J. Vovelle and S. Martin- Large-Time Behavior of Entropy Solutions to Scalar Conservation Laws on Bounded Domain

8/3/2019 J. Vovelle and S. Martin- Large-Time Behavior of Entropy Solutions to Scalar Conservation Laws on Bounded Domain

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123

Theory, Numerics, Applications

Sylvie Benzoni-Gavage Denis SerreEditors

Hyperbolic Problems:

Proceedings of the Eleventh International Conference

on Hyperbolic Problems held in Ecole Normale

Suprieure, Lyon, July 1721, 2006


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Sylvie Benzoni-GavageInstitut Camille JordanUMR CNRS 5208Universit Claude Bernard Lyon 143, Bd du 11 novembre 191869622 Villeurbanne cedex, [email protected]

Denis SerreUMPA, UMR CNRS 5669Ecole Normale Suprieure de Lyon46, alle dltalie69364 Lyon cedex 07, [email protected]

ISBN 978-3-540-75711-5 e-ISBN 978-3-540-75712-2

Library of Congress Control Number: 2007 937 290

Mathematics Subject Classification (2000): 35L40, 35L60, 35L65, 35L67, 35Q35, 35Q72, 35Q75,

65M06, 65M50, 83C55

c 2008 Springer-Verlag Berlin Heidelberg

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Large-Time Behavior of Entropy Solutions to

Scalar Conservation Laws on Bounded Domain

J. Vovelle and S. Martin

Summary. We study the large-time behavior of entropy solutions to scalar conser-vation laws on a one-dimensional bounded domain. We show that, if the boundarydata are stabilizing (in the sense that they allow for the existence of a station-ary solution), then the entropy solution converges to a stationary solution to theproblem.

1 Introduction

Let A Liploc(R). Let u(t) be the entropy solution to the following CauchyDirichlet problem

ut + (A(u))x = 0, t > 0, 0 < x < 1, (1)

u(x, 0) = u0(x), 0 < x < 1 (2)

u(0, t) = u1(t), u(1, t) = u2(t), t > 0. (3)

We investigate the asymptotic behavior [t +] of u(t). The data u0 andu are supposed to be measurable and bounded a.e.

1.1 Large-Time Behavior of the Entropy Solutionto the Cauchy Problem

The behavior as [t +] ofu(t), the entropy solution to the Cauchy problem

ut + (A(u))x = 0, x R, t > 0 (4)

u(x, 0) = u0(x), x R, (5)

has been extensively analyzed since the first works of Lax ([Lax57], and wealso refer to Ilin and Oleinik [IO60], DiPerna [DiP75], Liu and Pierre [LP84],DaFermos [Daf85], and Kim [Kim03]). In particular, if the flux A is strictly


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1070 J. Vovelle and S. Martin

convex and if u0 L1 L(R), the entropy solution converges to a so-calledN-wave with specific rates in L1(R) and L(R).

A related problem is the question of the stability of the solution travelingwaves of (4) and (5) or, more generally, the question of the stability of solu-tion as profiles to conservation laws. Let us refer to the works of Kawashimaand Matsumura [KM85], Matsumura and Nishihara [MN94], Liu and Nishi-hira [LN97], Freistuhler and Serre [FS98], and Serre [Ser04]. We state inparticular the following results of Freistuhler and Serre.

Theorem 1 (Freistuhler and Serre). Assume thatA is strictly convex. Letw be a (nontrivial) shock profile of (4). Then, for all u0 w + L

1(R),

u(t) w( + )L1

(R) 0, =

1

w+ wR u0 w.

Remark 1. It is a result of orbital stability; the value of is determined by theconservation of the mass. The result is false as soon as A is not convex.

1.2 Large-Time Behavior of the Entropy Solutionto the CauchyDirichlet Problem

The CauchyDirichlet problem is of course different from the Cauchy problemsince the boundary data can possibly act on the values of the solution at anytime and prevent any kind of convergence. There are much less references inthis topic: see Mascia and Terracina [MT99] (asymptotic behavior driven by

a source term) and also the work of Freistuhler and Serre [FS01] related tostability of boundary layers in the viscous approximation of a first-order scalarconservation law. We consider a.e. bounded measurable data and, without lossof generality, we actually consider data (hence solutions) with values in [0, 1].Our result is the following one: let

X0 := L(0, 1;[0, 1]), X := L(R+; [0, 1])

2

be endowed with the L1 norm. These are closed subspaces of L1(0, 1) andL1(R+)2, respectively (because the limit of a converging sequence in Lp isalso the limit of a a.e. converging subsequence), hence Banach spaces. Onthe space X0 X, we define the flow S(t) associated to the resolution of theCauchyDirichlet problem (1)(2)(3) (see Sect. 2). We then introduce the

notion of stationary solution and stabilizing boundary data.

