pairs and the entropy condition 51.3. Existence and uniqueness for scalar conservation laws 91.4. Compensated compactness 101.5. Existence and uniqueness for systems of conservation laws 131.6. Measure-valued solutions 131.7. Summary and outlook 14

Chapter 2. Entropy stable methods of first order 172.1. Finite volume and finite difference methods 182.2. Entropy conservative and entropy stable methods 182.3. Entropy stable methods for scalar equations 212.4. Entropy conservative schemes for systems 232.5. Entropy stable schemes for systems 252.6. Summary 27

Chapter 3. Convergence theory for finite volume methods 293.1. Strong convergence of finite volume schemes 303.2. Multi-dimensional measure-valued solutions of systems 343.3. Convergence of entropy stable methods 38

Chapter 4. Higher-order accurate entropy stable methods 394.1. High-order entropy conservative methods 394.2. High-order entropy stable methods for scalar equations: the ELW scheme 40

ix

x CONTENTS

4.3. Reconstruction based entropy stable schemes: scalar equations 424.4. Reconstruction based entropy stable schemes: systems of equations 444.5. Summary and outlook 47

Chapter 5. ENO stability properties 495.1. The ENO reconstruction method 505.2. The ENO sign property 525.3. Upper bound on jumps 575.4. The ENO interpolation method 605.5. A conjecture on the total variation of ENO reconstruction 64

Chapter 6. The TECNO scheme 676.1. The TECNO scheme for scalar conservation laws 676.2. The TECNO scheme for systems of equations 696.3. Numerical experiments: scalar conservation laws 706.4. Systems of conservation laws 73

Chapter 7. Conclusions 83Future work 83

Appendix A. Young measures 85

Bibliography 87

Introduction

A large class of phenomena in fields such as meteorology, oceanography, aerodynamicsand plasma physics can be modelled by conservation laws. If u(x, t) denotes the concentrationof some quantity, such as gas in a container or cars on a highway, then the net change of thetotal amount of u in a domain D can be expressed in terms of the integral equation

ddt

∫D

u(x, t) dx +

∫∂D

f (x, t) · n(x) dσ(x) =

∫D

g(x, t) dx.

Here, f (x, t) is a function specifying the net flux through a point x at time t, and g(x, t)specifies the production (or destruction) of the quantity at x, t. If g ≡ 0, then there is no netoverall change of the quantity u, and the equation above states that u is a conserved variable– it can only change due to flux through the boundary of D. In many cases, the flux functionf can be written as a function of u. Applying the divergence theorem to the equation, weobtain the conservation law

ut + div( f (u)) = 0.

Supplying the partial differential equation with appropriate initial data,

u(x, 0) = u0(x),

we obtain a Cauchy problem for u.An important subclass of such equations are hyperbolic conservation laws. Roughly

speaking, a conservation law is hyperbolic if information travels at a finite speed. Thus, con-trary to parabolic partial differential equations, local changes in the solutions of hyperbolicconservation laws have only local consequences.

Another fundamental difference from parabolic PDE is the appearance of discontinuitieseven for smooth initial data. Once discontinuities are present, the differential equation is nolonger meaningful in the classical sense. By interpreting the equation in a weaker sense, weallow for discontinuous solutions, but also allowing for the existence of multiple solutions tothe same Cauchy problem. Thus, to single out a solution we must impose some additionalselection criterion. For hyperbolic conservation laws this comes in the form of an entropy

xi

xii INTRODUCTION

condition, which asserts that there must be no production of entropy1. Along with some addi-tional conditions on the structure of f , the entropy condition implies uniqueness of solutionsto the conservation law. This unique solution is called the entropy solution [GR91].

Since the start of serious research on hyperbolic conservation laws in the 1940s, theoryand numerics have gone hand in hand. Indeed, several of the fundamental existence results forconservation laws were obtained through numerical approximation [Gli65, Lax71, BCP00].Early on it was shown that numerical schemes must in certain respects be consistent, not onlywith the conservation law itself, but also with respect to the entropy condition, in order toattain convergence to the entropy solution [Lax71]. Such schemes were later termed entropystable. The study of entropy stable schemes is the main topic of this thesis.

By the start of the 1980s, the convergence theory for first-order accurate numericalschemes for scalar conservation laws was well-developed [CM80]. Through the 1980s, thefocus shifted towards developing schemes with a higher (than first) order of accuracy. Amongthe many different approaches are the ENO and WENO methods, which combine stable first-order schemes with higher-order reconstruction methods, resulting in an overall high-orderaccurate scheme [HEOC87, SO88, LOC94, JS96]. Although these schemes have enjoyedwidespread popularity in the industrial and research communities, there is a lack of rigorousstability analysis of high-order accurate schemes [Shu99, Section 4.3].

The main purpose of this thesis is to address this lack of stability results for high-orderaccurate schemes. We construct high-order accurate entropy stable finite difference schemesfor both scalar conservation laws and systems of equations. The entropy dissipation rateof these schemes gives a weak bound on the gradient of the solution; this allows us to provestrong convergence to the entropy solution for scalar equations and convergence to an entropymeasure-valued solution for systems of equations.

The following is an outline of the thesis. The introductory Chapter 1 gives a brief intro-duction to the theory of hyperbolic conservation laws, with emphasis on the role of entropyconditions. We give an introduction to the method of compensated compactness, and discussthe concept of measure-valued solutions as introduced by DiPerna [DiP85].

In Chapter 2 we discuss the theory of first-order accurate, entropy stable finite differ-ence methods for both scalar conservation laws and systems of hyperbolic conservation laws.We demonstrate the construction of first-order accurate entropy conservative and entropy sta-ble methods, which will serve as building blocks for the higher-order accurate methods insubsequent chapters.

Chapter 3 deals with general convergence theory for numerical approximations to hy-perbolic conservation laws. By a careful analysis of previous work, we derive sufficientconditions, as weak as possible, for the (strong) convergence of numerical methods to theentropy solution. This is then contrasted with less restrictive conditions for convergence toentropy measure-valued solutions. We apply the theory discussed here to prove convergence

1According to the second law of thermodynamics, entropy should never decrease, but due to a difference insign, we require (mathematical) entropy never to increase.

INTRODUCTION xiii

of the schemes from Chapter 2 to the entropy solution for one-dimensional convex scalarconservation laws.

In Chapter 4 we discuss higher-order extensions of the entropy stable schemes fromChapter 3. As a first example we introduce a high-order accurate, entropy stable schemewhich converges for scalar equations, but which exhibits oscillations around shocks. Toameliorate this, we explore finite difference schemes using nonoscillatory reconstruction pro-cedures and derive a sufficient condition for entropy stability for both scalar equations andsystems. As we show in Chapter 5, the ENO reconstruction procedure satisfies precisely thiscondition.

Chapter 6 is the main contribution of the thesis. We apply the theory developed in Chap-ter 4 to obtain nonoscillatory, high-order accurate, entropy stable schemes, which we refer toas TECNO schemes. Using the convergence theory from Chapter 3, we prove that the schemeconverges to the entropy solution for scalar equations, and converges to a measure-valued so-lution for systems of equations. Finally, we compare our scheme to existing schemes in aseries of numerical experiments.

xiv INTRODUCTION

Notation

We introduce here notation which is used throughout the thesis. The canonical innerproduct on Rn is x · y = xT y =

∑nk=1 xkyk, with the associated norm |x| =

√x · x. The

subset Br(x) of Rn denotes the open ball with center x ∈ Rn and radius r > 0, and the unithypersphere in Rn is denoted S n−1 := x ∈ Rn : |x| = 1. We write for R+ := [0,∞)the nonnegative real axis. The identity function on a space X is denoted idX(x) := x. Thesubscript is dropped when the domain is clear from the context. We denote the indicatorfunction of a set U by

1U(x) =

1 if x ∈ U,0 if x < U.

If A ⊂ Rn is Lebesgue measurable then we denote its Lebesgue measure by meas(A).We denote by Ck(Rn,Rm) the set of k times continuously differentiable functions from

Rn to Rm. If m = 1 then we write Ck(Rn) := Ck(Rn,R). The space of smooth, compactlysupported functions on U ⊂ Rn is denoted D(U) := C∞c (U). The dual space D′(U) is thespace of distributions on U. The Jacobian of a function f ∈ C1(Rn,Rm) is denoted by a prime,f ′(u) =

d fdu

∣∣∣u. Partial derivatives are denoted by subscripts, ux = ∂u

∂x . If ukk∈N ⊂ L∞(Rn) is aweak-∗ convergent sequence then we denote its limit by u := w-∗ lim uk.

For symmetric matrices A, B ∈ Rn×n, we mean by A 6 B that the matrix B− A is positivesemi-definite (positive definite if there is strict inequality). A function η ∈ C2(Rn) is convexif η′′(u) > 0 for all u, strictly convex if η′′(u) > 0 for a.e. u and uniformly convex if η′′(u) > mfor all u for an m > 0.

For a grid function ui : i ∈ Z we denote its jump and average by [[u]]i+1/2 := ui+1 − ui

and ui+1/2 = 12 (ui + ui+1).

CHAPTER 1

Hyperbolic conservation laws

In this chapter we summarize the important existence, uniqueness and regularity resultsrelated to hyperbolic conservation laws. Because of the appearance of discontinuities in so-lutions of such equations, the differential equation is not meaningful in the classical sense.We will look at two different, weaker solution concepts: weak solutions and measure-valuedsolutions. To obtain uniqueness we must introduce entropy conditions that single out a “phys-ically relevant” solution.

In Section 1.4 we give a short review of compensated compactness, a method for show-ing strong convergence of a sequence of approximate solutions. The framework introduced inthis section will be used in Chapter 3 to show convergence of general finite volume methods,and in Chapter 4 to show convergence of the high-order accurate, entropy stable methodsintroduced there. In Section 1.6 we study the much weaker concept of measure-valued solu-tions, and argue why this should be viewed as a natural solution concept for general nonlinearevolution equations. We end the chapter with an overview of the rest of the thesis.

Most of the theorems in this chapter will be stated without proofs, with references whereappropriate.

1.1. Basic notions and examples

We consider systems of conservation laws,

ut + div( f (u)) = 0, (x, t) ∈ Rd × R+

u(·, 0) = u0,(1.1)

where u : Rd×R+ → U is the vector of conserved variables, taking values in some connected,nonempty set U ⊂ RN . The function f = ( f 1, . . . , f d) : U→ RN×d is the flux function, whichis assumed to be at least C2 on U. The conservation law is hyperbolic if for all u ∈ U andξ ∈ S d−1, the Jacobian ξ · f ′(u) =

∑dn=1 ξ

n d f n

du (u) has N real eigenvalues λ1(u) 6 · · · 6 λN(u)with linearly independent eigenvectors r1(u), . . . , rN(u) (the dependence on ξ is suppressed).It is strictly hyperbolic if the eigenvalues are distinct, λ1(u) < · · · < λN(u). An eigenpair(λk, rk) of ξ · f ′ is referred to as a wave family. The k-th wave family (λk, rk) is genuinelynonlinear if λ′k(u) · rk(u) , 0 for all u ∈ U, and is linearly degenerate if λ′k(u) · rk(u) = 0for all u ∈ U. If λ′k(u) · rk(u) vanishes only in parts of U then the conservation law is termednonconvex; such conservation laws will not be considered here. We refer to [GR91] for athorough introduction to the theory of hyperbolic conservation laws.

1

2 1. HYPERBOLIC CONSERVATION LAWS

It is easy to see that every scalar (N = 1) conservation law is hyperbolic; it is genuinelynonlinear iff f ′′(u) , 0 for all u ∈ U, and is linearly degenerate iff f is affine. A linear systemof equations with flux f (u) = Au is hyperbolic if A is diagonalizable with real eigenvalues.All wave families of linear systems are linearly degenerate.

1.1.1. Examples of hyperbolic conservation laws. We give here some examples ofhyperbolic conservation laws. The simplest example of a scalar conservation law is the linearadvection equation,

(1.2) ut + aux = 0,

whose unique solution is u(x, t) = u0(x − at). A common example of a nonlinear scalarconservation law is Burgers’ equation,

(1.3) ut +

(u2

2

)x

= 0.

The wave equation in divergence form is the linear system of conservation laws

(1.4)(u1

u2

)t+

(cu2

cu1

)x

= 0,

where c ∈ R is a constant. The system can be written in the form ut + Aux = 0 for adiagonalizable matrix A = RΛR−1 ∈ R2×2. Defining the characteristic variable u := R−1u, wehave the characteristic decomposition ut + Λux = 0. Each equation in this system is a linearadvection equation, which has a unique solution, and hence the system has a unique solution.

A simple but important example of a one-dimensional, nonlinear hyperbolic system ofconservation laws (d = 1, N > 1) is the shallow water equations, which model a body ofwater under the assumption that the horizontal length scales are much larger than the verticallength scale. The velocity w of the water can then be approximated by a depth-averagedvelocity, w(x, z) ≈ w(x). Denoting by h(x) the height (or depth) of the water column at x andby g the gravitational constant, one obtains the system

(1.5)(

hhw

)t+

(hw

12 gh2 + hw2

)x

= 0.

To model a fully three-dimensional body of water, a second spatial dimension may be added[LeV02]. Real-world effects such as a nontrivial bottom topography, the Coriolis force, windetc. may be incorporated by adding appropriate terms.

Another important example is the Euler equations of gas dynamics. Denoting by ρ thegas density, w its velocity, E total energy, and p the pressure, this system of equations is givenby

(1.6)

ρρwE

t

+

ρwρw2 + p(E + p)w

x

= 0.

1.1. BASIC NOTIONS AND EXAMPLES 3

The pressure p is given by an equation of state, which for ideal gases is

E =p

γ − 1+

12ρw2,

where γ > 1 is the adiabatic constant.

1.1.2. The formation of shocks. We give an example that illustrates the main difficultythat arises when dealing with hyperbolic conservation laws.

Consider the scalar, one-dimensional conservation law (1.1) (d = N = 1), and let theinitial data u0 ∈ C1(R) and flux function f ∈ C2(R) be given. We will show the local exis-tence of a classical solution of the Cauchy problem (1.1) using the method of characteristics.Assume for the moment that a classical solution u of (1.1) is given. For (x, t) ∈ R × R+, letthe backwards characteristic ξ(s) be defined by

ξ′(s) = f ′(u(ξ(s), s)) 0 < s < t

ξ(t) = x.

Using (1.1) we see that that ddt u(ξ(s), s) = 0, and hence u(x, t) = u0(ξ(0)). The characteristic

ξ can therefore be solved for explicitly as ξ(s) = ξ(0) + s f ′(u(ξ(s), s)), so it follows that

u(x, t) = u0(ξ(0)) = u0(x − t f ′(u(x, t))

).

Define Φ((x, t), u) := u − u0 (x − t f ′(u)). Since

Φ((x0, 0), u0(x0)) = 0 and∂Φ

∂u((x0, 0), u0(x0)) = 1

for every x0 ∈ R, the implicit function theorem gives the existence of an open set U ⊂ R2

containing (x0, 0) and a C1 function u : U → R such that

u(x, t) = u0(x − t f ′(u(x, t))

)∀ (x, t) ∈ U.

This function is a solution of (1.1); indeed, suppressing function arguments for the moment,

ut + f (u)x =− f ′(u)u′0

1 + tu′0 f ′′(u)+

f ′(u)u′01 + tu′0 f ′′(u)

= 0 ∀ (x, t) ∈ U

for small t. Given any bounded interval [a, b] ⊂ R we can repeat this process on all pointsx0 in a dense, countable subset of [a, b] and find that there is a T > 0 such that a classicalsolution of (1.1) exists in [a, b] × [0,T ].

Next, we demonstrate that in all but the most trivial cases, we cannot hope for the globalexistence of a classical solution. Define the characteristic ξ(x, t) = x + t f ′(u0(x)). From thediscussion above we know that

ux(ξ(x, t), t) =u′0(x)

1 + tu′0(x) f ′′(u0(x))=

u′0(x)

1 + t ddx f ′(u0(x))

.


Hence, if the function x 7→ ddx f ′(u0(x)) is ever negative, then

limt↑t0

ux(ξ(x, t), t) = ∞, t0 := −1

minx∈R

(ddx f ′(u0(x))

) ,and u is no longer differentiable.

1.1.3. Weak solutions. It is clear that we cannot expect solutions to be defined in theclassical sense. Instead, we interpret the differential equation in the sense of distributions.

Definition 1.1. A function u ∈ L∞(Rd × R+,RN) is a weak solution of (1.1) if it satisfies

(1.1) inD′(Rd × R+), that is, if

(1.7)∫Rd×R+

uϕt + f (u)div(ϕ) dxdt +

∫Rd

u0(x)ϕ(x, 0) dx = 0 ∀ ϕ ∈ D(Rd × R+).

Since the x-axis Rd × 0 as a subset of Rd × R+ has measure 0, the expression u(x, 0) =

u0(x) is not necessarily meaningful. However, it may be shown that weak solutions admit theinitial data in the following weak sense:

limT→0

1T

∫ T

0

∫Rd

(u(x, t) − u0(x)

)ψ(x) dxdt = 0 ∀ ψ ∈ D(Rd).

The following result provides an easy way of checking whether a given function is aweak solution.

Theorem 1.2. Let u : Rd×R+ → RN be piecewise differentiable with jump discontinuities

along d-dimensional smooth submanifolds of Rd ×R+ with normals (n,−σ) ∈ S d−1×R. Thenthe following are equivalent:

(i) u is a weak solution of (1.1).(ii) u satisfies (1.1) at its points of differentiability, and along each surface of disconti-

nuity it satisfies the Rankine-Hugoniot jump condition

(1.8) n · [[ f (u)]] − σ[[u]] = 0.

where

[[u]] = u+(x, t) − u−(x, t) and u±(x, t) = limh↓0

u(x ± nh, t).

We call the type of discontinuities as in the previous theorem shocks. In the particularcase d = 1, the types of discontinuities allowed in Theorem 1.2 are curves (γ(t), t) in the x-tplane, and (1.8) reads

[[ f (u)]] − σ[[u]] = 0,

where σ(t) = γ′(t).

1.2. ENTROPY PAIRS AND THE ENTROPY CONDITION 5

Example 1.3. Using the Rankine-Hugioniot condition it is not hard to come up withweak solutions. As an example, consider Burgers’ equation (1.3). We let the initial data begiven by

u0(x) =

0 if x < 01 if 0 6 x.

A Cauchy problem with piecewise constant data separated by a single jump discontinuity,like the above, is called a Riemann problem. A continuous solution of this Riemann problemis given by

u1(x, t) =

0 if x < 0x/t if 0 6 x < t1 if t 6 x.

However, using the Rankine-Hugoniot conditions (1.8), we can easily come up with moreweak solutions, for instance

u2(x, t) =

0 if x < t/21 if t/2 6 x

and u3(x, t) =

0 if x < t/41/2 if t/4 6 x < 3t/41 if 3t/4 6 x.

u2 and u3 trivially satisfy the conservation law where it is constant, and along their lines ofdiscontinuity x = t/2 and x = t/4, x = 3t/4, respectively, the Rankine-Hugoniot condition(1.8) holds, with shock speeds σ = 1/2 and σ = 1/4, σ = 3/4, respectively.

1.2. Entropy pairs and the entropy condition

The example in the previous section demonstrates that the distributional interpretationof the differential equation is too weak to obtain uniqueness. It is clear that we must enforceadditional constraints to single out a unique solution. These constraints come in the form ofentropy conditions.

Definition 1.4. A pair of functions (η, q) with η ∈ C2(RN) and q = (q1, . . . , qd) ∈C2(RN ,Rd) is called an entropy pair for (1.1) provided q′(u)T = η′(u) · f ′(u) for all u ∈ U,i.e. (qk)′(u)T = η′(u) · ( f k)′(u) for k = 1, . . . , d. An entropy pair (η, q) is convex if η isconvex.

Taking the inner product of (1.1) with η′(u) and using the chain rule, we see that smoothsolutions u satisfy the additional conservation law

(1.9) 0 = η′(u) ·(ut + f ′(u) grad(u)

)= η(u)t + div(q(u)).

In particular, integrating (1.9) over Rd × [0,T ] and assuming that lim|x|→∞ q(u(x, t)) = 0 fastenough (say, if u has compact support), we see that∫

Rdη(u(x,T )) dx =

∫Rdη(u0(x)) dx.

Hence, for smooth solutions, the total amount of entropy is constant in time.


The derivation of (1.9) of course only holds when u is differentiable. However, we maygeneralize it to the case of a piecewise differentiable function with jump discontinuities, usingthe same argument as in the proof of the Rankine-Hugoniot condition. If u has a discontinuityalong a surface with normal (n,−σ) ∈ S d−1 × R, then the right-hand side of (1.9) is a Diracδ-mass with weight

η(u)t + div(q(u)) =1

√1 + σ2

(n · [[q(u)]] − σ[[η(u)]]

).

In the one-dimensional case d = 1, the entropy production takes the form1

√1 + σ2

([[q(u)]] − σ[[η(u)]]

), σ := γ′(t)

whenever u has a discontinuity along a curve (γ(t), t).

Example 1.5. Returning to the example from the previous section, we see that u1 satisfiesthe entropy identity (1.9) everywhere for any entropy pair, as it is absolutely continuous. Foru2 and u3, we use the convex entropy pair η(u) = u2

2 , q(u) = u3

3 for concreteness. Then

η(u2)t + q(u2)x

∣∣∣∣x=t/2

=1√

54

(13−

12·

12

)=

1

6√

5> 0,

η(u3)t + q(u3)x

∣∣∣∣x=t/4

= η(u3)t + q(u3)x

∣∣∣∣x=3t/4

=1

96> 0.

Thus, u2 and u3 produce entropy across their discontinuities. Indeed, we shall soon show thatu2 and u3 produce entropy with respect to any convex entropy pair.

The stability condition that we impose is that entropy never increases. In the exampleabove, this would exclude the solutions u2 and u3, but not u1.

Definition 1.6. Let (η, q) be a convex entropy pair for (1.1). A function u ∈ L∞(Rd ×

R+,RN) is admissible with respect to (η, q) if

(1.10) η(u)t + div(q(u)) 6 0

inD′(Rd × R+), that is, if∫Rd×R+

η(u)ϕt + q(u)div(ϕ) dxdt +

∫Rdη(u0(x))ϕ(x, 0) dx > 0

for all 0 6 ϕ ∈ D(Rd ×R+). A weak solution of (1.1) is an entropy solution if it is admissiblewith respect to all convex entropy pairs.

Remark. Formally, we may integrate (1.10) in space and time to find that if |q(u)| → 0as |x| → ∞, then the total amount of entropy of an entropy solution decreases in time:

(1.11)∫Rdη(u(x, t)) dx 6

∫Rdη(u0(x)) dx.

If, say, η(u) = |u|p (p > 1) is an entropy for (1.1), then (1.11) provides an a priori Lp-stabilitybound on u(·, t) for all t > 0.

1.2. ENTROPY PAIRS AND THE ENTROPY CONDITION 7

As for weak solutions, we can derive a necessary and sufficient condition for (1.10) whenthe function u has isolated jump discontinuities.

Proposition 1.7 (Hopf [Hop69]). Let u : Rd × R+ → RN be piecewise differentiable

with jump discontinuities along surfaces with normals (n,−σ) ∈ S d−1 × R, and let (η, q) be aconvex entropy pair. Then the following are equivalent:

(i) u is admissible with respect to (η, q).(ii) u satisfies (1.9) at its points of differentiability, and along each surface of disconti-

nuity it satisfies

(1.12) n · [[q(u)]] − σ[[η(u)]] 6 0.

In the scalar, one-dimensional case, Proposition 1.7 admits a geometrical interpretation,as follows.

Theorem 1.8 (Oleinik [Ole57]). Consider the scalar, one-dimensional conservation law(d = N = 1), and let u be as in Proposition 1.7. Then the following are equivalent:

(i) (1.12) is satisfied for all convex entropy pairs (η, q).(ii)

(1.13)f (u+) − f (u−)

u+ − u−6

f (u) − f (u−)u − u−

for all u between u− and u+.

Proof. We apply the fundamental theorem of calculus to the quantities [[η(u)]] and [[q(u)]]and integrate by parts:

[[η(u)]] =

∫ u+

u−η′(u) du = η′(u)(u − u−)

∣∣∣u+

u=u− −

∫ u+

u−η′′(u)(u − u−) du

= η′(u+)[[u]] −∫ u+

u−η′′(u)(u − u−) du

and

[[q(u)]] =

∫ u+

u−η′(u) f ′(u) du = η′(u)( f (u) − f (u−))

∣∣∣u+

u=u− −

∫ u+

u−η′′(u)( f (u) − f (u−)) du

= η′(u+)[[ f (u)]] −∫ u+

u−η′′(u)( f (u) − f (u−)) du.

Hence,

[[q(u)]] − σ[[η(u)]] = η′(u+)([[ f (u)]] − σ[[u]]

)+

∫ u+

u−η′′(u)

(σ(u − u−) − ( f (u) − f (u−))

)du.

The first term vanishes because of the Rankine-Hugoniot condition (1.8). If the remainingintegral is to be nonpositive for all convex entropies η, then we must have

σ(u − u−) − ( f (u) − f (u−)) 6 0

for all u between u− and u+, which again using the Rankine-Hugoniot condition is precisely(1.13).


(1.13) is the Oleinik E-condition. It says that if u− < u+ then f (u) should lie above thestraight line connecting (u−, f (u−)) and (u+, f (u+)), while if u− > u+ then it should lie below.In the special case where f is strictly convex, we see that a shock is admissible if and onlyif u− > u+, since a convex function lies below all of its chords. This explains the resultsin Example 1.5, where u− = 0 < u+ = 1, and so any weak solution containing a shock isnonadmissible.