Definition 1. A function w X0 is said to be a stationary solution to (1)(2)(3) if there exists (u1, u2) X such that w = S(t)(w, (u1, u2)): we saythat w is a stationary solution associated to the boundary data (u1, u2). Aset of boundary data (u1, u2) X is said to be stabilizing if there exists astationary solution associated to it.


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Large-Time Behavior of Entropy Solutions 1071

Theorem 2. Suppose that A is a strictly convex function on [0, 1]. Let

(u1, u2) X

be a stabilizing set of boundary data. Then for all u0 X0, S(t)(u0, (u1, u2))converges to a stationary solution associated to (u1, u2) when t +.

We refer to [MV06] for a similar result in case of an nonautonomousequation (A depending on the variable x).

The proof of Theorem 2 uses the tools developed by Freistuhler and Serrefor the study of the stability of profiles to conservation laws, it also exploits theprinciple of comparison for sub- and supersolutions for the CauchyDirichletproblem (1)(2)(3).

2 Sub-, Supersolutions, and Flow

Definition 2. Let be an open subset of R. Let u0 X0, (u1, u2) X. Afunction u L((0, 1) R+) is an entropy subsolution to the problem (1)(2)(3) on if: for all k R, for all Cc ( R+), 0,

10

+0

(u k)+t + sgn+(u k)(A(u) A(k))xdxdt

+

10

(u0 k)+(x, 0)dx

++0

sgn+(u1 k)(A1(u1) A1(k))(0, t)dtt

+

+0

sgn+(u2 k)(A2(u2) A2(k))(1, t)dt 0, (6)

with A1(u) :=u0 (A

())+d and A2(u) :=u0 (A

())d. A supersolution on is defined similarly, using the entropy(u k). In case u is both a sub- andsupersolution on [0, 1], we say that u is an entropy solution.

Theorem 3. The entropy solution u of the CauchyDirichlet problem (1)(2)(3) is continuous in time with values in L1(0, 1) and thus defines a flowS(t) which has the following properties: it is monotone:

(u0, (u1, u2)) (v0, (v1, v2))

impliesS(t)(u0, (u1, u2)) S(t)(v0, (v1, v2))

for allt > 0; it is L1 nonexpansive:

S(t)(u0, (u1, u2)) S(t)(v0, (v))L1(0,1) u0 v0L1(0,1)

+ Lu1 v1L1(0,t) + Lu2 v2L1(0,t), (7)


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where L := AL; it preserves the space BV: if

u0BV(0,1), u1BV(R+), u2BV(R+) C,

thenS(t)(u0, (u1, u2))BV(0,1) C

. (8)

The property of monotony and (7) can be deduced from the followingresult of comparison (and more generally we refer to [BLN79, Ott93, Ott96]for the proof of Theorem 3).

Proposition 1. Let be an open subset of R. Let u, v L((0, 1) R+) be,respectively, an entropy sub- and supersolution to the problem (1)(2)(3) on. Then, for all C

c( R

+), 0,

10

+0

(u v)+t + sgn+(u v)(A(u) A(v))xdxdt

+

10

(u0 v0)+(x, 0)dx

+

+0

sgn+(u1 v1)(A1(u1) A1(v1))(0, t)dt

+

+0

sgn+(u2 v2)(A2(u2) A2(v2))(1, t)dt 0, (9)

We will use Proposition 1 to derive Lyapunov functionals for the evolu-

tion along the flow S, but first we have to analyze the stationary solutionsassociated to given stabilizing boundary data.

3 Stationary Solution

Let (u1, u2) X be stabilizing boundary data. Let (u1, u2) be the set ofstationary solutions associated to (u1, u2). The set (u1, u2) is nonempty byhypothesis: let w (u1, u2). The weak form of the equation shows that theflux q := A(w) is constant. Since A is strictly convex on [0, 1], the equationq = A(w) admits at most two constant solutions, say . By use of theentropy conditions in their weak form, we show that

(u1, u2)

wz := 1(z,1] + 1[0,z), z [0, 1]

(to sum up, the proof of this result uses Proposition 1 on = (0, 1) andamounts to compare an element of(u1, u2) with specific stationary solutionswith flux q = q). Then, by analysis of the boundary conditions, we prove thefollowing result.

In the following, and to give self-consistent proofs in this short chapter(see Sect. 4), we will assume:


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there is a nontrivial stationary shock in (u1, u2). (10)

This amounts to suppose < and that there is z (0, 1) such that wz (u1, u2). Notice that, although we do this hypothesis, Theorem 2 is true inits generality. Under hypothesis (10), the result of Theorem 2 is to be relatedto the result of Theorem 1. Indeed, in that case, we have

(u1, u2) =

wz := 1(z,1] + 1[0,z), z [0, 1]

,

i.e., every stationary solution is a shift of wz .

4 Convergence in Large Time

We give the proof of Theorem 2 under hypothesis (10). Let (u1, u2) X befixed stabilizing boundary data. We use the notations of Sect. 3. For u0 X0,we also denote (S(t)u0) the trajectory (S(t)(u0, (u1, u2)) since the boundarydata are fixed.

Lemma 1. Let u0 X0. The trajectory (S(t)u0) is relatively compact inL1(0, 1).

The relative compactness of the bounded sets of BV(0, 1) in L1(0, 1) andthe property (8) show that the result in case u0 has, additionally, a boundedvariation on (0, 1). We then deduce the lemma from the fact that S(t) is L1

nonexpansive. Indeed, if > 0, then there exists u0 X0 BV(0, 1) such thatu0 u0L1(0,1) < /2. We then have S(t)u0 S(t)u0L1(0,1) < /2 for allt > 0. Since {S(t)u0} is relatively compact in X0, there is a finite number ofballs of radii /2 covering it. We deduce that {S(t)u0} is covered by a finitenumber of balls of radii . Since is arbitrary and X0 is a Banach space,{S(t)u0} is relatively compact in X0.

A corollary of Lemma 1 is the fact that each limit set

(u0) :=t>0

{S()(u0); t}

is not empty. In a second step, we show that (u0) (u1, u2). To thispurpose, we first prove Lemma 2.

Lemma 2. Set

(X0) :=

u0X0

(u0) and X1 := {w X0, w a.e.}.

Then (X0) is a subset of X1 and X1 is invariant by the flow.


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That X1 is invariant by the flow follows from the result of comparison ofProposition 1 between S(t)u0 and the constant solutions and . To showthe inclusion (X0) X1, we also use comparison between S(t)u0 and theconstant solution (or ): for every Cc (R (0, +)), 0,10

+0

(S(t)u0 )+t + sgn+(S(t)u0 )(A(S(t)u0) A())xdxdt 0.

(11)Since < and , , are both constant solutions of the equation A(w) = qand A is convex, we have := A() > 0 and

sgn+(S(t)u0 )(A(S(t)u0) A()) (S(t)u0 )+.

Setting (x, t) := ex(t), Cc

((0, +), 0 in (11), we derive theinequality

, (t) :=

10

(S(t)u0 )+exdx

in D(0, +), from which follows the exponential decrease of . Similarly, weshow that

t

10

(S(t)u0 )exdx

decreases exponentially fast to 0. These two results show that (X0) X1.The next step of the proof is to use the set of solutions {wz, z [0, 1]} to

derive Lyapunov functionals. Indeed, the result of comparison of Proposition 1(used with = R and independent on x) shows that, for any z [0, 1],

the map u (wz u)+L1(R)

is nonincreasing along the trajectories. As a consequence of the LaSalleprinciple, we have, for any a (u0),

(wz S(t)a)+L1(R) = cst = (wz a)

+L1(R).

But by Lemma 2, we have

(wz S(t)a)+L1(R) =

z0

( S(t)a)dx

and we deduce

z

0 ( S(t)a)dx =

z

0 ( a)dx, z (0, 1),

i.e.,z0 (S(t)aa)dx = 0 for all z (0, 1). This shows that S(t)a = a and that a

is a stationary solution. We then conclude to the convergence of the trajectory.First, there is at most one element in (u0) for if a (u0), the trajectorycan only get closer and closer to a because a is a stationary solution and S(t)is nonexpansive. Second, (u0) is nonempty by compactness, it is thereforereduced to one element and (S(t)u0) converges to this element a (u1, u2).


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References

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[Bre83] Y. Brenier, Resolution dequa tions devolution quasilineaires en dimensionN despace a laide dequations lineaires en dimensionN+1, J. DifferentialEquations 50 (1983), no. 3, 375390.

[Daf85] C. M. Dafermos, Regularity and large time behaviour of solutions of aconservation law without convexity, Proc. Roy. Soc. Edinburgh Sect. A 99(1985), no. 3-4, 201239.

[DiP75] R.J. DiPerna, Decay and asymptotic behavior of solutions to nonlin-ear hyperbolic systems of conservation laws, Indiana Univ. Math. J. 24

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[FS01] H. Freistuhler and D. Serre, The L1-stability of boundary layers for scalarviscous conservation laws, J. Dynam. Differential Equations 13 (2001),no. 4, 745755.