1.2.1. Parametrized entropy pairs. A parametrized entropy pair is a pair of functionsη(u, k), q(u, k) for u, k ∈ RN such that (η(·, k), q(·, k)) is an entropy pair for all k and η(u, k) =

η(k, u), q(u, k) = q(k, u). In the scalar case (N = 1), it is easily seen that

(1.14) η(u, k) = |u − k|, q(u, k) = sgn(u − k)( f (u) − f (k))

is a parametrized entropy pair. This is the famous Kruzkov entropy pair. Kruzkov used this,along with his celebrated doubling of variables argument, to show L1-stability of the entropysolution of (1.1) with respect to the initial data.

It is straightforward to generalize the doubling of variables argument to systems of con-servation laws (N > 1); see [Tad03, Theorem 2.2]. However, a general construction offamilies of entropy pairs is only available for certain symmetric systems [Tad87b] and 2 × 2systems [Lax71, Section 3].

1.2.2. Entropy variables. The function u 7→ η′(u) has appeared several times in theprevious sections. This function is called the entropy variable and is denoted by 3(u) := η′(u).If η is strongly convex, η′′ > 0, then u 7→ 3(u) is invertible, and we denote its inverse by u(3).The mapping u 7→ 3 then induces a change of variables, and we can pose the conservationlaw in entropy variables:

u(3)t + f (u(3))x = 0.

This symmetrizes the system, in the sense that the matrix u′(3) =(η′′(u(3))

)−1 is symmetricpositive definite, and d

d3 f (u(3)) is symmetric. To see the latter, let the entropy potential bethe scalar function defined by ψ(3) = 3 · f (u(3)) − q(u(3)). It is easy to verify that ψ′(3) =

f (u(3)), and hence dd3 f (u(3)) = ψ′′(3) is symmetric. This symmetrization was first observed

by Godunov [God61], and was later developed by Mock [Moc80].

1.2.3. Examples of entropies. For a general scalar conservation law (N = 1), all convexfunctions η give rise to an entropy pair (η, q) by defining

q(u) =

∫ u

η′(s) f ′(s) ds.

Thus, scalar conservation laws are endowed with a rich family of entropies, an indispensabletool in the stability analysis of these equations. This extends to certain linear systems ofconservation laws. For the wave equation (1.4), we can let e.g. η(u) = (u1)2 + (u2)2, withcorresponding entropy flux q(u) = uT Au = 2cu1u2.

1.3. EXISTENCE AND UNIQUENESS FOR SCALAR CONSERVATION LAWS 9

The selection of entropies for nonlinear general systems of conservation laws is muchmore limited. For the shallow water equations, the only available entropy is given by the totalenergy of the solution,

(1.15) η(u) =12

(hw2 + gh2

), q(u) =

12

hw3 + gwh2.

The corresponding entropy variable 3 := η′(u) and entropy potential ψ(3) := 3· f (u(3))−q(u(3))are given by

(1.16) 3 =

(gh − w2

2w

), ψ(3) =

12

gwh2.

For the Euler equations (1.6), the relevant entropy pair is given by the entropy of thesolution,

η(u) =−ρsγ − 1

, q(u) =−ρwsγ − 1

,

where the thermodynamic entropy is defined by s := log(p) − γ log(ρ). The entropy variableand entropy potential are

(1.17) 3 =

γ−sγ−1 −

ρw2

2pρwp−ρp

, ψ(3) = ρw.

Harten [Har83] generalized this to an entire family of entropy pairs; however, only the aboveentropy pair simultaneously symmetrizes the Navier-Stokes equations when heat conductionis taken into account [HFM86].

1.3. Existence and uniqueness for scalar conservation laws

It was shown by Kruzkov in 1970 that for scalar conservation laws, there exists a uniqueentropy solution in the class of functions of bounded variation.

Theorem 1.9 (Kruzkov [Kru70]). Consider the scalar conservation law (1.1). For eachu0 ∈ L∞(Rd), there exists a unique entropy solution u of the corresponding Cauchy problem.This function satisfies

‖u(·, t)‖L∞(Rd) 6 ‖u0‖L∞(Rd) for all t > 0,

and it attains the initial data in the following L1 sense: there is a set T ⊂ [0,∞) of measurezero such that for all x0 ∈ R and r > 0,

limt↓0, t<T

∫Br(x0)

|u(x, t) − u0(x)| dx = 0.

If u, u : Rd × R+ → R are entropy solutions of (1.1) with initial datum u0, u0 ∈ L∞(Rd) ∩L1(Rd), respectively, then

(1.18)∫Rd|u(x, t) − u(x, t)| dx 6

∫Rd|u0(x) − u0(x)| dx for all t > 0.


In particular, if u0 ∈ BV(Rd), then

TV(u(·, t)) 6 TV(u0) for all t > 0.

The L1-stability with respect to initial data stated in the previous theorem may be gener-alized to arbitrary generalized entropy pairs; see [Tad03, Theorem 2.2].

If we restrict ourselves to uniformly convex flux functions, then it is enough to imposeadmissibility with respect to a single entropy pair, as seen in the following theorem.

Theorem 1.10 (Panov [Pan94]). Consider the scalar, one-dimensional conservation law(1.1). Assume that f is uniformly convex and u0 ∈ L∞(R), and let u ∈ L∞(R × R+) be a weaksolution that is admissible with respect to a strictly convex entropy pair (η, q). Then u is theentropy solution.

1.4. Compensated compactness

Kruzkov originally proved the existence part of Theorem 1.9 by showing that the solu-tions of a parabolic regularization of (1.1) converge strongly to an entropy solution as thediffusion coefficient goes to zero. The proof relies on a priori stability bounds on the regular-ized equation, most importantly a local bound on the total variation of the solution.

More generally, to show the convergence of a sequence of approximate solutions uεε>0,a common approach is to show that the sequence has uniformly bounded total variation, andhence, by Helly’s theorem, has a strongly convergent subsequence. However, this approach isnot always practicable; most importantly for us, one cannot hope for uniform total variationbounds for high-order accurate numerical methods.

The compensated compactness method, due to Murat and Tartar [Mur78, Tar79], seeksto overcome this difficulty by imposing a much weaker bound on the gradients of the approx-imating sequence. More precisely, it is asserted that the entropy residual η(uε)t +q(uε)x lies ina compact subset of H−1

loc(R × R+). Using Murat’s div-curl lemma, this gives enough controlon oscillations to be able to pass to the (strong) limit ε→ 0.

In this section we give a brief introduction to the compensated compactness method, asapplied to scalar one-dimensional conservation laws. The approach that we present relies onYoung measures, and a short summary of Young measures is given in Appendix A.

The following result, due to Murat [Mur78], is the main result in the theory of compen-sated compactness.

Lemma 1.11 (Div-curl lemma). Let Ω ⊂ R2 be open. Let Dε, Eε ∈ L2(Ω,R2) be suchthat Dε D, Eε E weakly in L2(Ω,R2) as ε→ 0, and assume that

(1.19) div(Dε)ε>0 and curl(Eε)ε>0 are precompact in H−1(Ω).

Then

(1.20) Dε · Eε → D · E inD′(Ω).

(Here, we have denoted div(D) = ∂D1∂y1

+ ∂D2∂y2

, curl(D) = ∂D2∂y1−

∂D1∂y2

and D · E = D1E1 + D2E2.)

1.4. COMPENSATED COMPACTNESS 11

The div-curl lemma is applied to the functions

(1.21) Dε :=(

f (uε)uε

), Eε :=

(−η(uε)q(uε)

)for some entropy pair (η, q). We then have

div(Dε) = uεt + f (uε)x, curl(Eε) = η(uε)t + q(uε)x.

By asserting that these quantities lie in a “nice” set, we are providing enough control onoscillations in uε to conclude that the weak convergence uε u is in fact strong.

The main result of this section, which we pose next, gives the connection between boundson the entropy residual η(uε)t + q(uε)x and the strong convergence of the approximating se-quence. The result was originally shown by Tartar [Tar79], who asserted precompactnessof these sequences for all entropy pairs (η, q). We pose two versions of the theorem, bothof which only require precompactness for two entropy pairs: if the flux function f is strictlyconvex, then we may choose any strictly convex entropy pair (Theorem 1.12); for generalflux functions we have to select a specific entropy pair (Theorem 1.13).

Theorem 1.12. Let Ω ⊂ R2 be open and bounded. Let uεε>0 be a bounded sequence inL∞(Ω) such that

(1.22) uεt + f (uε)xε>0 and η(uε)t + q(uε)xε>0 are precompact in H−1(Ω),

for a strictly convex entropy pair (η, q), and assume that f is strictly convex. Then uεε>0has a subsequence that converges pointwise a.e. and in Lp(Ω) for all p ∈ [1,∞) to a functionu ∈ L∞(Ω).

Proof. Let Dε, Eε be defined by (1.21). By the Fundamental Theorem of Young mea-sures, there is a subsequence (still indexed by ε) such that

Dε D :=(

fu

), Eε E :=

(−ηq

)weakly-∗ in L∞(Ω,R2),

and since Dε, Eε are L∞-bounded on a bounded domain, the weak convergence is also inL2(Ω,R2). The compactness criterium (1.19) follows from (1.22), so by the div-curl lemma,we have Dε · Eε D · E in D′(Ω). But then also Dε · Eε D · E weakly-∗ in L∞, whichusing the Young measure ν can be written as

(1.23) 〈νy, λ〉〈νy, q〉 − 〈νy, η〉〈νy, f 〉 = 〈νy, λq − η f 〉 a.e. y ∈ Ω,

or more compactly, u q − η f = λq − η f . This is the so-called Murat-Tartar commutatorrelation. Using this relation, we will show that the function

H(a, b) := (b − a)(q(b) − q(a)

)−

(η(b) − η(a)

)(f (b) − f (a)

)satisfies the conditions of Lemma A.3, and consequently, by Lemma A.2, the convergenceuε → u is strong.


Clearly, H(a, a) = 0. We have

∂H∂b

(a, b) = q(b) − q(a) + (b − a)q′(b) − (η(b) − η(a)) f ′(b) − η′(b)( f (b) − f (a))

and∂2H∂b2 (a, b) = η′′(b)((b − a) f ′(b) − ( f (b) − f (a))) + f ′′(b)((b − a)η′(b) − (η(b) − η(a)))

= η′′(b)∫ b

af ′(b) − f ′(s) ds + f ′′(b)

∫ b

aη′(b) − η′(s) ds.

Hence, ∂H∂b (a, a) = 0 for all a ∈ R, and if b > a then ∂2H

∂b2 (a, b) > 0, with equality only for b ina set of measure zero, since f and η are strictly convex. Hence, H(a, b) > 0 for all a < b, andas H(b, a) = H(a, b), we have H(a, b) > 0 for all a , b. Last, we have for a.e. y ∈ Ω∫

R2H d(νy × νy) =

∫R

∫R

(b − a)(q(b) − q(a)

)−

(η(b) − η(a)

)(f (b) − f (a)

)dνy(b)dνy(a)

= 2(〈νy, idq〉 − 〈νy, id〉〈νy, q〉 + 〈νy, η f 〉 − 〈νy, η〉〈νy, f 〉

)= 0,

where we have used Fubini’s theorem, the fact that νy(R) = 1, and (1.23). Hence, H satisfies(A.1).

Theorem 1.13 (Chen-Lu [CL89]). Let Ω ⊂ R2 be open and bounded. Let uεε>0 bea bounded sequence in L∞(Ω) such that (1.22) is satisfied with the entropy pair η = f ,q(u) =

∫ uf ′(s)2 ds. Assume that

(1.24) meass ∈ R : f ′′(s) = 0

= 0.

Then uεε>0 has a subsequence that converges pointwise a.e. and in Lp(Ω) for all p ∈ [1,∞)to a function u ∈ L∞(Ω).

Proof of Theorem 1.13. The proof follows that of Theorem 1.12 closely. The Murat-Tartar commutator relation (1.23) holds as before, but in this case the function H is

H(a, b) := (b − a)(q(b) − q(a)

)−

(f (b) − f (a)

)2.

Clearly, H(a, a) = 0. By the Cauchy-Schwarz inequality we have for every a < b

( f (b) − f (a))2 =

(∫ b

af ′(s) ds

)2

6 (b − a)∫ b

af ′(s)2 ds = (b − a)(q(b) − q(a)),

so H(a, b) > 0. But there is equality in the Cauchy-Schwarz inequality if and only if f ′

is constant on (a, b), which by (1.24) it cannot be. Hence, H(a, b) > 0 for all a < b, andsince H(a, b) = H(b, a), the first part of (A.1) follows. The second part follows by the sameargument as in Theorem 1.12, so in conclusion, ν is a function, i.e. νy = δu(y) for a.e. y ∈ Ω

for a u ∈ L∞(Ω).

Definition 1.14. Functions satisfying (1.24) are called genuinely nonlinear.

1.6. MEASURE-VALUED SOLUTIONS 13

1.5. Existence and uniqueness for systems of conservation laws

For general N × N systems of hyperbolic conservation laws there is no global well-posedness result like Kruzkov’s Theorem 1.9. However, some partial results exist. Laxshowed in his seminal paper [Lax57] existence and uniqueness of the entropy solution ofthe one-dimensional Riemann problem under the assumption that the two states are suffi-ciently close. Glimm [Gli65] proved existence of a weak solution to the Cauchy problemfor all u0 ∈ D, the L1-closure of the set of all piecewise constant functions with “sufficientlysmall” total variation. This was accomplished by showing that his random choice methodis stable with respect to what is now called the Glimm functional. More recently, Bressanet al. [BLY99, BCP00] showed that the front tracking method is stable with respect to theGlimm functional, and that it converges to the unique entropy solution for all u0 ∈ D. Wesum this up as follows.

Theorem 1.15. Let d = 1 and assume that the system (1.1) is strictly hyperbolic andthat all wave families are either genuinely nonlinear or nonlinearly degenerate. Then for allu0 ∈ L1 with sufficiently small total variation, there exists a unique entropy solution to theCauchy problem (1.1).

1.6. Measure-valued solutions

Summarizing the remarks in the previous section, there is a lack of global, “large data”existence and uniqueness results for systems of hyperbolic conservation laws. We describehere the alternative approach of measure-valued solutions, initiated by DiPerna in his seminalpaper [DiP85]. Unlike weak solutions, which are (almost everywhere defined) functions,a measure-valued solution is a Young measure, a mapping which to each point assigns aprobability distribution, describing the likely value at that point. This should be seen as anatural solution concept for evolution equations: for instance, one rarely, if ever, know theprecise initial conditions, as all measuring processes have some inherent uncertainty.

When a sequence of functions converges pointwise to a function u, then the Young mea-sure ν associated with the sequence is a unit mass concentrated at u, ν(x,t) = δu(x,t). In thiscase 〈ν(x,t), g(λ)〉 = g(〈ν(x,t), λ〉) = g(u(x, t)) for all (x, t) ∈ Rd × R+ and g ∈ C(RN). Hence, ifu were a weak solution of (1.1) then

〈ν, λ〉t + div(〈ν, f 〉) = 0 inD′(Rd × R+).

This expression is well-defined also when ν is not everywhere a point mass (although in thiscase the relation 〈ν, f (λ)〉 = f (〈ν, λ〉) is not satisfied). We take this as the definition of ameasure-valued solution of (1.1).

Definition 1.16 (DiPerna [DiP85]). A Young measure ν defined on Rd×R+ is a measure-valued solution of (1.1) if

(1.25) 〈ν, λ〉t + div(〈ν, f 〉) = 0 inD′(Rd × R+

).


If (η, q) is a convex entropy pair then a measure-valued solution ν of (1.1) is admissible withrespect to (η, q) if

(1.26) 〈ν, η〉t + div(〈ν, q〉) 6 0 inD′(Rd × R+).

It is called an entropy measure-valued solution of (1.1) if it is admissible with respect to allconvex entropy pairs (η, q).

To see that the concept of (entropy) measure-valued solutions is weaker than that of weaksolutions, it is enough to note that definitions (1.25), (1.26) are linear in ν; hence, if ν0 and ν1

are two (entropy) measure-valued solutions, then so is

νθ := (1 − θ)ν0 + θν1, θ ∈ (0, 1).

The definition (1.7) of weak solutions, on the other hand, is in general nonlinear in u, andso an analogously defined function uθ will not be a weak solution. See Schochet [Sch89] forfurther examples of measure-valued solutions.

It was shown in [DiP85] that for scalar conservation laws, an entropy measure-valuedsolution that has initial Dirac mass u0(x) coincides with the corresponding entropy solutionat all later times – what is known as weak-strong uniqueness. This was proved by showingthat if ν is an entropy measure-valued solution and u is the entropy solution of the Cauchyproblem (1.1), then for all parametrized entropy pairs (η, q) (cf. Section 1.2.1) the followingstability bound holds:

〈νy, η(u(y), λ)〉t + div(〈νy, q(u(y), λ)〉

)6 0 inD′(Rd × R+).

Letting (η, q) be the Kruzkov entropy pair (1.14), then the above implies that the functionV(t) :=

∫Rd 〈ν(x,t), |λ − u(x, t)|〉 dx is nonincreasing, analogous to the stability bound (1.18) for

entropy solutions. In particular, if the initial data u0 is assumed in a sufficiently strong sense,then V(t) = 0 for all times, and hence ν(x,t) = δu(x,t) for almost all (x, t).

More recently, Brenier, De Lellis and Szekelyhidi showed weak-strong uniqueness formeasure-valued solutions of the incompressible Euler equations [BLS11]. They also provedthat if there is a Lipschitz continuous solution of the hyperbolic system (1.1), then any entropymeasure-valued solution of (1.1) coincides with the classical solution.

1.7. Summary and outlook

Based on the discussion in this chapter, we may draw the following conclusions. Thetheory of existence, uniqueness and regularity for scalar conservation laws is mature and well-developed. The availability of a large class of entropies enables a full stability analysis, henceuniqueness of entropy solutions. Convergence frameworks such as functions of total variationand compensated compactness allows for passing to the limit in approximate solutions, thusimplying existence of entropy solutions.

For hyperbolic systems of equations, much less is known. No existence or uniquenessresults for “large data” are available for general systems. The weaker concept of measure-valued solutions makes it easier to show existence of solutions, with the possible disadvantageof allowing for pathological solutions.

1.7. SUMMARY AND OUTLOOK 15

In view of these remarks, it seems that the most that one can hope for from a numericalapproximation of (1.1) is that it

• is consistent with respect to entropy conditions,• is high-order accurate,• converges strongly to the entropy solution for scalar conservation laws,• converges to a measure-valued solution for systems of conservation laws.

The remainder of this thesis is devoted to developing numerical schemes satisfying theseproperties.

CHAPTER 2

Entropy stable methods of first order

As has been demonstrated in the previous chapter, entropy conditions play a crucial rolein the stability analysis of hyperbolic conservation laws. In this chapter we consider finitevolume methods which satisfy a discrete version of the entropy admissibility criterion (1.10)for convex entropy pairs. Such schemes will be called entropy stable methods. Entropy sta-bility was studied first in Lax’ important 1971 paper [Lax71]. There, he established that theLax-Friedrichs method is entropy stable, and as a corollary, that, should the method con-verge pointwise, then the limit is the unique entropy solution. This result was generalized byHarten, Hyman and Lax in 1976, who showed that all monotone schemes for scalar conser-vation laws are entropy stable [HHLK76]. Osher developed in his 1984 paper [Osh84] thetheory of E-schemes, of which monotone methods are a subset. E-schemes are designed pre-cisely to be entropy stable with respect to all convex entropies, and as they are total variationdiminishing, they converge to the entropy solution.

As was shown in [Osh84], all E-schemes are at most first-order accurate. The worktowards obtaining higher-order accurate entropy stable schemes was initiated by Tadmor[Tad84], [Tad87a]. He found a convenient relation that guarantees that a method is entropyconservative, meaning that it satisfies a discrete version of the entropy equality (1.9). Suchschemes are automatically second-order accurate, and methods of higher (than second) orderhave been developed more recently in [LMR03]. However, the lack of diffusion in entropyconservative methods will lead to oscillations in the vicinity of shocks. By adding numericaldiffusion we get rid of such oscillations, but generally at the cost of accuracy. In Chapter 4we will design higher-order numerical diffusion operators. In the present chapter we restrictourselves to first-order schemes.

In Section 2.2 we review the theory of entropy conservative and entropy stable schemes.In Sections 2.3–2.5 we demonstrate how to construct, and give concrete examples of, entropystable schemes for scalar equations and for systems. The interested reader is referred toTadmor’s review paper [Tad03] for further details.

For simplicity we only consider the one-dimensional hyperbolic conservation law (1.1),d = 1, although everything can easily be generalized to multi-dimensional equations.

17

18 2. ENTROPY STABLE METHODS OF FIRST ORDER

2.1. Finite volume and finite difference methods

In its simplest, one-dimensional form, a finite volume (finite difference) method for (1.1)is a method of the form

(2.1)ddt

ui +Fi+1/2 − Fi−1/2

∆x= 0,

here written in the semi-discrete form. The numerical flux function Fi+1/2 is a function of2m (m ∈ N) variables, Fi+1/2 = F(ui−m+1, . . . , ui+m), and we require that it is consistent, i.e.,that F(u, . . . , u) = f (u) for all u ∈ U. The spatial domain x ∈ R is partitioned into cellsIi = [xi−1/2, xi+1/2) with uniform grid size xi+1/2 − xi−1/2 ≡ ∆x and we approximate ui(t) ≈1

∆x

∫Ii

u(x, t) dx (finite volume) or ui(t) ≈ u(xi, t) (finite difference). The initial data is sampledas ui(0) := 1

∆x

∫Ii

u0 dx (finite volume) or ui(0) := u0(xi) (finite difference).In a practical implementation the temporal dimension must also be discretized, but for

clarity of exposition we will leave it continuous. In numerical experiments we integrate intime using strong stability preserving third-order Runge-Kutta methods from [SO88, GS98].These methods consist of subsequent applications and convex combinations of forward Eulersteps,

un+1i = un

i −∆t∆x

(Fn

i+1/2 − Fni−1/2

).

Standard stability analysis dictates that ∆t should be chosen such that the CFL number∆t∆x maxi∈Z | f ′(ui)| is less than 1; see [SO88, LeV02].

2.2. Entropy conservative and entropy stable methods

Recall that an entropy pair (η, q) for (1.1) is a pair of functions η, q : RN → R such thatq′(u) = η′(u)T f ′(u). In the remainder we will assume that η is strictly convex.

Definition 2.1. Let (η, q) be a convex entropy pair. We say that the finite volume method(finite difference method) (2.1) is entropy conservative if computed solutions satisfy the dis-crete entropy equality

(2.2)ddtη(ui) +

Qi+1/2 − Qi−1/2

∆x= 0,

for some 2m-point numerical flux function Qi+1/2 = Q(ui−m+1, . . . , ui+m) such that Q(u, . . . , u) =

q(u). We say that it is entropy stable if computed solutions satisfy the discrete entropy in-equality

(2.3)ddtη(ui) +

Qi+1/2 − Qi−1/2

∆x6 0.

Clearly, (2.2) is a discrete version of the entropy equality (1.9), whereas (2.3) is a discreteversion of the entropy stability criterion (1.10). Summing (2.3) over i ∈ Z and integrating

2.2. ENTROPY CONSERVATIVE AND ENTROPY STABLE METHODS 19

over t ∈ [0,T ], we see that the total amount of entropy of an entropy stable scheme decreasesin time:

(2.4)∑

i

η(ui(T )) ∆x 6∑

i

η(ui(0)) ∆x,

which is a discrete version of (1.11).Presently we find sufficient conditions for entropy conservation and entropy stability.

Recall from Section 1.2.2 that the entropy potential is defined as ψ(3) := 3 · f (u(3)) − q(u(3)),where 3(u) = η′(u) is the entropy variable, and that it satisfies ψ′(3) = f (u(3)). As was seen inSection 1.2, the entropy identity (1.9) follows by multiplying the conservation law by entropyvariables:

0 = 3 · (ut + f (u)x) = η(u)t + q(u)x.

The use of the chain rule in the computation 3 · f (u)x = q(u)x may be stated in terms of theentropy potential, as follows:

3 · f (u)x = (3 · f (u))x − 3x · f (u)

= (3 · f (u) − ψ(3))x − (3x · f (u) − 3x · ψ′(3))= q(u)x,

where we in the second line have added and subtracted ψ(3)x = 3x · ψ′(3). We mimic this

use of the chain rule for the scheme (2.1). Denote 3i = 3(ui), ψi = ψ(3i), etc., and let ui+1/2 =12 (ui + ui+1) and [[u]]i+1/2 = ui+1 − ui. Then, multiplying the flux part of (2.1) by 3i, we obtain

3i ·Fi+1/2 − Fi−1/2

∆x=3i+1/2 · Fi+1/2 − 3i−1/2 · Fi−1/2

∆x−

[[3]]i+1/2 · Fi+1/2 + [[3]]i−1/2 · Fi−1/2

2∆x

=

(3i+1/2 · Fi+1/2 − ψi+1/2

)−

(3i−1/2 · Fi−1/2 − ψi−1/2

)∆x

−

([[3]]i+1/2 · Fi+1/2 − [[ψ]]i+1/2

)+

([[3]]i−1/2 · Fi−1/2 − [[ψ]]i−1/2

)2∆x

.

Hence, if we define Qi+1/2 = Q(ui, ui+1) := 3i+1/2 · Fi+1/2 − ψi+1/2, then (2.1) multiplied by 3i isthe same as

(2.5)ddtη(ui) +

Qi+1/2 − Qi−1/2

∆x=

ri+1/2 + ri−1/2

2∆x,

where ri+1/2 := [[3]]i+1/2 · Fi+1/2 − [[ψ]]i+1/2 is the local entropy production.If we denote 3(s) := 3i + s[[3]]i+1/2, then

ri+1/2 = [[3]]i+1/2 · Fi+1/2 −

∫ 1

0

ddsψ(3(s)) ds

=

∫ 1

0[[3]]i+1/2 · Fi+1/2 − [[3]]i+1/2 · ψ

′(3(s)) ds

=

∫ 1

0[[3]]i+1/2 ·

(Fi+1/2 − f (u(3(s)))

)ds.