[GM83] Y. Giga and T. Miyakawa, A kinetic construction of global solutions of first order quasilinear equations, Duke Math. J. 50 (1983), no. 2, 505515.

[IO60] A.M. Ilin and O.A. Olenik, Asymptotic behavior of solutions of theCauchy problem for some quasi-linear equations for large values of the

time, Mat. Sb. (N.S.) 51 (93) (1960), 191216.[Kim03] Y.J. Kim, Asymptotic behavior of solutions to scalar conservation laws

and optimal convergence orders to N-waves, J. Differential Equations 192(2003), no. 1, 202224.

[KM85] S. Kawashima and A. Matsumura, Asymptotic stability of traveling wavesolutions of systems for one-dimensional gas motion, Comm. Math. Phys.101 (1985), no. 1, 97127.

[Lax57] P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl.Math. 10 (1957), 537566.

[LN97] T.-P. Liu and K. Nishihara, Asymptotic behavior for scalar viscous con-servation laws with boundary effect, J. Differential Equations 133 (1997),no. 2, 296320.

[LP84] T.-P. Liu and M. Pierre, Source-solutions and asymptotic behavior inconservation laws, J. Differential Equations 51 (1984), no. 3, 419441.

[LPT94] P.-L. Lions, B. Perthame, and E. Tadmor, A kinetic formulation of mul-tidimensional scalar conservation laws and related equations, J. Amer.Math. Soc. 7 (1994), no. 1, 169191.

[MV06] S. Martin, J. Vovelle, Large-time behaviour of the entropy solution of ascalar conservation law with boundary conditions, accepted for publicationin Quart. J. Mech. Appl. Math.

[MN94] A. Matsumura and K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Comm.Math. Phys. 165 (1994), no. 1, 8396.

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Contents

Part I Plenary Lectures

General Relativistic Hydrodynamics andMagnetohydrodynamics: Hyperbolic Systemsin Relativistic AstrophysicsJ.A. Font . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

On Approximations for Overdetermined Hyperbolic EquationsS.K. Godunov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Stable Galaxy Configurations

Y. Guo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35

Dissipative Structure of Regularity-Loss Typeand ApplicationsS. Kawashima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Dissipative Hyperbolic Systems: the Asymptotic Behaviorof SolutionsS. Bianchini, B. Hanouzet, and R. Natalini . . . . . . . . . . . . . . . . . . . . . . . 59

Part I I Invited Lectures

Higher Order Numerical Schemes for Hyperbolic Systems

with an Application in Fluid DynamicsV. Dolejs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

A Penalization Technique for the Efficient Computationof Compressible Fluid Flow with ObstaclesG. Chiavassa and R. Donat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89


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VIII Contents

Exact Solutions to Supersonic Flow onto a Solid WedgeV. Elling and T.-P. Liu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Resonance and NonlinearitiesT. Gallou et . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Lp-Stability Theory of the Boltzmann Equation Near VacuumS.-Y. Ha, M. Yamazaki, and S.-B. Yun . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

A Relaxation Scheme for the Two-Layer ShallowWater SystemR. Abgrall and S. Karni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Group Dynamics of Phototaxis: Interacting Stochastic

Many-Particle Systems and Their Continuum LimitD. Bhaya, D. Levy, and T. Requeijo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Vacuum Problem of One-Dimensional CompressibleNavierStokes EquationsH.-L. Li, J. Li, and Z. Xin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Stability and Instability Issues for Relaxation Shock ProfilesC. Mascia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

A New hp-Adaptive DG Scheme for Conservation Laws Basedon Error ControlA. Dedner and M. Ohlberger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Elliptic and Centrifugal Instabilities in Incompressible FluidsF. Gallaire, D. Gerard-Varet, and F. Rousset . . . . . . . . . . . . . . . . . . . . . . 199

On Compressible CurrentVortex SheetsY. Trakhinin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

On the Motion of Binary Fluid MixturesK. Trivisa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

On the Optimality of the Observability Inequalities forKirchhoff Plate Systems with Potentials in UnboundedDomainsX. Zhang and E. Zuazua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233


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Contents IX

Part I II Mini-Symposium on Hydraulics

High Order Finite Volume Methods Applied to SedimentTransport and Submarine AvalanchesD. Bresch, M.J.C. Daz, E.D. Fernandez-Nieto, A.M. Ferreiro,

and A. Mangeney . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

On a Well-Balanced High-Order Finite Volume Scheme forthe Shallow Water Equations with Bottom Topography andDry AreasJ.M. Gallardo, M. Castro, C. Pares, and J.M. Gonzalez-Vida . . . . . . . 259