Hence, if the above integrand is everywhere nonpositive, then (2.5) implies that the methodis entropy stable. Schemes that satisfy this are the so-called E-schemes.

Definition 2.2 (Osher [Osh84]). An E-scheme is a scheme (2.1) whose numerical flux Fsatisfies

(2.6) (3i+1 − 3i) ·(Fi+1/2 − f (u(3(s)))

)6 0 ∀ s ∈ [0, 1]

(where 3(s) := 3i + s(3i+1 − 3i)) for all 3i, 3i+1 and all convex entropy pairs (η, q).

In the scalar case (N = 1), (2.6) is equivalent to

sgn(ui+1 − ui)(Fi+1/2 − f (u)

)6 0 ∀ u between ui and ui+1,

since [[3]]i+1/2 = 1η′′(ξ) [[u]]i+1/2 for some ξ between ui and ui+1. The latter condition is indepen-

dent of the entropy pair, and hence holds for all entropy pairs. Osher showed in [Osh84]that E-schemes for scalar conservation laws converge strongly to the unique entropy solution.However, it was also shown that E-schemes are at most first order accurate. Hence, shouldwe simultaneously wish for entropy stability and high-order accuracy, we cannot demandentropy stability with respect to all convex entropy pairs.

Going back to (2.5), it is obvious that if ri+1/2 = 0 for all i then the scheme is entropyconservative. This is precisely Tadmor’s entropy conservation condition.

Theorem 2.3 (Tadmor [Tad87a]). If F satisfies

(2.7) [[3]]i+1/2 · Fi+1/2 = [[ψ]]i+1/2

for all i, then the scheme (2.1) is entropy conservative with numerical entropy flux

Qi+1/2 = 3i+1/2 · Fi+1/2 − ψi+1/2.

Moreover, in the scalar case (N = 1) it is second-order accurate in smooth regions of u.

Proof. Entropy conservation follows directly from (2.5) and the definition of ri+1/2. Ac-curacy in this case means that

Fi+1/2 − Fi−1/2

∆x= f (u(x))x

∣∣∣x=xi

+ O(∆x2).

Indeed, by the trapezoidal rule,

Fi+1/2 =Fi+1/2[[3]]i+1/2

[[3]]i+1/2

=[[ψ]]i+1/2

[[3]]i+1/2

=1

[[3]]i+1/2

∫ 3i+1

3i

ψ′(3) d3

=1

[[3]]i+1/2

∫ 3i+1

3i

f (u(3)) d3 =f (ui) + f (ui+1)

2−

[[3]]2i+1/2

12ψ′′(ξi+1/2)

for a ξi+1/2 between 3i and 3i+1. Hence,

Fi+1/2 − Fi−1/2

∆x=

f (ui+1) − f (ui−1)2∆x

+ O(∆x2) = f (u(x))x

∣∣∣x=xi

+ O(∆x2).

2.3. ENTROPY STABLE METHODS FOR SCALAR EQUATIONS 21

From (2.5) it is obvious that if ri+1/2 6 0 for all i, then the method is entropy stable.Rather than stating this as a theorem, we state the result in terms of the numerical diffusionof the scheme. Let Fi+1/2 be any numerical flux, and let Fi+1/2 be a flux satisfying (2.7). Byinspecting Fi+1/2 − Fi+1/2, it is not hard to show that there is a matrix Di+1/2 such that

(2.8) Fi+1/2 = Fi+1/2 − Di+1/2[[3]]i+1/2.

Inserting into ri+1/2, we obtain the following result.

Theorem 2.4 (Tadmor [Tad87a]). Assume that Di+1/2 > 0. Then the scheme with flux(2.8) is entropy stable, with the numerical entropy flux

Qi+1/2 = 3i+1/2 · Fi+1/2 − ψi+1/2 − 3i+1/2 · Di+1/2[[3]]i+1/2.

More precisely, it satisfies the entropy dissipation estimate

(2.9)ddtη(ui) +

Qi+1/2 − Qi+1/2

∆x= −

[[3]]i+1/2 · Di+1/2[[3]]i+1/2 + [[3]]i−1/2 · Di−1/2[[3]]i−1/2

2∆x.

Proof. We have

ri+1/2 = [[3]]i+1/2 ·(Fi+1/2 − Di+1/2[[3]]i+1/2

)− [[ψ]]i+1/2

= −[[3]]i+1/2 · Di+1/2[[3]]i+1/2 6 0

because F satisfies (2.7); hence, entropy stability follows from (2.5).

This results suggests a way of designing entropy stable schemes: first, find an entropyconservative numerical flux F through the relation (2.7), and then add numerical diffusion inthe form (2.8). This will be a recurring theme throughout this thesis.

From here on we shall denote all entropy conservative numerical flux functions by F,and entropy stable numerical fluxes by F.

2.3. Entropy stable methods for scalar equations

In the scalar, one-dimensional case (N = d = 1) the construction of an entropy conser-vative numerical flux is trivial using the entropy conservation criterion (2.7): given a convexentropy pair (η, q), define Fi+1/2 by

Fi+1/2 =

[[ψ]]i+1/2

[[3]]i+1/2

if ui , ui+1

f (ui) if ui = ui+1.

Thus, there is a unique three-point entropy conservative scheme for any entropy pair [Tad87a].

Example 2.5. Consider the linear advection (1.2). For simplicity we choose the squareentropy η(u) = u2

2 . Then

3(u) = u, q(u) =u2

2, ψ(u) =

u2

2.

We obtain the entropy conservative flux Fi+1/2 =12

[[u2]]i+1/2

[[u]]i+1/2

= ui+1/2.


Example 2.6. Consider Burgers’ equation (1.3). Again we can take the square entropyη(u) = u2

2 . We then easily find that

3(u) = u, q(u) =u3

3, ψ(u) =

u3

6.

Thus, a flux for Burgers’ equation that is entropy conservative with respect to η must satisfy[[u]]i+1/2Fi+1/2 = [[u3]]

6 , which is satisfied if and only if

Fi+1/2 =u2

i + uiui+1 + u2i+1

6.

Incidentally, this flux is also entropy conservative with respect to the entropy η(u) := (u−k)2

2for any fixed k ∈ R.

If Di+1/2 is any nonnegative number, then the flux (2.8) is entropy stable with respect to(η, q). By the mean value theorem there is a ξi+1/2 between ui and ui+1 such that [[3]]i+1/2 =

3′(ξi+1/2)[[u]]i+1/2 = η′′(ξi+1/2)[[u]]i+1/2. Since η is convex, we have Pi+1/2 := η′′(ξi+1/2)Di+1/2 > 0,and so (2.8) admits the equivalent formulation

(2.10) Fi+1/2 = Fi+1/2 − Pi+1/2[[u]]i+1/2.

We may now choose Pi+1/2 to be any nonnegative number. For instance, we may choose theLax-Friedrichs diffusion

Pi+1/2 :≡12

maxj∈Z

(| f ′(u j)|).

or the Local Lax-Friedrichs diffusion

Pi+1/2 :=12

max(| f ′(ui)|, | f ′(ui+1)|

).

It is clear that the construction of a first-order accurate entropy stable method for scalarconservation laws is more or less trivial. What remains, however, is to choose for whichentropy pair the scheme should be entropy stable. As shown in (2.4), a scheme that is entropystable with respect to (η, q) has nonincreasing total entropy

∑i η(ui(t))∆x. Thus, setting η(u) =

|u|r for an r ∈ [1,∞) gives an Lr-stable scheme,∑i

|ui(t)|r ∆x 6∑

i

|ui(0)|r ∆x.

By a specific choice of η, this extends to the case r = ∞.

Proposition 2.7. Suppose that u0 ∈ L∞(Rd) has compact support. Let a, b ∈ R such thata < infx u0(x) 6 supx u0(x) < b and let (η, q) be the uniformly convex entropy pair with

(2.11) η(u) :=1

u − a+

1b − u

.

Then any scheme that is entropy stable with respect to (η, q) satisfies a < ui(t) < b for alli ∈ Z, t > 0 and ∆x > 0.

2.4. ENTROPY CONSERVATIVE SCHEMES FOR SYSTEMS 23

Proof. By a translation we may assume that q(0) = 0, and hence the total entropy stabilityestimate (2.4) holds. For i ∈ Z and t > 0, denote by u∆x

i (t) the solution at (xi, t) computed withgrid size ∆x > 0. Since η(u) > 0 for all u ∈ (a, b), (2.4) implies that η

(u∆x

i (t))6 E∆x(0)

∆x < ∞

for all i ∈ Z, where E∆x(t) :=∑

j η(u∆xj (t))∆x. If a∆x and b∆x are the unique numbers such that

a < a∆x < b∆x < b and η(a∆x) = η

(b∆x) =

E∆x(0)∆x , then necessarily a∆x 6 u∆x

i (t) 6 b∆x for alli ∈ Z, t > 0 and ∆x > 0. In particular, a < u∆x

i (t) < b for all i ∈ Z, t > 0 and ∆x > 0.

Choosing the entropy (2.11) thus automatically gives an L∞-bound, an important prop-erty in many applications. Unfortunately, the singularities of η at u = a and u = b makes theanalysis of the resulting scheme difficult.

2.4. Entropy conservative schemes for systems

For systems of conservation laws (N > 1), (2.7) is a scalar equation for N unknownsF1

i+1/2, . . . , FN

i+1/2, and so in general there is no unique solution. Tadmor [Tad87a] derived one

general solution Fi+1/2 as follows: given ui, ui+1, let 3(s) = 3i + s[[3]]i+1/2. Then

[[ψ]]i+1/2 =

∫ 1

0

ddsψ(3(s)) ds = [[3]]i+1/2 ·

∫ 1

0ψ′(3(s)) ds = [[3]]i+1/2 ·

∫ 1

0f (u(3(s))) ds.

Hence, the numerical flux

(2.12) Fi+1/2 :=∫ 1

0f (u(3(s))) ds

is automatically entropy conservative. As an example we can consider the linear wave equa-tion (1.4) with the entropy pair η(u) = (u1)2 + (u2)2, q(u) = 2cu1u2. Then the entropy variableis simply 3(u) = u, which inserted into (2.12) gives

(2.13) Fi+1/2 =

∫ 1

0A(ui + s[[u]]i+1/2) ds =

12

(Aui + Aui+1) ,

which when inserted into the scheme (2.1) gives the central schemeddt

ui +f (ui+1) − f (ui−1)

2∆x= 0.

Unfortunately, the integral (2.12) may be very hard to evaluate for even the simplestnonlinear systems. Instead, we shall take a more direct approach, relying on explicit case-by-case computations. As an example we consider the shallow water equations (1.5), with theenergy (1.15) as an entropy. Inserting the entropy variable and entropy potential (1.16) into(2.7), we find that an entropy conservative flux Fi+1/2 = (F1

i+1/2, F2

i+1/2) must satisfy

F1[[gh]] −12

F1i+1/2[[w

2]] + F2[[w]] =12

g[[h2w]]

(where we have suppressed subindices for the moment). Using the identity

(2.14) [[ab]] = a[[b]] + [[a]]b


and grouping together jumps in the same variable, we obtain

g[[h]](F1 − hw

)+ [[w]]

(F2 − F1w −

12

gh2

)= 0.

Hence, if

(2.15) Fi+1/2 =

(F1

i+1/2

F2i+1/2

)=

hi+1/2wi+1/2

hi+1/2w2i+1/2 + 1

2 gh2i+1/2

then Fi+1/2 satisfies (2.7) and hence is entropy conservative. However, this is not the only so-lution. For instance, with a different choice of F1 we can obtain another entropy conservativeflux,

(2.16) Fi+1/2 =

(F1

i+1/2

F2i+1/2

)=

(hwi+1/2

hwi+1/2wi+1/2 + 12 ghihi+1

).

Lemma 2.8 ([Fjo09, FMT09]). The scheme (2.1) with F given by either (2.15) or (2.16)is second-order accurate and entropy conservative with respect to the energy (1.15).

For the Euler equations (1.6), the process is more involved since the entropy is a log-arithmic function of the conserved variables. Ismail and Roe [IR09] derived an entropyconservative numerical flux based on the logarithmic average

alni+1/2 :=

[[a]]i+1/2

[[log a]]i+1/2

.

Define the parameter vectors z as

(2.17) z =

z1

z2

z3

:=√ρ

p

1up

.Then the entropy conservative flux of [IR09] is Fi+1/2 =

(F1

i+1/2, F2

i+1/2, F3

i+1/2

)Twith

(2.18)

F1i+1/2 := z2

i+1/2(z3)lni+1/2

F2i+1/2 :=

z3i+1/2

z1i+1/2

+z2

i+1/2

z1i+1/2

F1i+1/2

F3i+1/2 :=

12

z2i+1/2

z1i+1/2

γ + 1γ − 1

(z3)lni+1/2

(z1)lni+1/2

+ F2i+1/2

.

2.5. ENTROPY STABLE SCHEMES FOR SYSTEMS 25

2.5. Entropy stable schemes for systems

Given an entropy conservative flux Fi+1/2, we add diffusion in the form (2.8) to obtainentropy stability. Any positive matrix Di+1/2 will do; we give here two choices, based onexisting finite volume schemes. Recall that the Rusanov flux is defined as

Fi+1/2 :=f (ui) + f (ui+1)

2−

ci+1/2

2[[u]]i+1/2

for ci+1/2 = maxk=1,...,N (|λk(ui)|, |λk(ui+1)|), where λk(u) is the k-th eigenvalue of f ′(u). TheRoe flux is defined as

Fi+1/2 :=f (ui) + f (ui+1)

2−

12

Ri+1/2|Λi+1/2|R−1i+1/2[[u]]i+1/2,

where Ri+1/2 = (r1(ui+1/2), . . . , rN(ui+1/2)) and Λi+1/2 = diag(λ1(ui+1/2), . . . , λN(ui+1/2)) are the ma-trices of eigenvectors and eigenvalues, respectively, of f ′(ui+1/2), and |Λ| := diag(|λ1|, . . . , |λN |).Here, ui+1/2 is an appropriately chosen intermediate state [Roe81]. Both of the diffusion oper-ators

ci+1/2

2 [[u]]i+1/2 and 12 Ri+1/2|Λi+1/2|R−1

i+1/2[[u]]i+1/2 are of the generic form

(2.19) Ri+1/2Ai+1/2R−1i+1/2[[u]]i+1/2

for a nonnegative diagonal matrix Ai+1/2. We wish to manipulate this expression to arrive atan expression of the form Di+1/2[[3]]i+1/2. Heuristically, we may write

(2.20) [[u]]i+1/2 ≈ u3(3i+1/2)[[3i+1/2]],

where u3 = dud3

1. The matrix u3(3) is symmetric positive definite whenever η is strictly convex,but something more can be said of it.

Lemma 2.9 (Barth [Bar98]). Let B = RAR−1 be a diagonalizable matrix and let S ∈RN×N be a symmetric positive definite matrix such that BS is symmetric. Then the columnsof R may be scaled such that S = RRT . In particular, BS = RART .

Since the s.p.d. matrix u3(3) right-symmetrizes f ′(u) = R(u)Λ(u)R(u)−1 (where u = u(3);see Section 1.2.2), we find that the columns of the eigenvector matrix R(u) may be scaledsuch that u3(3) = R(u)R(u)T for all u = u(3). We take Ri+1/2 to be that matrix R(u(3i+1/2)), andfind that (2.19) can be written as

Ri+1/2Ai+1/2R−1i+1/2[[u]]i+1/2 ≈ Ri+1/2Ai+1/2R−1

i+1/2u3(3i+1/2)[[3]]i+1/2

= Ri+1/2Ai+1/2R−1i+1/2Ri+1/2RT

i+1/2[[3]]i+1/2

= Ri+1/2Ai+1/2RTi+1/2[[3]]i+1/2.

As Ai+1/2 is a nonnegative diagonal matrix, the diffusion matrix Di+1/2 := Ri+1/2Ai+1/2RTi+1/2

isclearly also nonnegative. Hence, the flux

(2.21) Fi+1/2 = Fi+1/2 − Ri+1/2Ai+1/2RTi+1/2[[3]]i+1/2

1The mean value theorem does not necessarily hold for vector-valued functions. Nonetheless, an intermediatestate 3i+1/2 satisfying (2.20) with equality may be found in many cases. In general, the difference between the twosides in (2.20) is O(|[[3]]|2), and will be ignored.


is entropy stable. It remains to select the eigenvalue matrix Ai+1/2. We make three choices,inspired by the Lax-Friedrichs, Local Lax-Friedrichs and Roe schemes, respectively. Below,we denote the identity matrix in RN×N by Id.ELF: Let Ai+1/2 = 1

2 cId, with c := max j∈Z, k=1,...,N(|λk(u j)|

). This gives the diffusion matrix

(2.22) Di+1/2 =12

cRi+1/2RTi+1/2.

ELLF: Let Ai+1/2 = 12 ci+1/2Id, with ci+1/2 := maxk=1,...,N

(|λk(ui)|, |λk(ui+1)|

). This gives the

diffusion matrix

(2.23) Di+1/2 =12

ci+1/2Ri+1/2RTi+1/2.

ERoe: Let Ai+1/2 = 12 |Λi+1/2|, with Λi+1/2 := diag(λ1(ui+1/2), . . . , λN(ui+1/2)) and ui+1/2 = ui+ui+1

2(or any other intermediate value). This gives the diffusion matrix

(2.24) Di+1/2 =12

Ri+1/2|Λi+1/2|RTi+1/2.

One might also consider so-called polynomial viscosity matrices; see [DPRV99].For concreteness we give the explicit form of an entropy stable diffusion operator for the

shallow water equations (1.5). In this case, Lemma 2.9 takes the following form.

Lemma 2.10 ([Fjo09]). Denote

u3 :=dud3

=1g

[1 ww w2 + gh

].

(i) Define the following scaled version of the eigenvector matrix R of f ′(u):

(2.25) R =1√2g

[1 1

w −√

gh w +√

gh

].

Then RRT = u3.(ii) For ui, ui+1 ∈ R+ × R we have [[u]]i+1/2 = (u3)i+1/2[[3]]i+1/2, where

(u3)i+1/2 := u3(3(ui+1/2)) =1g

[1 wi+1/2

wi+1/2 w2i+1/2 + ghi+1/2

]is u3 evaluated at ui+1/2 =

(hi+1/2, hi+1/2wi+1/2

).

Proof. The results follow by insertion.

Therefore, the entropy stable scheme for the shallow water equations will be the schemewith flux (2.21), with Fi+1/2 being any of the entropy conservative fluxes (2.15) and (2.16),Di+1/2 being any of the three choices (2.22), (2.23) and (2.24), and Ri+1/2 being given by (2.25)with h = hi+1/2, w = wi+1/2.

2.6. SUMMARY 27

2.6. Summary

We may summarize this chapter as follows. Entropy stability gives an a priori bound onthe total entropy production of a scheme. Our strategy of constructing entropy stable schemesis to combine an entropy conservative flux with appropriate diffusion operators, expressed interms of the entropy variable. Using Tadmor’s entropy conservation condition, it is easy toconstruct entropy conservative schemes for scalar equations, and with a bit more work alsofor systems of equations.

Entropy stability is obtained by adding an appropriate diffusion term. We propose threesuch terms, inspired by the Lax-Friedrichs, Local Lax-Friedrichs and Roe schemes, respec-tively.

CHAPTER 3

Convergence theory for finite volume methods

In his pioneering 1983 paper [DiP83], DiPerna demonstrated how to use compensatedcompactness to show convergence of finite volume methods to the entropy solution. In par-ticular, DiPerna used Murat’s Lemma (Lemma 3.1) to show that under certain assumptions,the sequences of measures

ut + f (u)x∆x>0 and η(u)t + q(u)x∆x>0 ,

u being the approximate solution generated by the scheme, are precompact in H−1(Ω). Com-bined with Theorems 1.12 or 1.13, this proves strong (pointwise and Lp) convergence of thefinite volume approximation.

At the time of the publication of [DiP83], it was already well-known that all monotonemethods converge to the entropy solution; cf. Crandall and Majda [CM80]. The novelty inDiPerna’s compensated compactness technique, however, lies in its generality. By avoidingstability requirements such as monotonicity or requiring that the method is total variationdiminishing or an E-scheme, it is applicable to a much broader class of numerical methods.The downside is that it is only applicable to scalar, one-dimensional conservation laws (andto certain systems of two equations [DiP83]). The alternative approach of H-measures isapplicable to multi-dimensional scalar conservation laws; see Panov [Pan10].

As DiPerna’s paper is only concerned with first-order methods, certain assumptions arenot posed as weakly as they might be. The main purpose of the present chapter is to revisit andrefine his proof of convergence, sharpening its hypotheses to the absolute necessity. This willenable us to prove convergence of high-order accurate methods for scalar, one-dimensionalconservation laws in Chapter 4. In Section 3.2 we investigate necessary conditions for the(weak-∗) convergence of finite volume methods to measure-valued solutions. As this is aweaker solution concept, it should not come as a surprise that these conditions are weakerthan the conditions for convergence to the entropy solution. Moreover, since we do not usecompensated compactness, we are able to handle multi-dimensional equations on unboundeddomains.

Finally, Section 3.3 demonstrates how the convergence framework in Section 3.1 can beapplied to prove convergence of the first-order accurate entropy stable methods described inSection 2.3.

29

30 3. CONVERGENCE THEORY FOR FINITE VOLUME METHODS

3.1. Strong convergence of finite volume schemes

We consider the scalar, one-dimensional conservation law (1.1). If ui(t) is the computedsolution, we define the function u(x, t) ≡ ui(t) for x ∈ Ii. Similarly, for a flux function Fi+1/2

we denote F(x, t) = Fi+1/2(t) for x ∈ Ii, so that its spatial distributional derivative is

Fx =∑

i

(Fi+1/2 − Fi−1/2

)δxi−1/2 ,

where δxi−1/2 denotes the Dirac unit mass at xi−1/2.We will show that the sets ut + f (u)x∆x>0 and η(u)t + q(u)x∆x>0 are precompact in H−1,

where u = u∆x is computed using the finite volume methods (2.1) with grid size ∆x. (For each∆x > 0 we shall suppress the dependence of the computed solution ui(t) on ∆x.) To this end,we must relate the terms f (u)x and q(u)x to their discretized counterparts

Fi+1/2 − Fi−1/2

∆xand

Qi+1/2 − Qi−1/2

∆x,

respectively. (The terms ut and η(u)t are not discretized, thus easing the analysis.) As theseare nonlinear functions of u, we will require a bound on the spatial variation of u to pass tothe limit ∆x→ 0.

The term q(u)x is not discretized explicitly, but many schemes have an inherent dis-cretization of this term. We shall refer to any function of 2m variables Q = Q(u1, . . . , u2m)satisfying the consistency relation Q(u, . . . , u) = q(u) for all u ∈ R as a numerical entropyflux; we denote Qi+1/2 = Q(ui−m+1, . . . , ui+m). For instance, for a 3-point scheme (m = 1) wecan set

Q(ui, ui+1) = 3i+1/2Fi+1/2 − ψi+1/2

(see Section 1.2.2 and Chapter 2). Clearly, any numerical flux function F is a numericalentropy flux for the entropy pair (id, f ), where id(u) = u denotes the identity function.

Our goal will be to show precompactness of the sequence of measures η(u)t +q(u)x∆x>0by studying the measures

ddtη(ui) +

Qi+1/2 − Qi−1/2

∆x.

Here and in the remainder, we assume that u is a scalar function.Let Ω ⊂ R × (0,T ) be an open, bounded set such that supp(u) ⊂ Ω for all ∆x > 0. We

will view η(u)t + q(u)x as both a functional on functions defined on Ω, as well as a measureon Ω. For 0 < t1 6 t2 and Borel-measurable A ⊂ R, denote by ηt + qx the measure

(ηt + qx)(A × [t1, t2]) =

∫ t2

t1

(∫Aη(u(x, t))tdx +

∑i

(q(ui(t)) − q(ui−1(t))

)δxi−1/2 (A)

)dt

and by ηt + Qx

(ηt + Qx)(A × [t1, t2]) =

∫ t2

t1

(∫Aη(u(x, t))tdx +

∑i

(Qi+1/2(t) − Qi−1/2(t)

)δxi−1/2 (A)

)dt.

3.1. STRONG CONVERGENCE OF FINITE VOLUME SCHEMES 31

We may also view ηt + qx as the functional

(ηt + qx)(ϕ) :=∫

Ω

ϕ d(ηt + qx)

for ϕ in, e.g., W−1,p(Ω). We need the following lemma, a version of which is due to Murat[Mur81].

Lemma 3.1 (Murat’s Lemma). Let Ω ⊂ Rn be open and bounded and let µ j j∈N be abounded sequence in W−1,p(Ω) for some 2 < p 6 ∞. Assume that µ j = χ j + π j, where χ j isprecompact in H−1(Ω) and π j is bounded inM(Ω). Then µ j is precompact in H−1(Ω).

Here, M(Ω) is the set of Radon measures on Ω; see Appendix A. The following aux-iliary lemma provides the means to prove H−1-precompactness of the sequence of entropyresiduals.

Lemma 3.2. Let Ω ⊂ R × [0,T ] be bounded, let (η, q) be an entropy pair and let Q be anumerical entropy flux satisfying the Lipschitz condition(3.1)|q(u0) − Q(u−m+1, . . . , um)| 6 C (|u−m+2 − u−m+1| + · · · + |um − um−1|) ∀ u−m+1, . . . , um.

Assume that a sequence of functions u = u∆x (∆x > 0) satisfies

(3.2) ∃ M > 0 : ‖u∆x‖L∞(Ω) 6 M ∀ ∆x > 0

(3.3) supp(u∆x) ⊂ Ω ∀ ∆x > 0

(3.4)∫ T

0

∑i

∣∣∣u∆xi+1 − u∆x

i

∣∣∣P∆xdt → 0 as ∆x→ 0 for some P ∈ [2,∞)

(3.5) |ηt + Qx| (Ω) 6 C ∀ ∆x > 0.

Then ηt + qx, viewed as a sequence of functionals parametrized by ∆x > 0, is precompact inH−1(Ω).

Proof of Lemma 3.2. The proof consists of applying Murat’s lemma to ηt + qx. Thisfunctional is bounded in W−1,∞(Ω), since

|(ηt + qx)(ϕ)| =∣∣∣∣∫

Ω

η(u)ϕt + q(u)ϕxdxdt∣∣∣∣ 6 (‖η(u)‖L∞(Ω) + ‖q(u)‖L∞(Ω)

)‖ϕ‖W1,1(Ω),

which is bounded, by (3.2) and the continuity of η and q.We decompose

(ηt + qx)(ϕ) =

∫Ω

ϕ d(ηt + Qx)︸︷︷︸E1(ϕ)

+

∫Ω

ϕ d((ηt + qx) − (ηt + Qx))︸︷︷︸E2(ϕ)

.

For the first term we have

|E1(ϕ)| 6 ‖ϕ‖L∞(Ω) |ηt + Qx| (Ω) 6 C‖ϕ‖L∞(Ω)


by (3.5). Hence, E1 is bounded inM(Ω). It remains to show that E2 is precompact in H−1(Ω).Indeed,

|E2(ϕ)| =∣∣∣∣∣∫

Ω

ϕ d (q(u)x − Qx)∣∣∣∣∣

=

∣∣∣∣∣∣∣∫ T

0

∑i

(ϕ(xi+1/2, t) − ϕ(xi−1/2, t)

) (q(ui) − Qi+1/2

)dt

∣∣∣∣∣∣∣ (summation by parts)

6 ‖ϕx‖L2(Ω)

∫ T

0

∑i

|q(ui) − Q(ui−m+1, . . . , ui+m)|2 ∆xdt

1/2

(Holder inequality)

6 C‖ϕx‖L2(Ω)

∫ T

0

∑i

|ui+1 − ui|2∆xdt

1/2

(by (3.1))

6 C‖ϕx‖L2(Ω)|Ω|(P/2)′

∫ T

0

∑i

|ui+1 − ui|P∆xdt

1/P

(Holder inequality using P/2 > 1)

→ 0 (by (3.3), (3.4)).

In conclusion, Murat’s lemma implies that ηt + qx is precompact in H−1(Ω).

Remarks.• In practice, assumption (3.4) will be proved by showing

(3.6)∫ T

0

∑i

|ui+1 − ui|Pdt 6 C,

which clearly implies (3.4).• The connection between (3.4), (3.5) and (3.6) is as follows. We may estimate the

left-hand side of (3.5) using the entropy residual (2.5):

|ηt + Qx|(Ω) 6∫ T

0

∑i

|ri+1/2| dt =

∫ T

0

∑i

|[[3]]i+1/2Fi+1/2 − [[ψ]]i+1/2| dt

6

∫ T

0

∑i

|[[3]]i+1/2||Fi+1/2 − f (u)i+1/2| + |3i+1/2[[ f (u)]]i+1/2 − [[q]]i+1/2| dt

6 C∫ T

0

∑i

|ui+1 − ui|2 dt,

by the Lipschitz continuity of Fi+1/2 and the definition of q and 3. Thus, should thesufficient condition (3.6) with P = 2 hold, then both (3.4) and (3.5) are satisfied.This is the bound asserted by DiPerna [DiP83, p. 59]. For P > 2, however, onecannot conclude (3.5) on the basis of (3.4) or (3.6).

• For a p-th order accurate numerical scheme, the entropy residual will be of theorder of |ui+1 − ui|

p+1 (see Chapter 4 and [Lax71]). Thus, it is essential to consider

3.1. STRONG CONVERGENCE OF FINITE VOLUME SCHEMES 33

the general case P > 2, as one cannot hope for (3.6) with P = 2 for a high-orderaccurate scheme.

We can now state the main theorem of this section, where we make the connection be-tween the abstract convergence Lemma 3.2 and entropy stable methods.

Theorem 3.3. Let Ω ⊂ R × [0,T ] be bounded and (η, q) be an entropy pair such thateither

(i) both η and f are strictly convex, or(ii) η = f and f is genuinely nonlinear.

Let u be computed by the finite difference scheme (2.1), and assume that there is a numericalentropy flux Qi+1/2 such that

(3.7)ddtη(ui) +

Qi+1/2 − Qi−1/2

∆x6 0.

Suppose that

(A1) ∃ M > 0 : ‖u‖L∞(Ω) 6 M ∀ ∆x > 0

(A2) supp(u) ⊂ Ω ∀ ∆x > 0

(A3)∫ T

0

∑i

|ui+1 − ui|P∆xdt → 0 as ∆x→ 0 for some P ∈ [2,∞),

and that F and Q satisfy the Lipschitz conditions

(A4F) | f (u0) − F(u−m+1, . . . , um)| 6 C (|u−m+2 − u−m+1| + · · · + |um − um−1|)

(A4Q) |q(u0) − Q(u−m+1, . . . , um)| 6 C (|u−m+2 − u−m+1| + · · · + |um − um−1|)

for all u−m+1, . . . , um ∈ [−M,M]. Then there is a subsequence of u∆x∆x>0 which convergesa.e. in Ω and in Lp(Ω) for all 1 6 p < ∞ to a weak solution of (1.1) which is admissible withrespect to (η, q). In particular, in case (i), the whole sequence converges and the limit is theunique entropy solution.

Proof. We apply Lemma 3.2 to the entropy pairs (id, f ) and (η, q) to show that the se-quences ut + f (u)x∆x>0 and ηt + qx∆x>0 are precompact in H−1(Ω). Convergence thenfollows from Theorem 1.12 (case (i)) or Theorem 1.13 (case (ii)).

Assumptions (A1), (A2), (A3) are precisely (3.2), (3.3), (3.4). Assumptions (A4F),(A4Q) are the same as (3.1) for the numerical entropy fluxes F and Q, respectively. (3.5)trivially holds for (id, f ) since

|ut + Fx|(Ω) =

∫ T

0

∑i

∣∣∣∣∣ ddt

ui +Fi+1/2 − Fi−1/2

∆x

∣∣∣∣∣∆xdt = 0


(by (2.1)), and for (η, q) since

|ηt + Qx|(Ω) =

∫ T

0

∑i

∣∣∣∣∣ ddtη(ui) +

Qi+1/2 − Qi−1/2

∆x

∣∣∣∣∣ ∆xdt

= −

∫ T

0

∑i

(ddtη(ui) +

Qi+1/2 − Qi−1/2

∆x

)∆xdt (by (3.7))

=∑

i

η(ui(0)) ∆x −∑

i

η(ui(T )) ∆x

6 2diam(Ω)‖η‖L∞([−M,M]) 6 C

since u is L∞-bounded on a compact domain and η is continuous. Hence, Lemma 3.2 impliesthe compactness property (1.22), so in case (i) and (ii) we may apply Theorem 1.12 and 1.13,respectively, implying the existence of a strongly convergent subsequence of u∆x>0. Bythe Lax-Wendroff theorem, the limit is a weak solution. Admissibility with respect to (η, q)follows similarly.

In particular, in case (i), the limit is the unique entropy solution, by Panov’s Theorem1.10. By the uniqueness of the limit, every convergent subsequence converges to the samelimit, and hence the whole sequence converges to the entropy solution.

3.2. Multi-dimensional measure-valued solutions of systems

The compensated compactness method, as utilized in the previous section, is applicableonly to one-dimensional, scalar conservation laws on bounded domains. What it lacks ingenerality, it makes up for in strength: we are able to show strong convergence to a distri-butional solution under the relatively weak assumptions (A1)–(A4Q). In this section we takethe opposite approach: by weakening our assumptions further, we will be able to prove theweaker result of (weak) convergence to a measure-valued solution for general systems in anynumber of space dimensions on unbounded domains.

We will restrict ourselves to two-dimensional conservation laws (d = 2). The gen-eralization to higher dimensions is straightforward, and is left out for notational simplic-ity. Thus, consider the two-dimensional conservation law (1.1); we denote the compo-nents of the spatial variable by (x, y) ∈ R2. The spatial domain is partitioned into cellsIi, j := [xi−1/2, xi+1/2) × [y j−1/2, y j+1/2), and the solution u(x, y, t) is approximated by cell averagesui, j(t) ≈ 1

∆x∆y

∫Ii, j

u(x, y, t) dxdy or by point values ui, j(t) ≈ u(xi, y j, t). A prototypical finitevolume (finite difference) scheme is of the form

(3.8)ddt

ui, j +F x

i+1/2, j − F xi−1/2, j

∆x+

Fyi, j+1/2

− Fyi, j−1/2

∆y= 0.

The numerical flux function(F x

i+1/2, j, Fyi, j+1/2

)is assumed to be consistent with (1.1), i.e.,(

F x(u, . . . , u), Fy(u, . . . , u))

= f (u). We will denote ∆xui, j := ui+1, j − ui, j and ∆yui, j :=ui, j+1 − ui, j.

3.2. MULTI-DIMENSIONAL MEASURE-VALUED SOLUTIONS OF SYSTEMS 35

Lemma 3.4. Let N > 1, let Ω ⊂ R2 × [0,∞), and let u = u∆x,∆y be computed from (3.8).Assume that

(B1) ∃ M > 0 : ‖u∆x,∆y‖L∞(Ω) 6 M ∀ ∆x,∆y

(B3) ∃ P ∈ [2,∞) :∫ T

0

∑i, j

(∣∣∣∆xu∆x,∆yi, j

∣∣∣P +∣∣∣∆yu∆x,∆y

i, j

∣∣∣P) ∆x∆ydt → 0 as ∆x,∆y→ 0.

and that for all u−m+1, . . . , um,

(B4F)

| f x(u0) − F x(u−m+1, . . . , um)| 6 C (|u−m+2 − u−m+1| + · · · + |um − um−1|)| f y(u0) − Fy(u−m+1, . . . , um)| 6 C (|u−m+2 − u−m+1| + · · · + |um − um−1|)

Then the Young measure associated with u∆x,∆y∆x,∆y>0 is a measure-valued solution to (1.1).

Proof. (B1) gives the existence of the Young measure ν; cf. Theorem A.1. Denote h :=max(∆x,∆y). For all ϕ ∈ C2

c (Ω), we have

∫Ω

〈ν, λ〉ϕt + 〈ν, f 〉 · ∇ϕ dxdt = limh→0

∫ T

0

∑i, j

∫Ii, j

(−

(ddt

ui, j

)ϕ + f (ui, j) · ∇ϕ

)dxdt

= limh→0

∫ T

0

∑i, j

∫Ii, j

(F xi+1/2, j − F x

i−1/2, j

∆x+

Fyi, j+1/2

− Fyi, j−1/2

∆y

)ϕ + f (ui, j) · ∇ϕ

dxdt

= limh→0

∫ T

0

∑i, j

(F xi+1/2, j − F x

i−1/2, j

∆x+

Fyi, j+1/2

− Fyi, j−1/2

∆y

)ϕi, j∆x∆y + f (ui, j) ·

∫Ii, j

∇ϕ dx

dt

= limh→0

∫ T

0

∑i, j

((f x(ui, j) − F x

i+1/2, j

)∇ϕx

∆x∆y +(

f y(ui, j) − Fyi, j+1/2

)∇ϕy

∆x∆y

+ f (ui, j) ·∫

Ii, j

(∇ϕ − ∇ϕ

)dx

)dt

where we have denoted

ϕi, j :=1

∆x∆y

∫Ii, j

ϕ dxdy and ∇ϕ :=(∇ϕx, ∇ϕy)

=

(ϕi+1, j − ϕi, j

∆x,ϕi, j+1 − ϕi, j

∆y

).


The first term vanishes as h→ 0, since∫ T

0

∑i, j

(∣∣∣ f x(ui, j) − F xi+1/2, j

∣∣∣ ∣∣∣∇ϕx∣∣∣ +

∣∣∣∣ f y(ui, j) − Fyi, j+1/2

∣∣∣∣ ∣∣∣∇ϕy∣∣∣)∆x∆ydt

6 C∫ T

0

∑i, j

(∣∣∣∆xui, j

∣∣∣ ∣∣∣∇ϕx∣∣∣ +

∣∣∣∆yui, j

∣∣∣ ∣∣∣∇ϕy∣∣∣)∆x∆ydt

6 C(∫ T

0

∑i, j

(∣∣∣∆xui, j

∣∣∣P +∣∣∣∆yui, j

∣∣∣P) ∆x∆ydt)1/P(∫

Ω

∣∣∣∣∣∂ϕ∂x

∣∣∣∣∣P′ +

∣∣∣∣∣∂ϕ∂y

∣∣∣∣∣P′)1/P′

→ 0

by (B4F), Holder’s inequality and (B3). The second term also goes to zero because∑i, j

(∇ϕx, ∇ϕy)

1Ii, j → ∇ϕ

pointwise.

Completely analogously, we can show convergence to an entropy measure-valued solu-tion, provided the approximate solution satisfies a discrete entropy inequality

(3.9)ddtη(ui, j) +

Qxi+1/2, j − Qx

i−1/2, j

∆x+

Qyi, j+1/2

− Qyi, j−1/2

∆y6 0

for a numerical entropy flux(Qx

i+1/2, j,Qyi, j+1/2

)consistent with a convex entropy pair (η, q), i.e.,(

Qx(u, . . . , u), Qy(u, . . . , u))

= q(u).

Lemma 3.5. Assume that u satisfies (2.3) for a numerical entropy flux Q. Assume that(B1), (B3) are satisfied, and that for all u−m+1, . . . , um,

(B4Q)

|qx(u0) − Qx(u−m+1, . . . , um)| 6 C (|u−m+2 − u−m+1| + · · · + |um − um−1|)|qy(u0) − Qy(u−m+1, . . . , um)| 6 C (|u−m+2 − u−m+1| + · · · + |um − um−1|)

Then the Young measure ν associated with u∆x,∆y∆x,∆y>0 is admissible with respect to (η, q),i.e., it satisfies (1.26).

Remark. We may allow for a nonzero term on the right-hand side which vanishes weaklyin L2(Ω) as ∆x,∆y → 0. However, for the sake of simplicity we do not pose this generalresult.

3.2. MULTI-DIMENSIONAL MEASURE-VALUED SOLUTIONS OF SYSTEMS 37

Proof of Lemma 3.5. The proof follows that of Lemma 3.4 closely. For all ϕ ∈ C1c (Ω),

we have∫Ω

〈ν, η〉ϕt + 〈ν, q〉 · ∇ϕ dxdt = limh→0

∫ T

0

∑i, j

∫Ii, j

(−

(ddtη(ui, j)

)ϕ + q(ui, j) · ∇ϕ

)dxdt

> limh→0

∫ T

0

∑i, j

∫Ii, j

(Qxi+1/2, j − Qx

i−1/2, j

∆x+

Qyi, j+1/2

− Qyi, j−1/2

∆y

)ϕ + q(ui, j) · ∇ϕ

dxdt

= limh→0

∫ T

0

∑i, j

(Qxi+1/2, j − Qx

i−1/2, j

∆x+

Qyi, j+1/2

− Qyi, j−1/2

∆y

)ϕi, j∆x∆y + q(ui, j) ·

∫Ii, j

∇ϕ dx

dt

= limh→0

∫ T

0

∑i, j

((qx(ui, j) − Qx

i+1/2, j

)∇ϕx

∆x∆y +(qy(ui, j) − Qy

i, j+1/2

)∇ϕy

∆x∆y

+ q(ui, j) ·∫

Ii, j

(∇ϕ − ∇ϕ

)dx

)dt

with the same notation as in Lemma 3.4. As before, the first term vanishes as h→ 0, since∫ T

0

∑i, j

(∣∣∣qx(ui, j) − Qxi+1/2, j

∣∣∣ ∣∣∣∇ϕx∣∣∣ +

∣∣∣∣qy(ui, j) − Qyi, j+1/2

∣∣∣∣ ∣∣∣∇ϕy∣∣∣)∆x∆ydt

6 C∫ T

0

∑i, j

(∣∣∣∆xui, j

∣∣∣ ∣∣∣∇ϕx∣∣∣ +

∣∣∣∆yui, j

∣∣∣ ∣∣∣∇ϕy∣∣∣)∆x∆ydt

6 C(∫ T

0

∑i, j

(∣∣∣∆xui, j

∣∣∣P +∣∣∣∆yui, j

∣∣∣P) ∆x∆ydt)1/P(∫

Ω

∣∣∣∣∣∂ϕ∂x

∣∣∣∣∣P′ +

∣∣∣∣∣∂ϕ∂y

∣∣∣∣∣P′)1/P′

→ 0

The second term vanishes for the same reason as in Lemma 3.4.


3.3. Convergence of entropy stable methods

As an illustration of the framework presented in Section 3.1, we shall prove convergenceof the first-order accurate, entropy stable schemes for scalar, one-dimensional conservationlaws developed in Section 2.3. The approach will be the same for higher-order accurateschemes in Chapter 6, although the technical details will be somewhat more involved.

Theorem 3.6. Let f be strictly convex and let (η, q) be any entropy pair with η′′(u) > η0∀ u for some η0 > 0. Let u0 ∈ L∞(R) have compact support. Assume that the flux (2.8) isentropy stable with respect to (η, q), and that 1

c 6 Di+1/2 6 c for all i ∈ Z and t ∈ [0,T ], forsome c > 0. Assume that there is an M > 0 such that the solution computed by the schemesatisfies ‖u∆x‖L∞(R×[0,T ]) 6 M for all ∆x > 0. Then the computed solution u∆x∆x>0 convergespointwise a.e. in R × [0,T ] and in Lp(R × [0,T ]) for all 1 6 p < ∞ to the entropy solution of(1.1).

Proof. We show that conditions (A1) – (A4Q) of Theorem 3.3 are satisfied. The as-sumptions (A1) on uniform boundedness and (A2) on compact support hold by hypothesis.Integrating the entropy dissipation estimate (2.9) over i ∈ Z, t ∈ [0,T ] gives∑

i

η(ui(T )) ∆x −∑

i

η(ui(0)) ∆x = −

∫ T

0

∑i

Di+1/2[[3(t)]]2i+1/2 dt.

Hence, we obtain the following bound on the spatial variation,∫ T

0

∑i

Di+1/2[[3(t)]]2i+1/2 dt 6 C.

Since |[[u]]| 6 ‖u′(3)‖L∞ |[[3]]| = 1‖η′′(u)‖L∞

|[[3]]|, we have

C >∫ T

0

∑i

Di+1/2[[3(t)]]2i+1/2 dt >

η20

c

∫ T

0

∑i

[[u(t)]]2i+1/2 dt,

so the weak total variation bound (A3) is satisfied with P = 2. The Lipschitz conditions(A4F) and (A4Q) hold by the L∞ bound on u and the boundedness of Di+1/2. The result nowfollows from Theorem 3.3 (i).

Remark. The only real, non-technical assumption in Theorem 3.6 is the L∞-bound onu∆x. Although choosing the entropy (2.11) would have provided such a bound, this particularentropy is not continuous, and hence Theorem 3.3 is not applicable.

CHAPTER 4

Higher-order accurate entropy stable methods

By Theorem 2.4, an entropy stable numerical flux can be written in the generic form

Fi+1/2 = Fi+1/2 − Di+1/2[[3]]i+1/2,

where F is an entropy conservative flux function, 3 is the entropy variable and Di+1/2 is anypositive matrix. In this chapter we investigate whether higher-order accurate schemes canadmit such a representation. More precisely, we will design such schemes by replacing Fi+1/2

by a high-order accurate entropy conservative flux, and deriving sufficient conditions on Di+1/2

for the scheme to be entropy stable and high-order accurate.

4.1. High-order entropy conservative methods

The entropy conservative schemes from Chapter 2, being 3-point schemes, are limitedto second-order accuracy. However, following the procedure of LeFloch, Mercier and Rohde[LMR03], we can use these fluxes as building blocks to obtain 2k-th order accurate entropyconservative fluxes for any k ∈ N. These consist of linear combinations of second-orderaccurate entropy conservative fluxes F, and have the form

(4.1) F2ki+1/2 =

k∑r=1

αkr

r−1∑s=0

F(ui−s, ui−s+r).

Theorem 4.1 ([LMR03], Theorem 4.4). For k ∈ N, assume that αk1, . . . , α

kk solve the k

linear equations

2k∑

r=1

rαkr = 1,

k∑r=1

r2s−1αkr = 0 (s = 2, . . . , k),

and define F2k by (4.1). Then the finite difference scheme with flux F2k is• 2k-th order accurate, in the sense that for sufficiently smooth solutions u we have

F2k (ui−k+1, . . . , ui+k) − F2k (ui−k, . . . , ui+k−1)∆x

= f (u)x

∣∣∣∣u=ui

+ O(∆x2k

).

• entropy conservative – it satisfies the discrete entropy identity

ddtη(ui(t)) +

Q2ki+1/2− Q2k

i−1/2

∆x= 0

39

40 4. HIGHER-ORDER ACCURATE ENTROPY STABLE METHODS

where

(4.2) Q2ki+1/2 =

k∑r=1

αkr

r−1∑s=0

Q(ui−s, ui−s+r).

As an example, the fourth-order (k = 2) version of the entropy conservative flux (4.1) is

(4.3) F4i+1/2 =

43

F(ui, ui+1) −16

(F(ui−1, ui+1) + F(ui, ui+2)

)and the sixth-order (k = 3) version

F6i+1/2 =

32

F(ui, ui+1) −3

10

(F(ui−1, ui+1) + F(ui, ui+2)

)+

130

(F(ui−2, ui+1) + F(ui−1, ui+2) + F(ui, ui+3)

).

(4.4)

4.2. High-order entropy stable methods for scalar equations: the ELW scheme

Given the construction of arbitrarily high-order accurate entropy conservative fluxes inthe previous section, we may now form the numerical flux

(4.5) Fi+1/2 = F2ki+1/2 − Di+1/2[[3]]i+1/2.

If D ≡ 0 then this gives a 2k-th order accurate scheme, but without any numerical diffu-sion, computed solutions will have oscillations around shocks. Instead, we may require thatDi+1/2 = O([[3]]p−1

i+1/2). This gives a scheme with truncation error O(∆x2k + |[[3]]|p) = O(∆xp),

provided 2k > p, and so is p-th order accurate, but still has at least some numerical diffusion.For instance, we can let Di+1/2 = ci+1/2|[[3]]i+1/2|

p−1, with ci+1/2 being determined by the localwave speeds, say,

ci+1/2 =12

max(| f ′(ui)|, | f ′(ui+1)|

).

We call the scheme with flux

(4.6) Fi+1/2 = F2ki+1/2 − ci+1/2

∣∣∣[[3]]i+1/2

∣∣∣p−1[[3]]i+1/2,

with k such that 2k > p, the ELWp (Entropy stable Lax-Wendroff) scheme. Applying Theo-rem 2.4, we obtain the following.

Proposition 4.2. The ELWp scheme is formally p-th order accurate and entropy stable;it satisfies the entropy dissipation estimate

(4.7)ddtη(ui) +

Qi+1/2 − Qi−1/2

∆x= −

ci+1/2

∣∣∣[[3]]i+1/2

∣∣∣2k+1+ ci−1/2

∣∣∣[[3]]i−1/2

∣∣∣2k+1

2∆x6 0.

From the entropy dissipation estimate (4.7) we can deduce the following convergenceresult.

4.2. HIGH-ORDER ENTROPY STABLE METHODS FOR SCALAR EQUATIONS: THE ELW SCHEME 41

Proposition 4.3. Assume that f is strictly convex and that (η, q) is uniformly convex. Letu0 ∈ L∞(R) have compact support. Assume that ci+1/2 >

1c for all i ∈ Z for some c > 0, and

that there is an M > 0 such that the solution u∆x computed by the ELWp scheme satisfies‖u∆x‖L∞(R×[0,T ]) 6 M for all ∆x > 0. Then u∆x converges pointwise a.e. in R × [0,T ] and inLp(R × [0,T ]) for all 1 6 p < ∞ to the entropy solution of (1.1).

Proof. As in Theorem 3.6, we show that conditions (A1)–(A4Q) of Theorem 3.3 (i) aresatisfied. Conditions (A1), (A2) hold by hypothesis. For (A3), we integrate (4.7) over i ∈ Z,t ∈ [0,T ] to find that

∞ > C >∫ T

0

∑i

ci+1/2

∣∣∣[[3]]i+1/2

∣∣∣2k+1dt >

η0

c

∫ T

0

∣∣∣[[u]]i+1/2

∣∣∣2k+1dt

where η0 > 0 is such that η′′(u) > 1η0

for all u. Hence, (A3) holds with P = 2k + 1. TheLipschitz conditions (A4F), (A4Q) follow from the Lipschitz continuity of F and the L∞-bound on u.

4.2.1. Numerical experiment: Linear advection. Consider the one-dimensional linearadvection equation (1.2). We use initial data u0(x) = sin(πx), compute on the periodic domainx ∈ [−1, 1) with 50 gridpoints up to t = 1, and we use a CFL number of 0.9. The computedis shown in Figure 4.1. There is very little loss of accuracy, and it is difficult to discern thecomputed solution from the exact solution.

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

x

u

Exact

ELW3

−0.56 −0.54 −0.52 −0.5 −0.48 −0.46 −0.44

0.98

0.985

0.99

0.995

1

x

u

Exact

ELW3

Figure 4.1. Linear advection equation with sinusoidal initial data. Right:closeup around x = −0.5.

Next, we compute the experimental order of accuracy for the third- and fourth-orderELW schemes. As shown in Table 4.2, the expected order of accuracy is reached immediatelyon this smooth test problem.


ELW3 ELW4

L1 L∞ L1 L∞

n error order error order error order error order50 4.14e-3 – 3.58e-3 – 7.31e-4 – 6.04e-4 –100 5.25e-4 2.98 4.61e-4 2.96 4.62e-5 3.98 3.88e-5 3.96200 6.58e-5 3.00 5.79e-5 2.99 2.89e-6 4.00 2.44e-6 3.99400 8.22e-6 3.00 7.24e-6 3.00 1.81e-7 4.00 1.53e-7 4.00600 2.44e-6 3.00 2.15e-6 3.00 3.57e-8 4.00 3.03e-8 4.00800 1.03e-6 3.00 9.06e-7 3.00 1.13e-8 4.00 9.59e-9 4.00

1000 5.26e-7 3.00 4.64e-7 3.00 4.63e-9 4.00 3.93e-9 4.00Table 4.2. Experimental order of accuracy with ELW3 and ELW4. n de-notes the number of gridpoints.

4.2.2. Numerical experiment: Burgers’ equation. Consider Burgers’ equation (1.3)with initial data u0(x) = (1 + sin(πx))/2. This Cauchy problem develops a discontinuity att = 2

π≈ 0.636. We compute in the domain x ∈ [−1, 1] up to t = 2 using periodic boundary

conditions. The solution computed with the fourth-order ELW4 scheme on meshes with 50,100 and 200 grid points is shown in Figure 4.3(a). For comparison, we plot the solutioncomputed with the fourth-order entropy conservative scheme (4.3) in Figure 4.3(b). Notethat this is the entropy conservative flux used in the ELW4 scheme (4.6). The ELW diffusionoperator clearly has a smoothening effect on the underlying entropy conservative scheme. TheELW scheme still has oscillations in the wake of the shock, although it is reasonably accuratein the rest of the domain. As the grid size is lowered, the magnitude of the oscillations awayfrom the shock decreases.

4.3. Reconstruction based entropy stable schemes: scalar equations

Although the ELW scheme is entropy stable, high-order accurate and displays the correctorder of convergence in numerical examples, it exhibits large oscillations around shocks. Thismotivates the use of higher-order nonoscillatory reconstruction instead of the rather naivechoice of diffusion in (4.6). In this section we derive necessary conditions on the reconstruc-tion procedure to obtain an entropy stable scheme for scalar equations. The procedure isgeneralized to systems in the next section.

For our purposes, a p-th order reconstruction procedure is a map Ri : 3 j j∈Z 7→ 3i(x) thattakes a set of point values 3 j j∈Z, and for each cell Ii produces a (p − 1)-th order polynomial3i(x) which (a) interpolates the point value 3i, and (b) approximates the underlying functionto order O(∆xp). (These notions are made more precise in Chapter 5; see also [Shu99]). Inparticular, we are interested in the cell interface values 3±i := 3i(xi±1/2). Note that since the en-tropy conservative fluxes from Section 4.1 are finite difference schemes, we are interpolatingover point values, not cell averages. The initial data is sampled correspondingly.

4.3. RECONSTRUCTION BASED ENTROPY STABLE SCHEMES: SCALAR EQUATIONS 43

−1 −0.5 0 0.5 1

0.2

0.4

0.6

0.8

1

Exact

ELW4, nx=50

−1 −0.5 0 0.5 1

0.2

0.4

0.6

0.8

1

Exact

ELW4, nx=100

−1 −0.5 0 0.5 1

0.2

0.4

0.6

0.8

1

Exact

ELW4, nx=200

(a) ELW4 scheme.

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

Exact

EC4, nx=50

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

Exact

EC4, nx=100

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

Exact

EC4, nx=200

(b) Fourth-order entropy conservative scheme.

Figure 4.3. Burgers’ equation. The exact solution is shown in red.

Schemes of the form (4.5) will generally be of order O(∆x) whenever Di+1/2 = O(1).Instead of the diffusion operator Di+1/2[[3]]i+1/2 we shall consider diffusion terms of the formDi+1/2〈〈3〉〉i+1/2, where 〈〈3〉〉i+1/2 := 3−i+1 − 3

+i and 3±i are the cell interface values of a p-th order

reconstructed function 3i(x) := Ri(3 j j∈Z). This gives a numerical flux of the form

(4.8) F pi+1/2

= F2ki+1/2 − Di+1/2〈〈3〉〉i+1/2,

where Di+1/2 > 0 is yet to be determined. Since Di+1/2〈〈3〉〉i+1/2 = O(∆xp), the truncation errorof this method is of order O(∆x2k + ∆xp−1); if Di+1/2 is continuous with respect to u, then thetruncation error is O(∆x2k +∆xp). Hence, if k is chosen such that 2k > p then then the schemeis (formally) p-th order accurate. Note that the reconstructed variables are only used in thediffusion operator, not in the entropy conservative flux F2k.

To see if the scheme is entropy stable we write

Di+1/2〈〈3〉〉i+1/2 = Di+1/2

〈〈3〉〉i+1/2

[[3]]i+1/2

[[3]]i+1/2.

Thus, if〈〈3〉〉i+1/2

[[3]]i+1/2> 0, or equivalently,

(4.9) 〈〈3〉〉i+1/2 = Bi+1/2[[3]]i+1/2


for a Bi+1/2 > 0, then the diffusion term Di+1/2〈〈3〉〉i+1/2 can be written as a nonnegative termtimes [[3]]i+1/2, and hence the scheme is entropy stable. We will say that the reconstructionprocedure R satisfies the sign property if the reconstructed values 3+i , 3

−i+1 satisfy (4.9).

Example 4.4. We investigate which second-order reconstruction methods satisfy (4.9).Let ϕ be a slope limiter with the symmetry property ϕ(θ−1) = ϕ(θ)θ−1 for all θ , 0 (see[LeV02]). Define the quotients

θ−i =[[3]]i+1/2

[[3]]i−1/2

, θ+i =

[[3]]i−1/2

[[3]]i+1/2

.

We denote the “slope” in grid cell Ii by

σi =1

∆xϕ(θ−i )[[3]]i−1/2 =

1∆x

ϕ(θ+i )[[3]]i+1/2.

the second equality follows from the symmetry of ϕ. Hence, the reconstructed values atthe left and right cell interfaces of grid cell Ii are given by 3−i = 3i −

12ϕ(θ−i )[[3]]i−1/2 and,

respectively, 3+i = 3i + 12ϕ(θ+

i )[[3]]i+1/2. We obtain

〈〈3〉〉i+1/2 = 3i+1 − 3i −12

(ϕ(θ+

i ) + ϕ(θ−i+1))

[[3]]i+1/2 =

(1 −

12

(ϕ(θ+

i ) + ϕ(θ−i+1)))

[[3]]i+1/2.

Hence, since θ+i and θ−i+1 may be chosen independently of each other, the sign property (4.9)

is satisfied if and only if ϕ(θ) 6 1 for all θ ∈ R. It is easily seen that the minmod limiter, givenby

(4.10) ϕmm(θ) =

0 if θ < 0θ if 0 6 θ 6 11 otherwise

satisfies ϕ(θ) 6 1. In fact, the minmod limiter is the only symmetric TVD limiter that satisfiesthe sign property.

4.4. Reconstruction based entropy stable schemes: systems of equations

In this section we generalize the method described in the previous section to systemsof conservation laws. The generalization of the construction of the higher-order entropyconservative flux F2k is trivial, the only difference being that the expression (4.1) is vector-valued instead of scalar. The generalization of the diffusion term Di+1/2〈〈3〉〉i+1/2 is less obvious.As in Section 2.5, we consider diffusion matrices which are of the form

(4.11) Di+1/2 = Ri+1/2Ai+1/2RTi+1/2, Ri+1/2 invertible, Ai+1/2 > 0 diagonal.

Like in the previous section, we define our numerical flux as

(4.12) F pi+1/2

= F2ki+1/2 − Di+1/2〈〈3〉〉i+1/2,

4.4. RECONSTRUCTION BASED ENTROPY STABLE SCHEMES: SYSTEMS OF EQUATIONS 45

where Di+1/2 is now a nonnegative matrix and 〈〈3〉〉i+1/2 = 3i+1(xi+1/2) − 3i(xi+1/2) for some recon-structed function 3i(x), yet to be determined. The following preliminary lemma generalizesthe computations leading to the sign condition (4.9) to systems of equations.

Lemma 4.5. For each i ∈ Z, let Di+1/2 be given by (4.11). Let 3i(x) be a polynomial re-construction of the entropy variables in the cell Ii such that for each i, there exists a diagonalmatrix Bi+1/2 > 0 such that

(4.13) 〈〈3〉〉i+1/2 =(RT

i+1/2

)−1Bi+1/2RTi+1/2[[3]]i+1/2

Then the scheme with numerical flux (4.12) is entropy stable with numerical entropy flux

Qpi+1/2

= Q2ki+1/2 −

123i+1/2 · Di+1/2〈〈3〉〉i+1/2

and Q2k is defined in (4.2). More precisely, it satisfies the entropy dissipation estimate

(4.14)ddtη(ui) +

Qpi+1/2− Qp

i−1/2

∆x= −

[[3]]i+1/2 · Di+1/2〈〈3〉〉i+1/2 + [[3]]i−1/2 · Di−1/2〈〈3〉〉i−1/2

2∆x6 0.

Proof. Multiplying the finite difference scheme (2.1) by 3Ti and imitating the proof ofTheorem 2.4, we obtain

ddtη(ui) = −

1∆x

(Q2k

i+1/2 − Q2ki−1/2

)+

1∆x

(3

Ti Di+1/2〈〈3〉〉i+1/2 − 3

Ti Di−1/2〈〈3〉〉i−1/2

)= −

1∆x

(Qp

i+1/2− Qp

i−1/2

)−

12∆x

([[3]]T

i+1/2Di+1/2〈〈3〉〉i+1/2 + [[3]]Ti−1/2Di−1/2〈〈3〉〉i−1/2

).

Suppressing vector and matrix indices i + 1/2 for the moment, we have by (4.11) and (4.13)

[[3]]T D〈〈3〉〉 = [[3]]T RART R−T BRT [[3]] = [[3]]T RABRT [[3]] =(RT [[3]]

)TAB

(RT [[3]]

)> 0

(since Bi+1/2 > 0), and (4.14) follows.

Remark. If the reconstructed variables satisfies (4.13), then the numerical flux (4.12)admits the equivalent representation

F pi+1/2

= F2ki+1/2 −

12

Ri+1/2Ai+1/2Bi+1/2RTi+1/2[[3]]i+1/2.

This reveals the role of Bi+1/2 as limiting the amount of numerical diffusion: in smooth partsof the flow, we have Bi+1/2 ≈ 0, and we are left with the entropy conservative flux.

4.4.1. Reconstruction along scaled entropy variables. Lemma 4.5 provides sufficientconditions on the reconstruction for the scheme to be entropy stable. It remains to find areconstruction procedure that satisfies the crucial condition (4.13). Assume for the momentthat 3i, 3i+1 and 3+i , 3

−i+1 are given. Define the scaled entropy variables

(4.15) w±i := RTi±1/23i, w±i := RT

i±1/23±i .

The condition (4.13) now reads

(4.16) 〈〈w〉〉i+1/2 = Bi+1/2〈〈w〉〉i+1/2, Bi+1/2 > 0 diagonal,


where 〈〈w〉〉i+1/2 = w−i+1 − w+i = RT

i+1/2〈〈3〉〉i+1/2 and 〈〈w〉〉i+1/2 = w−i+1 − w+

i = RTi+1/2

[[3]]i+1/2. Thus,each component of wi must satisfy the property (4.9).

Let R be a reconstruction procedure that satisfies the sign property (4.9). As the valueswe are reconstructing, the scaled entropy variables, are centred at cell interfaces, we mustmodify the reconstruction method somewhat. For a fixed l ∈ 1, . . . ,N, denote the l-thcomponent of w and w by z and z, respectively. For each i, define the point value αi

i = z−i , andinductively

αij+1 = αi

j + 〈〈z〉〉 j+1/2 ( j = i, i + 1, . . . )

αij−1 = αi

j − 〈〈z〉〉 j−1/2 ( j = i, i − 1, . . . )

(recall that 〈〈z〉〉 j+1/2 = z−j+1 − z+j , the l-th component of 〈〈w〉〉 j+1/2). Similarly, we define βi

i = z+i

and

βij+1 = βi

j + 〈〈z〉〉 j+1/2 ( j = i, i + 1, . . . )

βij−1 = βi

j − 〈〈z〉〉 j−1/2 ( j = i, i − 1, . . . ).

Then α and β retain the cell interface jumps of z,

(4.17) [[αi]] j+1/2 = [[βi]] j+1/2 = 〈〈z〉〉 j+1/2 for all j.

As β and α have the same jump at a cell interface, we have

(4.18) αi+1j = βi

j for all j.

Let Φij(x) := R j

(αi

kk∈Z)

and Ψij(x) := R j

(βi

kk∈Z)

be the reconstructions of αi and βi incell I j. We obtain left and right reconstructed values:

z−i := Φii(xi−1/2) and z+

i := Ψii(xi+1/2).

This process is repeated in each grid cell Ii and for each component of w±i . Finally, we definethe reconstructed values 3±i by

(4.19) 3±i :=

(RT

i±1/2

)−1w±i ,

cf. (4.15).

Lemma 4.6. If the reconstruction procedure R satisfies the sign property (4.9), then thereconstructed values (4.19) satisfy (4.13).

Proof. By construction it suffices to show that the reconstructed scaled entropy variablesw±i satisfy (4.16). Again, denote the l-th component of wi and wi by zi and zi, respectively.Because of (4.18), the piecewise polynomial function Ψi is precisely equal to Φi+1. Hence,

〈〈z〉〉i+1/2 = z−i+1 − z+i = Φi+1

i+1(xi+1/2) − Ψii(xi+1/2) = Ψi

i+1(xi+1/2) − Ψii(xi+1/2),

which by assumption has the same sign as βii+1 − β

ii, which by definition equals 〈〈z〉〉i+1/2.

4.5. SUMMARY AND OUTLOOK 47

Remarks. We make a few remarks regarding the computational complexity of the recon-struction procedure detailed above.

• Although the definition (4.19) of the reconstructed entropy variables depends onthe inverse of RT

i+1/2, it is not necessary to compute this inverse. Indeed, inserting

(4.19) into the diffusion operator Di+1/2〈〈3〉〉i+1/2 of our scheme, we get

Di+1/2〈〈3〉〉i+1/2 = Ri+1/2Ai+1/2RTi+1/2

(RT

i+1/2

)−1〈〈w〉〉i+1/2 = Ri+1/2Ai+1/2〈〈w〉〉i+1/2.

• Most reconstruction procedures R encountered in practice, such as the TVD, ENOand WENO procedures, satisfy the additional property of being linear with respectto constants:

(4.20) Ri

(c + 3 j j∈Z

)= c + Ri

(3 j j∈Z

)∀ c ∈ R.

Let K ∈ Z be a fixed index in the mesh. Since

βij = βK

j +(βi

i − βKi)

for all j,

it follows from the linearity property (4.20) that

Ψij(x) = ΨK

j (x) +(βi

i − βKi)

for all j ∈ Z and x ∈ I j.

In fact, we will only be interested in the cell interface jumps, which by the discus-sion above can be written as

z−i+1 − z+i = ΨK

i+1(xi+1/2) − ΨKi (xi+1/2).

Hence, it suffices to reconstruct on only one mesh βKj j∈Z for all grid cells, as

opposed to reconstructing on different meshes for each cell. This greatly increasesthe efficiency of the reconstruction procedure, and should be contrasted to the morestandard approach of reconstructing locally over meshes R−1

i+1/2u j j for j close to i;

see [Shu99].• Finally, most reconstruction procedures depend only on the (un)divided differences

[βKj , . . . , β

Kj+p] of the data points. By (4.17), we have [βK

j , βKj+1] = [z+

j , z+j+1], so the

data points z+j j∈Z may be inserted directly in the first step of the computation of

(un)divided differences. Hence, the values βKj j∈Z do not even have to be computed.

4.5. Summary and outlook

We have presented high-order accurate entropy conservative and entropy stable schemesfor both scalar conservation laws and systems of conservation laws. The main idea is to com-bine high-order accurate entropy conservative fluxes, described in Section 4.1, with entropystable diffusion operators, described in subsequent sections.

The ELW scheme, presented in Section 4.2, is high-order accurate, entropy stable andconverges for convex scalar conservation laws. However, the scheme has unacceptably largeoscillations near shocks. Instead of tuning the diffusion coefficient to particular problems, we


considered nonoscillatory reconstruction procedures. The main conclusion from Section 4.3was that if the reconstruction procedure satisfies the sign property – namely, that the jump3−i+1 − 3

+i in reconstructed values at the cell interface has the same sign as the jump 3i+1 − 3i

in point values – then the resulting flux (4.8) is high-order accurate and entropy stable. Thiswas generalized to systems of equations in Section 4.4, where we described a reconstructionprocedure along the components of the so-called scaled entropy variables. This proceduregives an entropy stable scheme, provided the underlying, scalar reconstruction procedure Rsatisfies the sign property.

We found that the only symmetric second-order TVD limiter satisfying the sign propertyis the minmod limiter. However, the question of the existence of higher-order accurate recon-struction procedures satisfying the sign property was left open. In Chapter 5 we will see thatthe ENO procedure satisfies the sign property. This will allow us to construct non-oscillatory,arbitrarily high-order accurate finite difference schemes in Chapter 6.

CHAPTER 5

ENO stability properties

The ENO, or Essentially Non-Oscillatory, reconstruction procedure was introduced in1987 by Harten et. al. [HEOC87] in the context of accurate simulations for piecewise smoothsolutions of nonlinear conservation laws. The ENO procedure generates a piecewise polyno-mial approximation of a function from its cell averages. The essence of the ENO procedureis its ability to accurately recover discontinuous functions. The starting point is a collectionof cell averages uii∈Z over consecutive intervals Ii = [xi−1/2, xi+1/2),

(5.1) ui :=1|Ii|

∫Ii

u(x)dx,

from which one can form the piecewise constant approximation of the underlying functionu(x),

u(x) :=∑

i

ui1Ii (x).

But the averaging operator u 7→ u is limited to first order accuracy, whether u is smooth ornot. The purpose of the ENO procedure is to reconstruct a higher order approximation of u(x)from its given cell averages,

uii∈Z 7→ u(x)for a piecewise polynomial function u approximating the underlying function u to orderO(hp). In the context of finite volume methods for hyperbolic conservation laws, this recon-structed function is inserted in the numerical flux function to obtain a scheme of the genericform

(5.2)ddt

ui +1

∆x(Fi+1/2 − Fi−1/2

)= 0, Fi+1/2 := F

(u+

i , u−i+1

)where u±i = u(xi±1/2).

Since its introduction, the ENO procedure and its extensions, [SO89, Har89, Shu90],have been used with a considerable success in computational fluid dynamics; we refer tothe review article of Shu [Shu99] and the references therein. Moreover, ENO and its vari-ous extensions, in particular, with subcell resolution scheme (ENO-SR), [Har89], have beenapplied to problems in data compression and image processing in [Har89, CZ02, CDM03,ACDD05, ACD+08] and references therein.

There are only a few rigorous results about the global accuracy of the ENO procedure.In [ACDD05], the authors proved the second-order accuracy of ENO-SR reconstruction of

49

50 5. ENO STABILITY PROPERTIES

piecewise-smooth C2 data. Multi-dimensional global accuracy results for the so-called ENO-EA method were obtained in [ACD+08]. Despite the extensive literature on the constructionand implementation of ENO and its variants for the last 25 years, we are not aware of anyglobal, mesh independent stability properties.

In the present chapter we introduce and prove the sign-property of the ENO procedure,which says that the jump u−i+1 − u+

i of reconstructed values u at the cell interfaces xi+1/2 alwayshas the same sign as the jump ui+1− ui in cell averages. Moreover, we provide an upper boundto the jump in reconstructed values relative to the jump in cell averages. The importance ofstability bounds on the values u+

i , u−i+1 will be seen in the finite volume method (5.2); it is pre-

cisely these values that are used in the finite volume method. That being said, we emphasizethat the results in this chapter are purely results on reconstruction and interpolation theory,and no further assumptions or connections to numerical methods for differential equationswill be made.

The results presented in this chapter is joint work with Eitan Tadmor and have beenpublished in [FMT12b].

5.1. The ENO reconstruction method

The ENO procedure is a reconstruction procedure, taking cell averages ui and construct-ing a piecewise (p − 1)-th order polynomial function u(x),

u(x) =∑

i

ui1Ii (x) 7→ u(x) :=∑

i

ui(x)1Ii (x),

which addresses the following requirements:

Accurate: It approximates u(x) up to order p in the sense that

(5.3) u(x) = u(x) + O(hp), h = maxi|Ii|

whenever u is sufficiently smooth in a neighborhood of x.Conservative: It is conservative in sense of conserving the original cell averages,

(5.4)1|Ii|

∫Ii

ui(x)dx = ui.

Non-oscillatory: It avoids “Gibbs phenomena”.

The accuracy requirement suggests applying a standard polynomial interpolation. Totake into account the conservation requirement, this interpolation is performed on the prim-itive of u; differentiating the resulting interpolant, it will be seen that the conservation re-quirement is fulfilled. Last, the rather vague criterion of being non-oscillatory is addressedby performing the interpolation over a variable, data-dependent stencil; it is the latter that isthe heart of the ENO method.

Let U(x) :=∫ x−∞

u(s) ds denote the primitive of u(x). Although the underlying functionu(x) might be unavailable, its primitive U(x) is computable at the cell interfaces x j+1/2 through

5.1. THE ENO RECONSTRUCTION METHOD 51

the relation

(5.5) U j+1/2 =

∫ x j+1/2

−∞

u(s)ds =

j∑k=−∞

∫ xk+1/2

xk−1/2

u(s)ds =

j∑k=−∞

|Ik |uk.

If U(x) is any function that interpolates U j+1/2 j∈Z, so that U(x j+1/2) = U j+1/2 ∀ j, then (5.5)implies that its derivative u(x) := U′(x) satisfies (5.4). If F moreover satisfies U(x) = U(x) +

O(hp+1) then u satisfies the accuracy requirement (5.3). This leads us to defining Ui(x) asthe p-th order polynomial interpolating p + 1 point values Ur, . . . ,Ur+p and we set U(x) =∑

i Ui(x)1Ii (x). To address the non-oscillatory requirement, the (half-integer) left stencil indexr must be determined in a data-dependent manner.

When the underlying data is sufficiently smooth, the accuracy requirement can be metby interpolating the primitive U on any set of p + 1 point-values

Ur, . . . ,Ui−1/2,Ui+1/2, . . . ,Ur+p.

To satisfy the conservation property (5.4), the stencil of interpolation must include Ui−1/2 andUi+1/2. There are p such stencils, ranging from the leftmost stencil corresponding to an indexr = i − p + 1/2 to the rightmost stencil corresponding to an index r = i − 1/2. Since we areinterested in the approximation of piecewise smooth functions, we must carefully select thestencil in order to avoid spurious oscillations. The main idea behind the ENO procedure isthe use of a stencil with a data-dependent index, r = ri, which is adapted to the smoothness ofthe data. The choice of ENO stencil is accomplished in an iterative manner, based on divideddifferences of the data.Algorithm 5.1 (ENO reconstruction – stencil selection).Let point values of the primitive Ui−p+1/2, . . . ,Ui+p−1/2 be given. Then the left stencil indicesr1

i , . . . , rpi are determined as follows.

r1i ← i − 1/2

for k = 1→ p − 1 doif

∣∣∣U[xrk

i −1, . . . , xrki +k

]∣∣∣ < ∣∣∣U[xrk

i, . . . , xrk

i +k+1]∣∣∣ then

rk+1i ← rk

i − 1else

rk+1i ← rk

iend if

end for

The divided differences U[x j, . . . , x j+k] are a good measure of the k-th order of smooth-ness of U(x). Thus, the ENO procedure is based on data-dependent stencils which are chosenin the direction of smoothness, in the sense of preferring the smallest divided differences.


The Newton representation of the p-th degree interpolant Ui(x), based on point-valuesU(xrp

i), . . . ,U(xrp

i +p), is given by

(5.6) Ui(x) =

p∑k=0

U[xrk

i, . . . , xrk

i +k] k−1∏

m=0

(x − xrk−1

i +m

).

(We set r−1i = r0

i = r1i = i − 1/2.) Observe that in (5.6), we have summed the contributions of

stencils in their “order of appearance” in the stencil selection procedure, rather than the usualsum of stencils from left to right. Differentiating Ui, we obtain the ENO reconstruction:

Definition 5.2. For each cell Ii, let the stencil index rpi be computed according to Algo-

rithm 5.1; let Ui(x) be the interpolant of V over the stencil U jrp

i +pj=rp

i, and define

(5.7) ui(x) := U′i (x) =

p∑k=1

U[xrk

i, . . . , xrk

i +k] k−1∑

l=0

k−1∏m=0m,l

(x − xrk−1

i +m

).

Then the composite function u(x) =∑

i ui(x)1Ii (x) is called the p-th order ENO reconstruc-tion of uii∈Z.

5.2. The ENO sign property

We can now state the main result of this chapter.

Theorem 5.3 (The sign property). Fix an integer p > 1, and let u(x) =∑

i ui(x)1Ii (x)be the p-th order ENO reconstruction of uii∈Z. Let u+

i := ui(xi+1/2) and u−i+1 := ui+1(xi+1/2)denote the reconstructed point-values to the left and right of the cell interface xi+1/2. Then thefollowing sign property holds at all interfaces:

(5.8)

if ui+1 − ui > 0 then u−i+1 − u+

i > 0;if ui+1 − ui < 0 then u−i+1 − u+

i 6 0;if ui+1 − ui = 0 then u−i+1 − u+

i = 0.

Moreover, there is a constant Cp, depending only on p and on the mesh-ratio of neighboringgrid cells, max| j−i|6p

(|I j+1|/|I j|

), such that

(5.9) 0 6u−i+1 − u+

i

ui+1 − ui6 Cp.

The sign property tells us that at each cell interface, the jump of the reconstructed ENOpointvalues has the same sign as the jump in the underlying cell averages. The sign propertyis illustrated in Figure 5.1, which shows a third-, fourth- and fifth-order ENO reconstructionof randomly chosen cell averages. Even though the reconstructed polynomial may have largevariations within each cell, its jumps at cell interfaces always have the same sign as the jumpsof the cell averages. Moreover, the relative size of these jumps is uniformly bounded. We

5.2. THE ENO SIGN PROPERTY 53

remark that the inequality on the left-hand side of (5.9) is a direct consequence of the signproperty (5.8).

0 5 10 15−0.2

0

0.2

0.4

0.6

0.8

1

p = 3

Cell averages

Reconstruction

0 5 10 15−0.2

0

0.2

0.4

0.6

0.8

1

1.2

p = 4

Cell averages

Reconstruction

0 5 10 15−0.2

0

0.2

0.4

0.6

0.8

1

1.2

p = 5

Cell averages

Reconstruction

Figure 5.1. ENO reconstruction of randomly chosen cell averages.

It should be emphasized that the main Theorem 5.3 is valid for any order of ENO re-construction and for any mesh size. It is valid for non-uniform meshes and does in no waydepend on the underlying function u.

The proof of both the sign property and the related upper-bound (5.9) depends on thejudicious choice of ENO stencils in Algorithm 5.1, and it may fail for other choices of ENO-based algorithms. In particular, the popular WENO methods, which are based on upwind orcentral weighted ENO stencils [LOC94, JS96] fail to satisfy the sign property, as can easilybe confirmed numerically.

The aim of this section is to prove the sign property (5.8). To this end we derive a novelexpression of the interface jump, u−i+1− u+

i , as a sum of terms which involve (p+1)-th divideddifferences of V , and we show that each summand in this expression has the same sign asui+1 − ui. For the remainder of this chapter we will assume for notational convenience thati = 0.

We denote the j-th divided difference of the primitive U as

D[ r,r+ j] := U[xr, . . . , xr+ j], D[ r,r+ j] =D[ r+1,r+ j] − D[ r,r+ j−1]

xr+ j − xr, r = . . . ,−3/2,−1/2.

Thus, for example, by (5.5) we have D[−1/2,3/2] = U[x−1/2, x1/2, x3/2] = (u1 − u0)/(x3/2 − x−1/2). IfD[−1/2,3/2] = 0, or in other words, if u0 = u1, then it is easy to see that the ENO procedure willend up with identical stencils for I0 and I1, which in turn yields u+

0 = u−1 . We may thereforeassume that D[−1/2,3/2] , 0, and the sign property will be proved by showing that

(5.10) Sign property:

if D[−1/2,3/2] > 0 then u−1 − u+0 > 0;

if D[−1/2,3/2] < 0 then u−1 − u+0 6 0.


From (5.7), we find that the value of u+0 is

(5.11)

u+0 = u0(x1/2) =

p∑k=1

D[ rk0,r

k0+k]

k−1∑l=0

k−1∏m=0m,l

(x1/2 − xrk−1

0 +m

)=

p∑k=1

D[ rk0,r

k0+k]

k−1∏m=0

m,1/2−rk−10

(x1/2 − xrk−1

0 +m

)

The last equality follows from the fact that all but the one term corresponding to l = 1/2− rk−10

drop out.To simplify notation, we use

∏to denote a product which skips any of its zero factors,

∏

j∈J α j :=∏

j∈J:α j,0 α j. Thus, for example, a simple shift of indices in (5.11) yields

u+0 =

p∑k=1

D[ rk0,r

k0+k]

k−2

∏m=−1

(x1/2 − xrk−1

0 +1+m

).

In an similar fashion, we handle the ENO reconstruction at cell I1. Let r11, . . . , r

p1 be the

signature of that cell. Note that rk0 6 rk

1, since the ENO reconstruction at stage k in cell I1

cannot select a stencil further to the left than the one used in cell I0. If rk0 = rk

1, then the twointerpolation stencils are the same, and so u−1 − u+

0 = 0. Hence, we only need to consider thecase rk

0 < rk1 for all k. The reconstructed value of u1(x) = U′1(x) at x = x1/2 is given by

u−1 = u1(x1/2) =

p∑k=1

D[ rk1,r

k1+k]

k−1

∏m=0

(x1/2 − xrk−1

1 +m

)The jump in the values reconstructed at x = x1/2 is then given by

(5.12) u−1 − u+0 =

p∑k=1

D[ rk1,r

k1+k]

k−1

∏m=0

(x1/2 − xrk−1

1 +m

)− D[ rk

0,rk0+k]

k−2

∏m=−1

(x1/2 − xrk−1

0 +1+m

) .The following lemma provides a much needed simplification for the rather intimidating

expression (5.12), in terms of a key identity, which is interesting in its own right.

Lemma 5.4. The jump of the reconstructed point-values in (5.12) is given by

u−1 − u+0 =

rp1−1∑

r=rp0

D[ r,r+p+1](xr+p+1 − xr)p−1

∏m=0

(x1/2 − xr+m+1

).(5.13)

Proof. We proceed in two steps. In the first step, we consider the special case when thetwo stencils that are used by the ENO reconstruction in cells I0 and I1 are only one grid cellapart, say, rp

0 = rp1 − 1 = r and in this case, Lemma 5.4 claims that u−1 − u+

0 equals

(5.14) u1(x1/2) − u0(x1/2) = D[ r,r+p+1]

(xr+p+1 − xr

) p−1

∏m=0

(x1/2 − xr+m+1

).

5.2. THE ENO SIGN PROPERTY 55

Indeed, the interpolant of U(xr+1), . . . ,U(xr+p),U(xr), assembled in the specified order fromleft to right and then adding U(xr) at the end, is given by

U0(x) =

p−1∑k=0

D[ r+1,r+1+k]

k−1∏m=0

(x − xr+1+m) + D[ r,r+p]

p−1∏m=0

(x − xr+1+m) .

Similarly, the interpolant of U(xr+1), . . . ,U(xr+p+1), assembled in the specified order fromleft to right, is given by

U1(x) =

p−1∑k=0

D[ r+1,r+k+1]

k−1∏m=0

(x − xr+1+m) + D[ r+1,r+p+1]

p−1∏m=0

(x − xr+1+m)

(cf. (5.6)). Thus, their difference amounts to

U1(x) − U0(x) =(D[ r+1,r+p+1] − D[ r,r+p]

) p−1∏m=0

(x − xr+1+m)

= D[ r,r+p+1]

(xr+p+1 − xr

) p−1∏m=0

(x − xr+1+m) ,

which reflects the fact that U0 and U1 coincide at the p points xr+1, . . . , xr+p. Differentiationyields

u1(x) − u0(x) = D[ r,r+p+1]

(xr+p+1 − xr

) p−1∑l=0

p−1∏m=0m,l

(x − xr+1+m) .

At x = x1/2, all product terms on the right vanish except for l = −r − 1/2, since x1/2 belongs toboth stencils. We end up with

u1(x1/2) − u0(x1/2) = D[ r,r+p+1]

(xr+p+1 − xr

) p−1

∏m=0

(x1/2 − xr+m+1

).

This shows that (5.14) holds, verifying Lemma 5.4 in the case that the stencils associatedwith I0 and I1 are separated by just one cell.

In step two, we extend this result for arbitrary stencils, where I0 and I1 are associatedwith arbitrary offsets, rp

0 6 rp1 . Denote by ur the interpolant at points xr, . . . , xr+p, so that

u0 = urp0 and u1 = ur

p1 . Using the representation from the first step for the difference of

one-cell shifted stencils,(ur+1 − ur

)(x1/2), we can write the jump at the cell interface as a


telescoping sum,

(u1 − u0)(x1/2) =(ur

p1 − ur

p0 )

(x1/2)

=

rp1−1∑

r=rp0

(ur+1 − ur

)(x1/2)

=

rp1−1∑

r=rp0

D[ r,r+p+1](xr+p+1 − xr)p−1

∏m=0

(x1/2 − xr+m+1

),

and (5.13) follows.

Next, we apply Lemma 5.4 to conclude the proof of the sign property. To this end, weshow that each non-zero summand in (5.13) has the same sign as u1 − u0. Since

(5.15) sgn

(xr+p+1 − xr)p−1

∏m=0

(x1/2 − xr+m+1

) = (−1)r+p−1/2, r = −p + 1/2, . . . ,−1/2,

then in view of (5.10), it remains to prove the following essential lemma.

Lemma 5.5. Let rk0

pk=1 and rk

1pk=1 be the signatures of the ENO stencils associated with

cells I0 and, respectively, I1. Then the following holds:

if D[−1/2,3/2] > 0 then (−1)r+p−1/2D[ r,r+p+1] > 0, r = rp0 , . . . , r

p1 − 1,(5.16a)

if D[−1/2,3/2] < 0 then (−1)r+p−1/2D[ r,r+p+1] 6 0, r = rp0 , . . . , r

p1 − 1.(5.16b)

Since r runs over half-integers, (5.16) imply that each non-zero term in the sum (5.13)has the same sign as D[−1/2,3/2], and Theorem 5.3 follows from the sign property, (5.10).

Proof. We consider the case (5.16a) where D[−1/2,3/2] > 0; the case (5.16b) can be arguedsimilarly. The result clearly holds for p = 1, where rp

0 = rp1 − 1 = −1/2. The general

case follows by induction on p. Assuming that (5.16) holds for some p > 1, namely, that(−1)r+p−1/2D[ r,r+p+1] > 0 for r = rp

0 , . . . , rp1 − 1, we will verify that it holds for p + 1. Indeed,

(−1)r+p+1/2D[ r,r+p+2] = (−1)r+p+1/2D[ r+1,r+p+2] − D[ r,r+p+1]

xr+p+2 − xr

=(−1)r+p+1/2D[ r+1,r+p+2] + (−1)r+p−1/2D[ r,r+p+1]

xr+p+2 − xr

=|D[ r+1,r+p+2]| + |D[ r,r+p+1]|

xr+p+2 − xr> 0

(5.17)

for r = rp0 , . . . , r

p1 − 2, by the induction hypothesis. Thus, it remains to examine D[ r,r+p+2]

when r = rp+10 < rp

0 and r = rp+11 > rp

1 .

5.3. UPPER BOUND ON JUMPS 57

(a) The case rp+10 = rp

0 is already included in (5.17), so the only other possibility iswhen rp+1

0 = rp0−1. According to the ENO selection principle, this choice of extend-

ing the stencil to the left occurs when |D[ rp0−1,rp

0 +p]| < |D[ rp0 ,r

p0 +p+1]|. Consequently,

since (−1)rp+10 +p+1/2 = (−1)rp

0 +p−1/2, and by assumption (−1)rp0 +p−1/2D[ rp

0 ,rp0 +p+1] =

|D[ rp0 ,r

p0 +p+1]|, we have

(−1)rp+10 +p+1/2D[ rp+1

0 ,rp+10 +p+2] =

|D[ rp0 ,r

p0 +p+1]| − (−1)rp

0 +p−1/2D[ rp0−1,rp

0 +p]

xrp0 +p+1 − xrp

0−1> 0,

and (5.16a) follows.(b) The case rp+1

1 = rp1 − 1 is already covered in (5.17). The only other possibility is

therefore rp+11 = rp

1 . By the ENO selection procedure, this extension to the rightoccurs when |D[ rp

1−1,rp1 +p]| > |D[ rp

1 ,rp1 +p+1]|. But (−1)rp+1

1 +p−1/2 = −(−1)rp1 +p+1/2, and by

assumption, (−1)rp1 +p+1/2D[ rp

1−1,rp1 +p] = |D[ rp

1−1,rp1 +p]|; hence

(−1)rp+11 +p−1/2D[ rp+1

1 −1,rp+11 +p+1] =

(−1)rp1 +p−1/2D[ rp

1 ,rp1 +p+1] + |D[ rp

1−1,rp1 +p]|

xrp1 +p+1 − xrp

1−1> 0,

and (5.16a) follows.

5.3. Upper bound on jumps

In this section, we will prove (5.9), which establishes an upper bound on the size of thejump in reconstructed values in terms of the jump in the underlying cell averages. We needthe following lemma.

Lemma 5.6. Let rp0 , r

p1 be the (half-integer) offsets of the ENO stencils associated with

cell I0 and I1, respectively. Then

(5.18a)D[ r,r+p+1]

D[−1/2,3/2](−1)r+p−1/2 6 C r,p, r = rp

0 , . . . , rp1 − 1,

where the constants C r,p are defined recursively, starting with C r,1 = 1, and

(5.18b) C r,k+1 =2

xr+k+2 − xrmax

(Cr,k,Cr+1,k

)∀ r.

The quantity on the left in (5.18a) was shown to be bounded from below by zero inLemma 5.5; here we prove an upper bound. The constants Cr,p only depend on the grid sizes|I j|.

Proof. The result clearly holds for p = 1. We prove the general induction step passingfrom p 7→ p + 1. Using the recursion relation

D[ r,r+p+2] =D[ r+1,r+p+2] − D[ r,r+p+1]

∆x, ∆x := xr+p+2 − xr,


we have

0 6D[ r,r+p+2]

D[−1/2,3/2](−1)r+p+1/2 =

1∆x

(D[ r+1,(r+1)+p+1]

D[−1/2,3/2](−1)r+p+1/2 +

D[ r,r+p+1]

D[−1/2,3/2](−1)r+p−1/2

)6

C r+1,p + C r,p

∆x6 C r,p+1, r = rp

0 , . . . , rp1 − 2.

(5.19)

We turn to the remaining cases.

(a) As in Lemma 5.5, if rp+10 = rp

0 then the induction step is already covered in (5.19),so the only remaining case is r := rp+1

0 = rp0 − 1, corresponding to Lemma 5.5(a).

In this case, |D[ r,r+p+1]| 6 |D[ r+1,r+p+2]|, so

D[ r,r+p+2]

D[−1/2,3/2](−1)r+p+1/2 =

1∆x

D[ r+1,r+p+2] − D[ r,r+p+1]

D[−1/2,3/2](−1)r+p+1/2

62

∆xD[ r+1,r+p+2]

D[−1/2,3/2](−1)r+p+1/2 6

2C r+1,p

∆x6 C r,p+1.

(b) If rp+11 = rp

1 − 1 then the induction step is already covered in (5.19), so the onlyremaining case is r := rp+1

1 = rp1 , corresponding to Lemma 5.5(b). In this case,

|D[ r+1,r+p+2]| 6 |D[ r,r+p+1]|, so

D[ r,r+p+2]

D[−1/2,3/2](−1)r+p+1/2 =

1∆x

D[ r+1,r+p+2] − D[ r,r+p+1]

D[−1/2,3/2](−1)r+p+1/2

62

∆xD[ r,r+p+1]

D[−1/2,3/2](−1)r+p−1/2 6

2C r,p

∆x6 C r,p+1.

Using the explicit form (5.13) of the jump u−1 − u+0 , we get the following explicit expres-

sion of the upper-bound asserted in (5.9).

Theorem 5.7. Let u−1 and u+0 be the point-values reconstructed by the ENO Algorithm

5.1 at the cell interface x = x1/2+ and x = x1/2−, respectively. Then

u−1 − u+0

u1 − u06 Cp :=

1x3/2 − x−1/2

−1/2∑r=−p+1/2

C r,p

∣∣∣∣∣∣∣∣(xr+p+1 − xr)p−1

∏m=0

(x1/2 − xr+m+1

)∣∣∣∣∣∣∣∣ .

5.3. UPPER BOUND ON JUMPS 59

Proof. Let rp0 , r

p1 be the offsets of the ENO stencils associated with cell I0 and I1, respec-

tively. By Lemmas 5.4 and 5.6, we have

u−1 − u+0

u1 − u0=

1x3/2 − x−1/2

rp1−1∑

r=rp0

D[ r,r+p+1]

D[−1/2,3/2](−1)r+p−1/2

∣∣∣∣∣∣∣∣(xr+p+1 − xr)p−1

∏m=0

(x1/2 − xr+m+1

)∣∣∣∣∣∣∣∣6

1x3/2 − x−1/2

rp1∑

r=rp0

C r,p

∣∣∣∣∣∣∣∣(xr+p+1 − xr)p−1

∏m=0

(x1/2 − xr+m+1

)∣∣∣∣∣∣∣∣6

1x3/2 − x−1/2

−1/2∑r=−p+1/2

C r,p

∣∣∣∣∣∣∣∣(xr+p+1 − xr)p−1

∏m=0

(x1/2 − xr+m+1

)∣∣∣∣∣∣∣∣ .

When the mesh is uniform, |Ii| ≡ h, the expression for the upper bound C can be calcu-

lated explicitly. The recursion relation (5.18b) yields C r,p =2p

hp−1(p + 1)!, and the coefficient

of the (p + 1)-th order divided differences in (5.13) is∣∣∣∣∣∣∣∣(xr+p+1 − xr)p−1

∏m=0

(x1/2 − xr+m+1

)∣∣∣∣∣∣∣∣ = hp(p + 1)(−r − 1/2)!(p + r − 1/2)!.

Thus, we arrive at the following bound on the jump in reconstructed values.

Corollary 5.8. Assume that the mesh is uniform with mesh size |Ii| ≡ const. Let u−1 andu+

0 be the pointvalues reconstructed at the cell interface x = x1/2+ and x = x1/2−, respectively,by the p-th order ENO Algorithm. Then

u−1 − u+0

u1 − u06 Cp = 2p−1 1

p!

p−1∑k=0

k!(p − k − 1)!.(5.20)

Table 5.2 shows the upper bound (5.20) on (u−1 − u+0 )/(u1 − u0) for some values of p.

Returning to Figure 5.1, we see that although the jumps at the cell interfaces can get large,they cannot exceed Cp times the size of the jump in cell averages, regardless of the values inneighbouring cell averages.

It can be shown that the bound Cp given in Theorem 5.7 is sharp. Indeed, the worst-casescenarios for orders of accuracy p = 2, 3, 4, 5 are shown in Figure 5.3. The mesh in this figureis xi+1/2 = i, and the cell averages are chosen as

ui =

0 if i is odd1 if i is even and i 6 41 − 10−10 if i is even and i > 4.

The number 10−10 is chosen at random; any small perturbation will give the same effect. Thisperturbation ensures that cells Ii for i 6 4 interpolate over a stencil to the left of the cell


p Upper bound Cp

1 12 23 10/3 = 3.333 . . .4 16/3 = 5.333 . . .5 128/15 = 8.533 . . .6 208/15 = 13.866 . . .

Table 5.2. Upper bounds on relative jumps for uniform meshes.

0 1 2 3 4 5 6 7 8 9

−0.5

0

0.5

1

1.5

2

p = 2

Cell averages

Reconstruction

0 1 2 3 4 5 6 7 8 9

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

p = 3

Cell averages

Reconstruction

0 1 2 3 4 5 6 7 8 9

−2

−1

0

1

2

3

4

5

p = 4

Cell averages

Reconstruction

0 1 2 3 4 5 6 7 8 9−4

−3

−2

−1

0

1

2

3

4

5

p = 5

Cell averages

Reconstruction

Figure 5.3. Worst case cell interface jumps for p = 2, 3, 4, 5.

interface x = x4+1/2, and cells with i > 4 to the right of it. We see that for each p, the jump atx = 4 is precisely the bound Cp as given in Table 5.2.

5.4. The ENO interpolation method

The ENO Algorithm can be formulated as a nonlinear interpolation procedure. Thestarting point is a given collection of point-values ui = u(xi) at given grid-points xii∈Z. Thepurpose of the ENO procedure in this context is to recover a highly accurate approximation

5.4. THE ENO INTERPOLATION METHOD 61

of u(x) from its point-values ui = u(xi),

ENO: uii∈Z 7→ u(x) :=∑

k

uk(x)1Ik (x),

where now Ii = [xi−1/2, xi+1/2) and xi±1/2 := xi+xi±12 . Here, uk(x) are polynomials of degree p − 1

which interpolate the given data,ui(xi) = ui.

Moreover, the ENO interpolant u(x) is essentially non-oscillatory in the sense of recoveringu(x) to order O(hp). In particular, since u(x) may experience jump discontinuities, we wishto recover the point-values, u+

i := ui(xi+1/2) and u−i+1 := ui+1(xi+1/2), with high-order accuracy,∣∣∣u+i − u(xi+1/2−)

∣∣∣ +∣∣∣u−i+1 − u(xi+1/2+)

∣∣∣ = O(hp).

The ENO interpolant u(x) is based on the usual divided differences u[xi, . . . , xi+ j]i∈Z, definedby

u[xi] = ui, u[xi, . . . , xi+ j] :=u[xi+1, . . . , xi+ j] − u[xi, . . . , xi+ j−1]

xi+ j − xi.

We denote by `ki

pk=1 the left index of the ENO stencil associated with grid point xi. In

this case of ENO interpolation, these indices are non-positive integers, corresponding to theinteger indices of the prescribed gridpoints x j. These offsets are selected according to thefollowing ENO selection procedure.

Algorithm 5.9 (ENO interpolation).Let point values ui−p+1, . . . , ui+p−1 be given. Then the left stencil indices `1

i , . . . , `pi are deter-

mined as follows.`1

i ← ifor k = 1→ p − 1 do

if |u[x`ki −1, . . . , x`k

i +k−1]| < |u[x`ki, . . . , x`k

i +k]| then`k+1

i ← `ki − 1

else`k+1

i ← `ki

end ifend for

To simplify the notation we let d[ i,i+ j] abbreviate the divided differences u[xi, . . . , xi+ j].

We set ui(x) as the interpolant of 3 over the stencil uk`

pi +p−1

k=`pi

:

ui(x) =

p−1∑k=0

d[ `ki ,`

ki +k]

k−1∏m=0

(x − x`k−1

i +m

).

We refer to the composite function u(x) =∑

i ui(x)1Ii (x) as the p-th order ENO interpolationof uii∈Z.


In the following theorem we state the main stability result for this version of the ENOinterpolation procedure, analogous to the sign property of the ENO reconstruction procedurefrom cell averages.

Theorem 5.10 (The sign property for ENO interpolation). Fix an integer p > 1. Giventhe point-values ui, let u(x) be the p-th order ENO interpolant of these point-values, anddefine u+

i := ui(xi+1/2) and u−i+1 := ui+1(xi+1/2) denote the reconstructed point-values to the leftand right of the cell interface xi+1/2. Then the following sign property holds at all interfaces:

(5.21)

if ui+1 − ui > 0 then u−i+1 − u+

i > 0;if ui+1 − ui < 0 then u−i+1 − u+

i 6 0;if ui+1 − ui = 0 then u−i+1 − u+

i = 0.

Moreover, there is a constant cp, depending only on p and on the mesh-ratio max|i− j|6p(|x j+1−

x j|/|x j − x j−1|) of neighboring grid cells, such that

(5.22) 0 6u−i+1 − u+

i

ui+1 − ui6 cp.

The proof of Theorem 5.10 is very similar to the proof of Theorem 5.3. We thereforeonly sketch the arguments without details.

5.4.1. The sign property for the ENO interpolant. We focus on the jump across theinterface at x = x1/2. Let `k

0pk=1 and `k

1pk=1 be the ENO stencils associated with gridpoints x0

and x1, respectively. Recall that in this case of ENO interpolation, these offsets are integers,−p + 1 6 `k

0 6 0, −p + 2 6 `k1 6 1. Our first key step is to compute the jump at the

interface point x1/2 in the case where the ENO procedure are separated by just one point,namely, `p

0 = `p1 − 1. As before, we denote the reconstructed values at the interface x = x1/2

by u+0 = u0(x1/2) and u−1 = u1(x1/2).

Lemma 5.11. If ` := `p0 = `

p1 − 1 then

u−1 − u+0 = d[ `,`+p+1](x`+p+1 − x`)

p−1∏m=1

(x1/2 − x`+m).

By assembling a telescoping sum of several such stencils we obtain

Corollary 5.12. For general `p0 6 `

p1 , we have

(5.23) u−1 − u+0 =

`p1−1∑`=`

p0

d[ `,`+p+1](x`+p+1 − x`)p−1∏m=1

(x1/2 − x`+m).

Since u1 − u0 = (x1 − x0)d[ 0,1], we wish to show that the jump in reconstructed valuesat the cell interface has the same sign as d[ 0,1]. To this end we show that each summand in

5.4. THE ENO INTERPOLATION METHOD 63

(5.23) has the same sign as d[ 0,1]. Indeed, since

(5.24) sgn

(x`+p − x`)p−1∏m=1

(x1/2 − x`+m)

= (−1)`+p+1, −p + 1 6 ` 6 0,

it suffices to prove the following:

Lemma 5.13. If `p0 , `

p1 are selected according to the ENO stencil selection procedure,

then

if d[ 0,1] > 0 then (−1)`+p+1d[ `,`+p+1] > 0;

if d[ 0,1] < 0 then (−1)`+p+1d[ `,`+p+1] 6 0,

` = `p0 , . . . , `

p1 − 1.

Remark. The sign property for ENO interpolation does not depend on the specific valueof the cell interface point x1/2. Indeed, this value only appears in (5.24), which holds for anyvalue x1/2 ∈ (x0, x1). Hence, we can pose the following, stronger version of (5.21):

(5.25) ∀ xi+1/2 ∈ [xi, xi+1]

if ui+1 − ui > 0 then ui+1(xi+1/2) − ui(xi+1/2) > 0;if ui+1 − ui < 0 then ui+1(xi+1/2) − ui(xi+1/2) 6 0;if ui+1 − ui = 0 then ui+1(xi+1/2) − ui(xi+1/2) = 0.

Note that this does not easily translate to ENO reconstruction, because the cell interfacexi+1/2 appears explicitly in the definition of divided differences, and hence there is a nonlineardependence on xi+1/2.

5.4.2. Upper bounds on the relative jumps for ENO interpolant. Next, we show thecorresponding upper bound on u−1 − u+

0 for ENO reconstruction with point values.

Lemma 5.14. If `p0 , `

p1 are selected according to the ENO stencil selection procedure,

then

0 6d[ `,`+p+1]

d[ 0,1](−1)`+p+1 6 c `,p ` = `

p0 , . . . , `

p1 − 1,

where c `,p are defined recursively, starting with c `,1 = 1, and

c `,k+1 =2

x`+k − x`max(c `,k, c `+1,k).

When the mesh is uniform we can compute c `,p explicitly. If xi+1 − xi ≡ h then it isstraightforward to show that c `,p ≡ (2/h)p−1 1/p!. Moreover, the coefficient of the (p + 1)-thorder divided differences in (5.23) is∣∣∣∣∣∣∣(x`+p − x`)

p−1∏m=1

(x1/2 − x`+m)

∣∣∣∣∣∣∣ = hp p

∣∣∣∣∣∣∣p−1∏m=1

(1/2 − ` − m)

∣∣∣∣∣∣∣ .Thus, we arrive at the following bound on the jump in reconstructed values.


p Upper bound cp

1 12 23 3.54 65 10.3756 18.25

Table 5.4. Upper bounds on relative jumps for uniform meshes.

Theorem 5.15. Assume that the mesh is uniform with mesh size xi+1 − xi ≡ h, and let u±1/2

be the p-th order ENO interpolation to the left and right of x = x1/2. Then

u−1 − u+0

u1 − u06 cp := 2p−1 1

(p − 1)!

0∑`=−p+1

∣∣∣∣∣∣∣p−1∏m=1

(1/2 − ` − m)

∣∣∣∣∣∣∣ .Table 5.4 shows the upper bound on (u−1 − u+

0 )/(u1 − u0) for p 6 6. As for the ENOreconstruction procedure in Section 5.3, it may be shown that these bounds are sharp.

5.5. A conjecture on the total variation of ENO reconstruction

We have established optimal local lower and upper bounds on the jumps of reconstructedvalues in terms of the jumps in cell averages, |u−i+1− u+

i | 6 C|ui+1− ui| for a C > 0. It is obviousthat the converse cannot hold, since the jump in reconstructed values may be zero when thejump in cell averages is not. However, a global estimate of this form might hold, as stated inthe following conjecture.

Conjecture 5.16. Fix an integer p > 1 and let ui(x) be the p-th order ENO reconstructionof u j j∈Z in cell Ii. Assume that uii∈Z has compact support. Then there is a C > 0, dependingonly on p and on the mesh xi+1/2i∈Z, such that

(5.26)∑i∈Z

∣∣∣ui+1 − ui

∣∣∣p+16 C‖u‖p−1

`∞

∑i∈Z

(u−i+1 − u+

i

)(ui+1 − ui

).

By the sign property, each summand on the right-hand side of (5.26) is nonnegative.The conjecture has been verified numerically for a large array of stencil data, and for uniformmeshes the optimal constant seems to be C = 2p−1. However, the global nature of the estimatemakes a proof seem very difficult, and local estimates such as those in the previous sectionsare not sufficient. For instance, if

ui =

0 i 6 0i/n 1 6 i < n1 n 6 i,

5.5. A CONJECTURE ON THE TOTAL VARIATION OF ENO RECONSTRUCTION 65

0 2 4 6 8 10 12

0

0.2

0.4

0.6

0.8

1

p = 2

Cell averagesCell interfacesReconstruction

(a) Continuous, piecewise polynomial data.

0 5 10 15 20−1.5

−1

−0.5

0

0.5

1

1.5

p = 2

Cell averages

Cell interfaces

Reconstruction

(b) Worst-case scenario.

Figure 5.5. ENO reconstructed values and cell averages.

for an n ∈ N, then u−i+1 − u+i = 0 in all but two cells, but ui+1 − ui = 1/n for i = 0, . . . , n − 1.

Still, ∑i

∣∣∣ui+1 − ui

∣∣∣p+1=

1np 6

1n2 = ‖u‖p−1

`∞

∑i

(u−i+1 − u+

i

)(ui+1 − ui

).

The cell averages and reconstructed polynomials for p = 2, n = 10 are shown in Figure5.5(a). This example extends to the case where ui for i = 1, . . . , n is the cell average of apolynomial of order less than p: the left-hand side is O(1/np), while the right-hand side hasa fixed (independent of n) number of nonzero summands, each of which is O(1/n2).

The above examples indicate that the worst-case scenario for the estimate (5.26) is notwhen u is smooth. Indeed, from numerical experiments it seems that the worst-case scenario(that is, when equality is attained in (5.26)) is approached as n→ ∞ with the oscillatory data

ui =

0 i 6 0 or i > n1 i = 0 or i = 1 (mod 4)−1 i = 2 or i = 3 (mod 4).

The reconstructed polynomials when p = 2, n = 20 is plotted in Figure 5.5(b). In thisparticular example, the ratio between the left- and the right-hand sides of (5.26) is about1.82, and as n increases, the ratio approaches 2p−1 = 2. No other “worst cases” have beenfound. This indicates an upper bound of C = 2p−1.

CHAPTER 6

The TECNO scheme

In this chapter we conclude the construction of the high-order accurate, entropy stablefinite difference schemes that were introduced in Chapter 4. The two essential componentsof that construction was (i) high-order entropy conservative fluxes, as discussed in Section4.1, and (ii) a reconstruction procedure satisfying the sign property (4.9). As shown in theprevious chapter, the ENO procedure satisfies precisely this property.

The results presented in this chapter is joint work with Eitan Tadmor and have beenpublished in [FMT12a].

We proceed with the precise definition of our scheme and state its stability properties,first for scalar equations and then for systems of equations.

6.1. The TECNO scheme for scalar conservation laws

Consider the scalar, one-dimensional conservation law (1.1), endowed with a convexentropy pair (η, q). As before, we denote the entropy variable by 3 = 3(u) := η′(u).

Definition 6.1. For a p ∈ N, let k ∈ N be such that 2k > p. Let F be a two-point entropyconservative flux, and define F2k by (4.1). Let Di+1/2 > 0 and let 3±i := 3i(xi±1/2) be the cellinterface values of the p-th order ENO interpolation 3i(x) of the point values 3 j j∈Z. Then theTECNOp scheme is the finite difference scheme (2.1) with flux (4.8).

Theorem 6.2. The TECNOp scheme for scalar conservation laws is (formally) p-th orderaccurate and entropy stable, with the numerical entropy flux

Qpi+1/2

= Q2ki+1/2 − 3i+1/2Di+1/2〈〈3〉〉i+1/2.


(6.1)ddtη(ui) +

Qpi+1/2− Qp

i+1/2

∆x= −

Di+1/2[[3]]i+1/2〈〈3〉〉i+1/2 + Di−1/2[[3]]i−1/2〈〈3〉〉i−1/2

2∆x6 0.

If F and Q are Lipschitz continuous (i.e., satisfy (A4F), (A4Q) with p = 1) and Di+1/2 6 c fora c > 0, then F p and Qp are also Lipschitz continuous (i.e., satisfy (A4F), (A4Q)).

Proof. Accuracy follows from the fact that F2ki+1/2

is accurate to order 2k > p, and that〈〈3〉〉i+1/2 = O(∆xp) in smooth regions. Entropy stability follows analogously to Theorem 2.4;the term [[3]]i+1/2〈〈3〉〉i+1/2 is nonnegative due to the ENO sign property, Theorem 5.10.

67

68 6. THE TECNO SCHEME

If F is Lipschitz continuous then clearly F2k is. The Lipschitz continuity of F p followsfrom the upper bound on D, the continuity of 3 with respect to u and the upper bound (5.22)of the ENO interpolant, |〈〈3〉〉i+1/2| 6 cp|[[3]]i+1/2|. The proof for Qp is identical.

Remark. The generalization of the TECNO scheme to multi-dimensional conservationlaws (d > 1) on Cartesian meshes is straightforward: perform the one-dimensional recon-struction in each direction and then sum up, as described in Section 3.2. Note, however, thatthe convergence Theorem 6.3 only holds for the one-dimensional scheme, as it relies on com-pensated compactness. It remains to be investigated whether the proof of convergence canbe generalized to Panov’s convergence framework for multi-dimensional scalar conservationlaws [Pan10].

From (6.1) we can deduce that∫ T

0

∑i

[[3]]i+1/2〈〈3〉〉i+1/2 6 C,

provided Di+1/2 >1c > 0 for a c > 0. To be able to deduce a bound on the spatial variation of

the form (A3), we would need a bound of the form∑i

|[[3]]i+1/2|P 6 C

∑i

[[3]]i+1/2〈〈3〉〉i+1/2

for some P > 2. This is precisely Conjecture 5.16. Assuming for the moment that theconjecture holds, we can prove convergence in the following sense.

Theorem 6.3. Let f be strictly convex and let (η, q) be uniformly convex. Let u0 ∈ L∞(R)have compact support. Assume that 1

c 6 Di+1/2 6 c for all i ∈ Z and t ∈ [0,T ], for somec > 0, and that there is an M > 0 such that the solution computed by the scheme satisfies‖u∆x‖L∞(R×[0,T ]) 6 M for all ∆x > 0. Then, assuming that Conjecture 5.16 is true, the com-puted solution u∆x∆x>0 converges pointwise a.e. and in Lp(R × [0,T ]) for all 1 6 p < ∞ tothe entropy solution of (1.1).

Proof. As in Theorem 3.6, we show that conditions (A1)–(A4Q) of Theorem 3.3 (i) aresatisfied. Conditions (A1), (A2) hold by hypothesis. From (6.1) it follows that∑

i

η(ui(T )) ∆x −∑

i

η(ui(0)) ∆x = −

∫ T

0

∑i

Di+1/2[[3]]i+1/2〈〈3〉〉i+1/2 dt,

so 0 6∫ T

0

∑i Di+1/2[[3]]i+1/2〈〈3〉〉i+1/2 dt 6 C for a C > 0. Hence,

C >∫ T

0

∑i

Di+1/2[[3]]i+1/2〈〈3〉〉i+1/2 dt

>1c

∫ T

0[[3]]i+1/2〈〈3〉〉i+1/2 dt >

1cMp−1

∫ T

0

∑i

|[[3]]i+1/2|p+1 dt,

6.2. THE TECNO SCHEME FOR SYSTEMS OF EQUATIONS 69

by the lower bound on D, the L∞-bound on u and Conjecture 5.16. Uniform convexity ofη implies that |[[3]]i+1/2| > η0|[[u]]i+1/2| for some η0 > 0, so (A3) holds with P = p + 1. TheLipschitz conditions (A4F), (A4Q) follow from Theorem 6.2.

6.2. The TECNO scheme for systems of equations

Using the reconstruction procedure described in Section 4.4, we can construct a TECNOscheme for systems of equations. Thus, consider the system of equations (1.1) endowed withan entropy pair (η, q).

Definition 6.4. For a p ∈ N, let k ∈ N be such that 2k > p. Let F be a two-point entropyconservative flux; let Di+1/2 be of the form (4.11) and let the reconstructed entropy variables3±i defined as in Section 4.4. Then the TECNOp scheme is the finite difference scheme (2.1)with flux (4.12).

Theorem 6.5. The TECNOp scheme for systems of conservation laws is (formally) p-thorder accurate and entropy stable, with the numerical entropy flux

Qpi+1/2

= Q2ki+1/2 − 3i+1/2 · Di+1/2〈〈3〉〉i+1/2.


(6.2)ddtη(ui) +

Qpi+1/2− Qp

i+1/2

∆x= −

[[3]]i+1/2 · Di+1/2〈〈3〉〉i+1/2 + [[3]]i−1/2 · Di−1/2〈〈3〉〉i−1/2

2∆x6 0.

If F and Q are Lipschitz continuous (i.e., satisfy (A4F), (A4Q) with p = 1) and Di+1/2 6 c fora c > 0, then F p and Qp are also Lipschitz continuous (i.e., satisfy (A4F), (A4Q)).

Proof. Entropy stability follows from Lemma 4.5. The rest is completely analogous tothe scalar case.

We end this section with a proof of the fact that the TECNO scheme converges weakly-∗to a measure-valued solution. This result can be proven for the d-dimensional scheme, butfor the sake of simplicity we prove only the one-dimensional version.

Theorem 6.6. Let (η, q) be nonnegative and uniformly convex. Assume that 1c 6 Ai+1/2 6 c

for all i for some c > 0, and that the solution u∆x computed with the TECNOp scheme isuniformly L∞-bounded, i.e. ‖u∆x‖L∞(R×[0,T ]) 6 M for all ∆x > 0. Then, assuming Conjecture5.16 is true, the Young measure associated with the sequence u∆x is an entropy measure-valued solution.

Proof. We verify conditions (B1), (B3), (B4F), (B4Q) of Lemmas 3.4 and 3.5. The L∞-bound (B1) follows by hypothesis, and the Lipschitz conditions (B4F) and (B4Q) follow fromTheorem 6.5. We verify (B3) only for the one-dimensional (d = 1) case for the sake of clarity.From (6.2) it follows that∑

i

η(ui(T )) ∆x −∑

i

η(ui(0)) ∆x = −

∫ T

0

∑i

[[3]]i+1/2 · Di+1/2〈〈3〉〉i+1/2 dt,


so 0 6∫ T

0

∑i[[3]]i+1/2 · Di+1/2〈〈3〉〉i+1/2 dt 6 C for a C > 0. By construction of the reconstructed

values, there is for each i ∈ Z and t ∈ [0,T ] a diagonal matrix Bi+1/2 > 0 such that (4.13) issatisfied. Hence, suppressing indices for the moment,

[[3]]T D〈〈3〉〉 = [[3]]T RABRT [[3]] =(RT [[3]]

)TAB

(RT [[3]]

)= 〈〈w〉〉T AB〈〈w〉〉 >

1c〈〈w〉〉T B〈〈w〉〉

where 〈〈w〉〉i+1/2 = w−i+1 − w+i and w±i := Ri±1/23i are the scaled entropy variables defined in

(4.15). The diagonal entries of Bi+1/2 are precisely the ratio (5.22) coming from the ENO re-construction of the components of w. Hence, by Conjecture 5.16, we have (again suppressingindices)∑

i

〈〈w〉〉T B〈〈w〉〉 =∑

i

N∑n=1

Bn〈〈wn〉〉2 > C∑

i

N∑n=1

|〈〈wn〉〉|p+1

‖wn‖p−1L∞>

C

‖w‖p−1L∞

∑i

|〈〈w〉〉|p+1,

the last equality following from the equivalence of norms on RN . Since η is uniformly convexand bounded on the compact set u : |u| 6 M, there is an η0 > 0 such that 1

η06 η′′(u) 6 η0

for all u. The L∞-bound on u implies that ‖w‖L∞ 6√η0M. By the definition of w, we have

|〈〈w〉〉i+1/2|2 = [[3]]T

i+1/2Ri+1/2RTi+1/2[[3]]i+1/2 = [[3]]T

i+1/2u3(3i+1/2)[[3]]i+1/2 >1η0|[[3]]i+1/2|

2

since u3(3) = (η′′(u(3)))−1 > 1η0

. By the mean value theorem for vector-valued functions, wehave |[[3]]i+1/2| >

1η0|[[u]]i+1/2|. In conclusion,∑

i

|[[u]]i+1/2|p+1 6 C

∑i

[[3]]Ti+1/2Di+1/2〈〈3〉〉i+1/2

for a C depending only on p, η0, c, and M, so (B3) holds with P = p + 1.

6.3. Numerical experiments: scalar conservation laws

In this section and the next we compare the TECNO method with standard ENO methodsin a series of numerical experiments. We demonstrate that our methods perform as well orbetter through qualitative comparisons of computed solutions, and quantitative comparisonsof convergence rates.

We consider the TECNO method with Lax-Friedrichs-type ELFp diffusion operator,(2.22), and compare it with the ENO-LF finite difference scheme of [SO88]. We choosethe square entropy η(u) = u2.

6.3.1. Linear advection. Consider the one-dimensional linear advection equation (1.2).Since all discontinuities for this equation are contact discontinuities, entropy should be pre-served over time, i.e. ∫

R

η(u(x, t)) dx =

∫R

η(u0(x)) dx.

6.3. NUMERICAL EXPERIMENTS: SCALAR CONSERVATION LAWS 71

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

x

u(x

,1)

ELF3, nx=50

Exact

−0.54 −0.52 −0.5 −0.48 −0.46

0.988

0.99

0.992

0.994

0.996

0.998

1

1.002

1.004

x

u(x

,1)

ELF3, nx=50

Exact

Figure 6.1. Linear advection equation with sinusoidal initial data. Right:closeup around x = −0.5.

0 0.2 0.4 0.6 0.8 1

−10−4

−10−5

−10−6

−10−7

−10−8

−10−9

t

Re

lative

en

tro

py

ELF3, nx=100

ELF4, nx=100

ELF5, nx=100

ELF3, nx=200

ELF4, nx=200

ELF5, nx=200

Figure 6.2. Relative entropy over time

We use the same initial data as in Section 4.2.1. Figure 6.1 shows the solution computedwith the TECNO3 scheme on a mesh of 50 grid points. The solution is clearly computed toa high degree of accuracy. Other schemes perform qualitatively very similar and so are notdisplayed here.

As previously noted, total entropy should be preserved over time. However, the error inthe time stepping method will introduce numerical diffusion in the schemes. In Figure 6.2 wedisplay the relative change in total entropy,

E(t) − E(0)E(0)

, E(t) :=∫ 1

−1η(u(x, t)) dx.

The relative change is monotonously decreasing, meaning that there is no entropy production.Moreover, the entropy decay decreases with higher order of accuracy and smaller grid sizes.


Finally, we compute with both the TECNOp and ENO-LFp schemes on a series ofmeshes. The error and approximate orders of convergence are displayed in Table 6.3. Apartfrom a slight deterioration in the L∞-order of convergence for the fourth-order schemes, theschemes display the expected order of convergence.

TECNO3 ENO-LF3

L1 L∞ L1 L∞

n error order error order error order error order50 7.32e-4 – 6.09e-4 – 8.97e-4 – 7.46e-4 –

100 9.18e-5 2.99 7.52e-5 3.02 1.12e-4 2.99 9.32e-5 3.00200 1.15e-5 3.00 9.33e-6 3.01 1.41e-5 3.00 1.15e-5 3.02400 1.44e-6 3.00 1.16e-6 3.01 1.76e-6 3.00 1.42e-6 3.01600 4.26e-7 3.00 3.42e-7 3.01 5.22e-7 3.00 4.20e-7 3.01800 1.80e-7 3.00 1.44e-7 3.01 2.20e-7 3.00 1.77e-7 3.01

1000 9.21e-8 3.00 7.36e-8 3.00 1.13e-7 3.00 9.04e-8 3.01TECNO4 ENO-LF4

L1 L∞ L1 L∞



1000 1.61e-9 3.97 1.20e-8 3.23 1.68e-9 3.92 1.74e-8 3.22TECNO5 ENO-LF5

L1 L∞ L1 L∞



1000 7.58e-11 5.00 5.95e-11 5.00 7.60e-11 5.00 5.97e-11 5.00Table 6.3. Experimental order of accuracy with TECNO and ENO-LF.n denotes the number of gridpoints.

6.3.2. Burgers’ equation. We repeat the numerical experiment from Section 4.2.2. Thesolution computed with the TECNO3 scheme on meshes with 50, 100 and 200 grid points is

6.4. SYSTEMS OF CONSERVATION LAWS 73

−1 −0.5 0 0.5 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

u

Exact

ELF3, nx=50

−1 −0.5 0 0.5 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

xu

Exact

ELF3, nx=100

−1 −0.5 0 0.5 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x

u

Exact

ELF3, nx=200

Figure 6.4. Burgers’ equation with the TECNO3 scheme. The exact solu-tion is shown in red.

shown in Figure 6.4. The scheme approximates the solution well, apart from small oscilla-tions in the wake of the shock. Contrast this to the ELW scheme in Figure 4.3, which haslarge oscillations after the shock.

6.4. Systems of conservation laws

6.4.1. The wave equation. Consider the wave equation (1.4) with the entropy conser-vative flux (2.13). We consider the TECNO scheme with the ELF diffusion operator (2.22),which for p = 1 gives the numerical flux

Fi+1/2 =12

(Aui + Aui+1) −12

c(ui+1 − ui).

The TECNOp scheme is compared with the ENO-LFp scheme using flux splitting and recon-struction along characteristic directions. We use initial data

u1(x, 0) =

x − 1 if x < 01 − x if x > 0

u2(x, 0) = 0

on the domain x ∈ [−1, 1] with periodic boundary conditions. The solution features an initialjump discontinuity at x = 0 which breaks into two linear (contact) discontinuities. We use amesh of 50 points and a CFL number of 0.4. Computed solutions at time t = 1.5 are displayedin Figure 6.5. The two methods resolve the flow with a comparable level of accuracy.

6.4.2. Shallow water equations. Consider the one-dimensional shallow water equa-tions (1.5). For the TECNO schemes we use the two-point entropy conservative flux (2.15)together with the ELLF diffusion operator (2.23). We compare the TECNO scheme with theENO-LLF scheme using flux splitting in characteristic variables [Shu99]. The gravitationalconstant is set to g = 1.

A dambreak problem. We consider a dambreak problem for the shallow water equationswith initial data

h(x, 0) =

1.5 if |x| < 0.21 if |x| > 0.2

hw(x, 0) = 0


−1 −0.5 0 0.5 1−0.5

0

0.5

Exact

ENO−LF3

TeCNO3

−1 −0.5 0 0.5 1−0.5

0

0.5

Exact

ENO−LF4

TeCNO4

−1 −0.5 0 0.5 1−0.5

0

0.5

Exact

ENO−LF5

TeCNO5

Figure 6.5. The height for the wave equation with discontinuous initialdata, computed with the third, fourth and fifth order ELF and ENO-LFschemes at time t = 1.5 on a mesh of 50 points.

for x ∈ [−1, 1] with periodic boundary conditions The exact solution consists of two shocksseparated by two rarefactions. We display computed heights in Figure 6.6. For a CFL num-ber of 0.5, Figure 6.6(a) reveals that the TECNO schemes are comparable to the ENO-LLFschemes of corresponding order. If we increase the CFL number to 0.9 then there are largeoscillations in the ENO-LLF scheme (Figure 6.6(b)), while the TECNO scheme still givesreasonable results.

6.4.3. Euler equations. Consider the Euler equations (1.6). We use the TECNO schemewith entropy conservative flux given by (2.18) and the ELLF diffusion matrix (2.23). Theeigenvalues and eigenvectors of the Jacobian are computed at the arithmetic average of theleft and right states. The method is compared to the ENO-LLF scheme. In all experimentswe set γ = 1.4.

Sod shock tube. The Sod shock tube experiment is the Riemann problem

(6.3) u(x, 0) =

uL if x < 0uR if x > 0

with ρL

wL

pL

=

101 ,

ρR

wR

pR

=

0.1250

0.1

in the computational domain x ∈ [−5, 5]. The initial discontinuity breaks into a left-goingrarefaction wave, a right-going contact discontinuity and a right-going shock wave. Thecomputed density at time t = 2 is shown in Figure 6.7. There is little or no difference betweenthe two schemes. The results demonstrate that increasing p or the number of gridpointsincreases the accuracy of the computed solution.


−1 −0.5 0 0.5 10.95

1

1.05

1.1

1.15

1.2

1.25

Exact

ENO−LLF3

TeCNO3

−1 −0.5 0 0.5 10.95

1

1.05

1.1

1.15

1.2

1.25

Exact

ENO−LLF4

TeCNO4

−1 −0.5 0 0.5 10.95

1

1.05

1.1

1.15

1.2

1.25

Exact

ENO−LLF5

TeCNO5

(a) CFL = 0.5

−1 −0.5 0 0.5 10.95

1

1.05

1.1

1.15

1.2

1.25

Exact

ENO−LLF3

TeCNO3

−1 −0.5 0 0.5 10.95

1

1.05

1.1

1.15

1.2

1.25

Exact

ENO−LLF4

TeCNO4

−1 −0.5 0 0.5 10.8

0.9

1

1.1

1.2

1.3

Exact

ENO−LLF5

TeCNO5

(b) CFL = 0.9

Figure 6.6. Computed heights for the shallow water dam break problemwith third, fourth and fifth order ENO and TECNO5 schemes at t = 0.4 ona mesh of 50 points.

Lax shock tube. We consider the Euler equations in the computational domain [−5, 5]with Riemann initial data (6.3) given byρL

wL

pL

=

0.4450.6983.528

,ρR

wR

pR

=

0.50

0.571

.The computed density at t = 1.3 is shown in Figure 6.8. The results for ENO and TECNOschemes are very similar in this experiment. There are slight oscillations behind the shockfor the TECNO schemes.

Shock-Entropy wave interaction. This numerical example was proposed by Shu and Os-her in [SO89] and is a good test of a scheme’s ability to resolve a complex solution with bothstrong and weak shocks and highly oscillatory but smooth waves. The computational domainis x ∈ [−5, 5] and we use initial data

u(x, 0) =

uL if x < −4uR otherwise,


−5 0 5

0.2

0.4

0.6

0.8

1

Exact

ENO−LLF3

TeCNO3

−5 0 5

0.2

0.4

0.6

0.8

1

Exact

ENO−LLF4

TeCNO4

−5 0 5

0.2

0.4

0.6

0.8

1

Exact

ENO−LLF5

TeCNO5

(a) 50 gridpoints

−5 0 5

0.2

0.4

0.6

0.8

1

Exact

ENO−LLF3

TeCNO3

−5 0 5

0.2

0.4

0.6

0.8

1

Exact

ENO−LLF4

TeCNO4

−5 0 5

0.2

0.4

0.6

0.8

1

Exact

ENO−LLF5

TeCNO5

(b) 100 gridpoints

Figure 6.7. Density at t = 2 computed with ENO-LLF and TECNO for theSod shock tube.

with ρL

wL

pL

=

3.8571432.62936910.33333

,ρR

wR

pR

=

1 + 15 sin(5x)

01

.The approximate solutions are computed on a mesh of 200 grid points, corresponding to about7 grid points for each period of the entropy waves. The solution computed by the ENO-LFFand TECNO schemes are displayed in Figure 6.9. There are very minor differences betweenthe ENO and TECNO schemes of the same order. The test also illustrates that the higherorder schemes perform better than the low order schemes.

6.4.4. Numerical experiments for two dimensional Euler equations. We test the TECNOschemes for the two dimensional Euler equations

(6.4)

ρρwx

ρwy

E

t

+

ρwx

ρ(wx)2 + pρwxwy

(E + p)wx

x

+

ρwy

ρwxwy

ρ(wy)2 + p(E + p)wy

y

= 0.


−5 0 5

0.4

0.6

0.8

1

1.2

Exact

ENO−LLF3

TeCNO3

−5 0 5

0.4

0.6

0.8

1

1.2

Exact

ENO−LLF4

TeCNO4

−5 0 5

0.4

0.6

0.8

1

1.2

Exact

ENO−LLF5

TeCNO5

(a) 50 gridpoints

−5 0 5

0.4

0.6

0.8

1

1.2

Exact

ENO−LLF3

TeCNO3

−5 0 5

0.4

0.6

0.8

1

1.2

Exact

ENO−LLF4

TeCNO4

−5 0 5

0.4

0.6

0.8

1

1.2

Exact

ENO−LLF5

TeCNO5

(b) 100 gridpoints

Figure 6.8. Density at t = 1.3 computed with ENO-LLF and TECNO forthe Lax shock tube.

The density ρ, velocity field (wx,wy), pressure p and total energy E are related by the equationof state

E =p

γ − 1+ρ((wx)2 + (wy)2)

2.

The reader is referred to [FMT12b] for the two-dimensional entropy conservative flux for(6.4). We use Roe-type diffusion matrices Λx and Λy of the form (2.24) in the TECNOdiffusion operator.

Long-time vortex advection. We start by testing the TECNO schemes on a smooth testcase for the two-dimensional Euler equations, taken from Shu [Shu97]. The initial data is setin terms of velocity u and 3, temperature θ =

pρ

and thermodynamic entropy s = log p−γ log ρ:

wx = 1 − (y − yc)ϕ(r), wy = 1 + (x − xc)ϕ(r), θ = 1 −γ − 1

2γϕ(r)2, s = 0,

where (xc, yc) is the initial center of the vortex, r :=√

(x − xc)2 + (y − yc)2 and

ϕ(r) := εeα(1−τ2), τ :=rrc.


−5 0 5

1

1.5

2

2.5

3

3.5

4

4.5

Exact

ENO−LLF3

TeCNO3

−5 0 5

1

1.5

2

2.5

3

3.5

4

4.5

Exact

ENO−LLF4

TeCNO4

−5 0 5

1

1.5

2

2.5

3

3.5

4

4.5

Exact

ENO−LLF5

TeCNO5

0.5 1 1.5 2 2.5

3

3.5

4

4.5

Exact

ENO−LLF3

TeCNO3

0.5 1 1.5 2 2.5

3

3.5

4

4.5

Exact

ENO−LLF4

TeCNO4

0.5 1 1.5 2 2.5

3

3.5

4

4.5

Exact

ENO−LLF5

TeCNO5

Figure 6.9. Density at t = 1.8 computed with ENO-LLF and TECNOfor the Shu-Osher shock-entropy wave interaction problem. Bottom row:closeup of x ∈ [0.2, 2.5].

We set the parameters γ = 7/5, ε = 52π , α = 1/2, rc = 1 and (xc, yc) = (5, 5). The exact

solution to this initial value problem is simply

u(x, y, t) = u(x − t, y − t, 0).

In other words, the initial vortex, centered at (xc, yc) is advected diagonally with a velocity of1 in the x- and y-directions.

The computational domain is set to [0, 10] × [0, 10] and we use periodic boundary con-ditions to simulate the flow over the entire plane. We compute up to t = 100, during whichtime the vortex will have traversed through the domain 10 times and should end up exactlywhere it started. Figure 6.10 shows the computed density at the final time step on a meshof 200 × 200 points. Clearly, there is a significant gain in accuracy with increased order ofaccuracy, and the 3rd and 4th order TECNO schemes deviate only by a few percent fromthe exact solution. This experiment illustrates the robust performance of high-order TECNOschemes in resolving smooth solutions.

Vortex-shock interaction. This problem consists of a single left-moving shock whichinteracts with a right-moving vortex, and has been taken from [Shu97]. The initial shock hasthe values

u(x, 0) =

uL if x < 0.5uR otherwise,


2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

0 2 4 6 8 10

0.5

0.6

0.7

0.8

0.9

1

(a) TECNO2

0 2 4 6 8 10

0.5

0.6

0.7

0.8

0.9

1

(b) TECNO3

0 2 4 6 8 10

0.5

0.6

0.7

0.8

0.9

1

(c) TECNO4

Figure 6.10. TECNO schemes on the long-time vortex advection problem.Top: ρ at t = 100. Bottom: A slice in x-direction at y = 5 of TECNO(circles) and the exact solution (line).

with (ρL, pL, uL, 3L) = (1, 1,√γ, 0) and

ρR = ρL

(1 + βpR

β + pR

), pR = 1.3

wxR =√γ +√

2

1 − pR√γ − 1 + p(γ + 1)

, 3R = 0,

where β := γ+1γ−1 . The left state uL is then perturbed slightly by adding a vortex. The exact

values are specified by the perturbation in velocity, temperature and entropy:

w1 =y − yc

rcϕ(r), w2 = −

x − xc

rcϕ(r), θ = −

γ − 14αγ

ϕ(r)2, s = 0.

Here, r and ϕ are exactly as in the previous experiments. We set the free parameters to beε = 0.3, (xc, yc) = (0.25, 0.5), rc = 0.05 and α = 0.204. With these parameters, the jump inpressure across the shock wave is about twice as big as the magnitude of the vortex.

We compute on the domain [0, 1] × [0, 1] up to time t = 0.35. The domain is partitionedinto 200 × 200 grid points. The computed densities are plotted in Figure 6.11. The resultsshow that the TECNO schemes resolve both the shock as well as the smooth vortex well.There is a gain in resolution as higher order accurate schemes are employed. The results arecomparable to those obtained with standard ENO and WENO schemes in [Shu97].


0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

(a) TECNO2

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

(b) TECNO3

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

(c) TECNO4

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

(d) TECNO5

Figure 6.11. TECNO schemes on the shock-vortex interaction problem.The density is plotted at time t = 0.35 on a mesh of 200 × 200 points.

Cloud-shock interaction: The initial data for this test case is set to be

ρ =

3.86859 if x < 0.0510 if r < 0.151 otherwise

p =

167.345 if x < 0.0510 otherwise

wx =

11.2536 if x < 0.051 otherwise

wy ≡ 0,

where r :=√

x2 + y2. The computational domain is [0, 1] × [0, 1] with Neumann type non-reflecting boundary conditions. The exact solution in this case consists of the interactionof a right moving shock with a high density bubble, resulting in a complicated pattern thatincludes both bow and tail shocks as well as smooth regions in the center of the domain. Thecomputed densities on an mesh of 200×200 points at time t = 0.06 are shown in Figure 6.12.For the sake of comparison, a reference solution computed with the TECNO3 scheme on amesh of 1400×1400 points is also shown. The results illustrate that the TECNO schemes arestable and resolve the complex solution quite well. There is a clear gain in accuracy with theTECNO3 and TECNO4 schemes compared to the TECNO2 scheme.


(a) Reference (b) TECNO2

(c) TECNO3 (d) TECNO4

Figure 6.12. TECNO schemes on the cloud-shock interaction problem.The density at time t = 0.06 is plotted on a mesh of 200 × 200 points.A reference solution is also plotted for comparison.

CHAPTER 7

Conclusions

We have developed high-order accurate finite difference schemes for both scalar con-servation laws and systems of hyperbolic conservation laws. The entropy stability of thescheme provides a bound on the gradient which allows us to prove convergence to the en-tropy solution for scalar equations, and convergence to an entropy measure-valued solutionfor systems of equations. The scheme is non-oscillatory and robust with respect to differentequations and initial data. It is free of any tuning parameters and is straightforward to imple-ment. Finally, it is computationally cheap, both for one-dimensional scalar problems and formulti-dimensional systems of equations. We have seen that the scheme performs well on aseries of test problems compared to existing finite difference schemes.

Future work

We conclude with the following suggestions for future work:The convergence proof of the TECNO scheme relies on Conjecture 5.16 about the total

variation bound of the ENO procedure. A proof (or a counterexample) of this conjecturewould be interesting also in its own right.

The high-order accurate entropy conservative fluxes which the TECNO schemes are builton rely on a Cartesian mesh. To make the scheme more applicable to real-world problems, ageneralization to unstructured meshes would be necessary. Entropy conservative schemes onunstructured meshes would presumably take the form of finite element methods; see [Tad03,Section 9]. These would then be coupled with appropriately chosen diffusion operators toguarantee the stability and convergence of the method.

The schemes considered in this thesis are semi-discrete. It would be interesting to in-vestigate explicit time integration methods such that the resulting fully discrete scheme is en-tropy stable. The nonlinearity of the spatial discretization makes this a formidable challenge.For linear spatial discretizations, it has been shown that the third-order explicit Runge-Kuttascheme is entropy stable [Tad02], but preliminary investigations indicate that this does notcarry over to the fourth- or fifth-order scheme.

83

APPENDIX A

Young measures

In this chapter we review the theory of measure-valued functions, so-called Young mea-sures. We refer the reader to e.g. [MNRR96] for a more thorough introduction.

We denote by C0(Rn) the set of continuous functions g : Rn → R such that |g(λ)| → 0 as|λ| → ∞, and equip it with the supremum norm. Thus, C0(Rn) is the closure of D(Rn) withrespect to ‖ · ‖∞. The dual space of C0(Rn) is denotedM(Rn), and it may be shown that thisspace is isometrically isomorphic to the set of finite Radon measures on Rn (a Radon measureis a Borel measure that is inner regular and locally finite). We will not distinguish betweenthese equivalent definitions ofM and will frequently write for g ∈ C0(Rn), µ ∈ M(Rn)

〈µ, g〉 =

∫Rn

g(λ) dµ(λ).

A frequently occuring subset ofM(Rn) is the set of probability measures,

Prob(Rn) := µ ∈ M(Rn) : µ > 0, µ(Rn) = 1.

(By µ > 0 we mean µ(B) > 0 for all Borel sets B, or equivalently, 〈µ, g〉 > 0 for all nonnega-tive g ∈ C0(Rn).)

Let Ω ⊂ Rm be open. A Young measure is a weak-∗ measurable function ν : Ω →

Prob(Rn); we denote ν as being parameterized by its argument, νy := ν(y). Weak-∗ measur-ability means that the function y 7→ 〈νy, g〉 is measurable for all g ∈ C0(Rn). Given a Youngmeasure ν, we denote g(y) := 〈νy, g〉 and u(y) := 〈νy, id〉 =

∫Rn λ dνy(λ). (We frequently abuse

notation and write 〈νy, λ〉 =∫Rn λ dνy(λ).) The most obvious example of a Young measure is a

function. If u : Ω→ Rn is any measurable function, then νy := δu(y) defines a Young measurewhich satisfies g = g(u) for all g ∈ C(Rn).

The full power of Young measures is seen in the following Fundamental Theorem ofYoung measures: given any sequence of functions, it is enough to assume a uniform L∞

bound to guarantee (weak) convergence towards a Young measure.

Theorem A.1 (Tartar [Tar79]). Let Ω ⊂ Rm be open and let uk : Ω → Rn be a boundedsequence in L∞(Ω,Rn). Let K ⊂ Rn be the bounded set K =

⋃k∈N range(uk). Then there is a

subsequence, still denoted by uk, and a Young measure ν with supp(νy) ⊂ K such that

g(uk)∗ g in L∞(Ω)

for all g ∈ C(Rn), where g(y) = 〈νy, g〉 =∫Rn g(λ) dνy(λ).

85

86 A. YOUNG MEASURES

Theorem A.1 has been generalized to unbounded sequences of functions that decay “suf-ficiently fast” at infinity; see Ball [Bal89].

Lemma A.2 (DiPerna [DiP85], Corollary 2.1). Let Ω ⊂ Rm be open and let ukk∈N be abounded sequence in L∞(Ω,Rn), converging weakly-∗ to some u ∈ L∞(Ω,Rn). Let ν be theYoung measure associated with ukk. Then the following are equivalent:

(i) νy = δu(y) for a.e. y ∈ Ω.(ii) uk → u in Lp

loc(Ω,Rn) for every 1 6 p < ∞, and for every compact K ⊂ Rm there isa subsequence such that uk → u a.e. in K ∩Ω.

Proof. By extending uk and u by zero we may assume that Ω = Rm. If νy = δu(y) for a.e. ythen g(y) = g(u(y)) for a.e. y. Hence, g(uk)

∗ g(u) in L∞, so in particular,∫

K(u − uk)2ϕ dy =

∫K

(u2 + u2k − 2uku)ϕ dy→ 0

for every ϕ ∈ L1(K) and compact K ⊂ Rm. Setting ϕ = 1, we find that uk → u in L2loc.

Convergence in Lploc for all 1 6 p < ∞ follows from this and the L∞ bound on uk. Last,

pointwise convergence follows from a result due to Riesz [Bar66, pp. 71-73].Conversely, assume that uk → u pointwise a.e. Then g(uk(y)) → g(u(y)) = g(y) for

a.e. y ∈ Ω, or equivalently, 〈νy, g〉 = 〈δu(y), g〉 for all g ∈ C0(R). Hence, νy = δu(y).

Hence, to show that the convergence uk → u is strong, it suffices to show that νy is aDirac measure for a.e. y. The following auxiliary lemma will prove convenient in showingthis.

Lemma A.3 (Vecchi [MNRR96]). Let Ω ⊂ Rm be open and let ν : Ω → Prob(Rn) be aYoung measure. Then the following are equivalent:

(i) There is a u ∈ L∞(Ω,Rn) such that νy = δu(y) for a.e. y ∈ Ω.(ii) For every y ∈ Ω there is an H ∈ L1(Rn × Rn; νy × νy

)such that

(A.1) H(a, b)

> 0 if a , b= 0 if a = b

and∫Rn×Rn

H d(νy × νy) = 0 ∀ y ∈ Ω.

Proof. If ν = δu then H(a, b) = |a − b| satisfies (A.1).Let H satisfy (A.1), and assume conversely that there exist distinct points a, b ∈ supp(νy)

for a y ∈ Ω. Let U and V be disjoint neighbourhoods of a and b, respectively. ThenH(y1, y2) > 0 for all (y1, y2) ∈ U × V . Hence,

0 =

∫Rn×Rn

H d(νy × νy) >∫

U×VH d(νy × νy) > 0,

a contradiction. Hence, supp(νy) can contain at most one point, so either νy = 0 or νy = δu(y)for some u(y) ∈ Rn. But since νy(Rn) = 1, it cannot vanish everywhere, so we must haveνy = δu(y). The function u is measurable by the weak-∗measurability of ν, and is in L∞(Ω,Rn)since |u(y)| 6 supz supw∈supp(νz) |w| < ∞.

Bibliography

[ACD+08] F. Arandiga, A. Cohen, R. Donat, N. Dyn, and B. Matei. Approximation of piecewise smooth func-tions and images by edge-adapted (ENO-EA) nonlinear multiresolution techniques. Appl. Comput. Har.Anal., 24:225–250, 2008.

[ACDD05] F. Arandiga, A. Cohen, R. Donat, and N. Dyn. Interpolation and approximation of piecewise smoothfunctions. SIAM Journal on Numerical Analysis, 43(1):41–57, 2005.

[Bal89] J. Ball. A version of the fundamental theorem for Young measures. In M. Rascle, D. Serre, and M. Slem-rod, editors, PDEs and Continuum Models of Phase Transitions, volume 344 of Lecture Notes inPhysics, pages 207–215. Springer Berlin / Heidelberg, 1989.

[Bar66] R. G. Bartle. The elements of integration and Lebesgue measure. John Wiley & Sons, Inc., 1966.[Bar98] T. J. Barth. Numerical methods for gasdynamic systems on unstructured meshes. In M. Ohlberger, D.

Kroner, and C. Rohde, editors, An Introduction to Recent Developments in Theory and Numerics ofConservation Laws, volume 5 of Lecture Notes in Computational Science and Engineering, pages 195–285. Springer, Berlin, 1998.

[BCP00] A. Bressan, G. Crasta, and B. Piccoli. Well-posedness of the Cauchy problem for n×n systems of conser-vation laws, volume 146 of Memoirs of the American Mathematical Society. American MathematicalSociety, 2000.

[BLS11] Y. Brenier, C. De Lellis, and L. Szkelyhidi. Weak-strong uniqueness for measure-valued solutions.Communications in Mathematical Physics, 305:351–361, 2011.

[BLY99] A. Bressan, T.-P. Liu, and T. Yang. L1 stability estimates for n×n conservation laws. Archive for RationalMechanics and Analysis, 149:1–22, 1999.

[CDM03] A. Cohen, N. Dyn, and B. Matei. Quasilinear subdivison schemes with applications to ENO interpola-tion. Appl. Comput. Har. Anal., 15:89–116, 2003.

[CL89] G.-Q. Chen and Y.-G. Lu. A study to applications of the theory of compensated compactness. ChineseScience Bulletin, 34:15–19, 1989.

[CM80] M. G. Crandall and A. Majda. Monotone difference approximations for scalar conservation laws. Math.Comp., 34:1–21, 1980.

[CZ02] T. Chan and H. M. Zhou. ENO-wavelet transforms for piecewise smooth functions. SIAM Journal onNumerical Analysis, 40(4):1369–1404, 2002.

[DiP83] R. DiPerna. Convergence of approximate solutions to conservation laws. Archive for Rational Mechan-ics and Analysis, 82:27–70, 1983.

[DiP85] R. J. DiPerna. Measure-valued solutions to conservation laws. Archive for Rational Mechanics andAnalysis, 88:223–270, 1985.

[DPRV99] P. Degond, P.-F. Peyrard, G. Russo, and P. Villedieu. Polynomial upwind schemes for hyperbolic sys-tems. Comptes Rendus de l’Acadmie des Sciences - Series I - Mathematics, 328(6):479 – 483, 1999.

[Fjo09] U. S. Fjordholm. Structure preserving finite volume methods for the shallow water equations. Master’sthesis, University of Oslo, 2009.

87

88 BIBLIOGRAPHY

[FMT09] U. S. Fjordholm, S. Mishra, and E. Tadmor. Energy preserving and energy stable schemes for theshallow water equations. In ”Foundations of Computational Mathematics”, Proc. FoCM held in HongKong 2008, volume 363 of London Math. Soc. Lecture Notes Series, pages 93–139, 2009.

[FMT12a] U. S. Fjordholm, S. Mishra, and E. Tadmor. Arbitrarily high order accurate entropy stable essen-tially non-oscillatory schemes for systems of conservation laws. SIAM Journal on Numerical Analysis,50:544–573, 2012.

[FMT12b] U. S. Fjordholm, S. Mishra, and E. Tadmor. ENO reconstruction and ENO interpolation are stable.Foundations of Computational Mathematics, 2012.

[Gli65] J. Glimm. Solutions in the large for nonlinear hyperbolic systems of equations. Communications onPure and Applied Mathematics, 18(4):697–715, 1965.

[God61] S. K. Godunov. An interesting class of quasi-linear systems. Dokl. Akad. Nauk SSSR, 139:521–523,1961.

[GR91] E. Godlewski and P.-A. Raviart. Hyperbolic systems of conservation laws. Ellipses, 1991.[GS98] S. Gottlieb and C.-W. Shu. Total variation diminishing Runge-Kutta schemes. Mathematics of Compu-

tation, 67:73–85, 1998.[Har83] A. Harten. On the symmetric form of systems of conservation laws with entropy. Journal of Computa-

tional Physics, 49(1):151 – 164, 1983.[Har89] A. Harten. ENO schemes with subcell resolution. J. Comput. Phys., 83(1):148–184, 1989.[HEOC87] A. Harten, B. Engquist, S. Osher, and S. R. Chakravarthy. Uniformly high order accurate essentially

non-oscillatory schemes, III. J. Comput. Phys., 71(2):231–303, 1987.[HFM86] T.J.R. Hughes, L.P. Franca, and M. Mallet. A new finite element formulation for computational fluid

dynamics: I. symmetric forms of the compressible euler and navier-stokes equations and the secondlaw of thermodynamics. Computer Methods in Applied Mechanics and Engineering, 54(2):223 – 234,1986.

[HHLK76] A. Harten, J. M. Hyman, P. D. Lax, and B. Keyfitz. On finite-difference approximations and entropyconditions for shocks. Communications on Pure and Applied Mathematics, 29(3):297–322, 1976.

[Hop69] E. Hopf. On the right weak solution of the Cauchy problem for quasilinear equations of first order.Journal of Mathematics and Mechanics, 19:483–487, 1969.

[IR09] F. Ismail and P. L. Roe. Affordable, entropy-consistent Euler flux functions II: Entropy production atshocks. J. Comput. Phys., 228(15):5410–5436, 2009.

[JS96] G.-S. Jiang and C.-W. Shu. Efficient implementation of weighted ENO schemes. J. Comput. Phys.,126(1):202–228, 1996.

[Kru70] S. N. Kruzkov. First order quasilinear equations in several independent variables. Math USSR SB,10(2):217–243, 1970.

[Lax57] P. D. Lax. Hyperbolic systems of conservation laws II. Communications on Pure and Applied Mathe-matics, 10(4):537–566, 1957.

[Lax71] P. D. Lax. Shock waves and entropy. Contributions to nonlinear functional analysis, pages 603–634,1971.

[LeV02] R. J. LeVeque. Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, 2002.[LMR03] P. G. LeFloch, J. M. Mercier, and C. Rohde. Fully discrete, entropy conservative schemes of arbitrary

order. SIAM Journal on Numerical Analysis, 40(5):pp. 1968–1992, 2003.[LOC94] X.-D. Liu, S. Osher, and T. Chan. Weighted essentially non-oscillatory schemes. J. Comput. Phys.,

115(1):200 – 212, 1994.[MNRR96] J. Malek, J. Necas, M. Rokyta, and M. Ruzicka. Weak and measure-valued solutions to evolutionary

PDEs, volume 13 of Applied Mathematics and Mathematical Computation. Chapman & Hall, London,1996.

[Moc80] M.S Mock. Systems of conservation laws of mixed type. Journal of Differential Equations, 37(1):70 –88, 1980.

[Mur78] F. Murat. Compacit par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci, 5(3):489–507, 1978.

BIBLIOGRAPHY 89

[Mur81] F. Murat. L’injection du cone positif de H1 dans W1,q est compacte pour tout q < 2. J. Math. PuresAppl. (9), 60(3):309–322, 1981.

[Ole57] O. A. Oleinik. On discontinuous solutions of non-linear differential equations. Uspekhi Mat. Nauk.,12:3–73, 1957. English translation, Amer. Math. Soc. Trans., ser. 2, no.26, pp. 95-192.

[Osh84] S. Osher. Riemann solvers, the entropy condition, and difference approximations. SIAM Journal onNumerical Analysis, 21(2):pp. 217–235, 1984.

[Pan94] E. Yu. Panov. Uniqueness of the solution of the cauchy problem for a first order quasilinear equationwith one admissible strictly convex entropy. Mathematical Notes, 55:517–525, 1994.

[Pan10] E.Yu. Panov. Existence and strong pre-compactness properties for entropy solutions of a first-orderquasilinear equation with discontinuous flux. Archive for Rational Mechanics and Analysis, 195:643–673, 2010.

[Roe81] P.L Roe. Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys,43(2):357 – 372, 1981.

[Sch89] S. Schochet. Examples of measure-valued solutions. Communications in Partial Differential Equations,14(5):545–575, 1989.

[Shu90] C.-W. Shu. Numerical experiments on the accuracy of ENO and modified ENO schemes. J. Sci. Com-put., 5(2):127–149, 1990.

[Shu97] C.-W. Shu. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolicconservation laws. Technical report, 1997.

[Shu99] C.-W. Shu. High order ENO and WENO schemes for computational fluid dynamics. In T. J. Barth andH. Deconinck, editors, High-Order Methods for Computational Physics, volume 9 of Lecture Notes inComputational Science and Engineering, pages 439–582. Springer, 1999.

[SO88] C.-W. Shu and S. Osher. Efficient implementation of essentially non-oscillatory shock-capturingschemes. J. Comput. Phys., 77(2):439–471, 1988.

[SO89] C.-W. Shu and S. Osher. Efficient implementation of essentially non-oscillatory shock-capturingschemes, II. J. Comput. Phys., 83(1):32–78, 1989.

[Tad84] E. Tadmor. Numerical viscosity and the entropy condition for conservative difference schemes. Mathe-matics of Computation, 43(168):pp. 369–381, 1984.

[Tad87a] E. Tadmor. The numerical viscosity of entropy stable schemes for systems of conservation laws. I.Mathematics of Computation, 49:91–103, 1987.

[Tad87b] Eitan Tadmor. Entropy functions for symmetric systems of conservation laws. Journal of mathematicalanalysis and applications, 122:355–359, 1987.

[Tad02] E. Tadmor. From semi-discrete to fully discrete: stability of Runge-Kutta schemes by the energymethod. II. In D. Estep and S. Tavener, editors, ”Collected Lectures on the Preservation of Stabilityunder Discretization”, volume 109 of Proceedings in Applied Mathematics, pages 25–49, 2002.

[Tad03] E. Tadmor. Entropy stability theory for difference approximations of nonlinear conservation laws andrelated time dependent problems. Acta Numerica, 12:451–512, 2003.

[Tar79] L. Tartar. Compensated compactness and applications to partial differential equations. In Nonlinearanalysis and mechanics: Heriot-Watt Symposium, Vol. IV, pages 136–212. Pitman, 1979.

hyperbolic conservation laws - eth z · to the conservation law. this unique solution is called the...

Documents