A Simple Well-Balanced Model for Two-Dimensional CoastalEngineering ApplicationsF. Marche . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

A Model for Bed-Load Transport and MorphologicalEvolution in Rivers: Description and PertinenceA. Paquier and K. El Kadi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

Part IV Contributed Talks

Homogenization of Conservation Laws with OscillatorySource and Nonoscillatory Data

D. Amadori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .299

Short-Time Well-Posedness of Free-Surface Problemsin Irrotational 3D FluidsD.M. Ambrose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

Mathematical Study of Static Grain Deep-Bed Drying ModelsD. Aregba-Driollet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

Finite Volume Central Schemes for Three-DimensionalIdeal MHDP. Arminjon and R. Touma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

Finite Volume Methods for Low Mach Number Flows

under buoyancyP. Birken . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

Time Splitting with Improved Accuracyfor the Shallow Water EquationsA. Bourchtein and L. Bourchtein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339


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X Contents

Compact Third-Order Logarithmic Limiting for NonlinearHyperbolic Conservation LawsM. Cada, M. Torrilhon, and R. Jeltsch . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

A Finite Volume Grid for Solving Hyperbolic Problemson the SphereD. Calhoun, C. Helzel, and R.J. LeVeque . . . . . . . . . . . . . . . . . . . . . . . . . 355

Capturing Infinitely Sharp Discrete Shock Profiles with theGodunov SchemeC. Chalons and F. Coquel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

Propagation of Diffusing Pollutant by a Hybrid

EulerianLagrangian MethodA. Chertock, E. Kashdan, and A. Kurganov . . . . . . . . . . . . . . . . . . . . . . . 371

Nonlocal Conservation Laws with MemoryC. Christoforou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

Global Weak Solutions for a Shallow Water EquationG.M. Coclite, H. Holden, and K.H. Karlsen . . . . . . . . . . . . . . . . . . . . . . . 389

Structural Stability of Shock Solutions of Hyperbolic Systemsin Nonconservation Form via Kinetic RelationsB. Audebert and F. Coquel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

A Hyperbolic Model of Multiphase Flow

D. Amadori and A. Corli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

Nonlinear Stability of Compressible Vortex SheetsJ-F. Coulombel and P. Secchi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

Regularity and Compactness for the DiPernaLions FlowG. Crippa and C. De Lellis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

A Note on L1 Stability of Traveling Waves for aOne-Dimensional BGK ModelC.M. Cuesta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

The Weak Rankine Hugoniot InequalityB. Despres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

Numerical Investigations Concerning the Strategy of Controlof the Spatial Order of Approximation Along a FittedGasLiquid InterfaceC. Dickopp and J. Ballmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449


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Contents XI

Domain Decomposition Techniques and Hybrid MultiscaleMethods for Kinetic EquationsG. Dimarco and L. Pareschi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

A Shock Sensor-Based Second-Order Blended (Bx) UpwindResidual Distribution Scheme for Steady and UnsteadyCompressible FlowJ. Dobes and H. Deconinck. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

Artificial Compressibility Approximation for theIncompressible NavierStokes Equationson Unbounded DomainD. Donatelli and P. Marcati. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

Traveling-Wave Solutions for Hyperbolic Systemsof Balance LawsA. Dressel and W.-A. Yong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

A Hyperbolic-Elliptic Model for Coupled Well-PorousMedia FlowS. Evje and K.H. Karlsen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

High-Resolution Finite Volume Methods for ExtracorporealShock Wave TherapyK. Fagnan, R.J. LeVeque, T.J. Matula, and B. MacConaghy . . . . . . . . 503

Asymptotic Properties of a Class of Weak Solutions to the

NavierStokesFourier SystemE. Feireisl. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

A New Technique for the Numerical Solution of theCompressible Euler Equations with Arbitrary Mach NumbersM. Feistauer and V. Kucera. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523

Monokinetic Limits of theVlasov-Poisson/Maxwell-Fokker-Planck SystemL. Hsiao, F. Li, and S. Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

High-Resolution Methods and Adaptive Refinement forTsunami Propagation and Inundation

D.L. George and R.J. LeVeque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .541

Young Measure Solutions of Some Nonlinear MixedType EquationsH.-P. Gittel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551

Computing Phase Transitions Arising in Traffic Flow ModelingC. Chalons and P. Goatin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559


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XI I Contents

Dafermos Regularization for Interface Couplingof Conservation LawsB. Boutin, F. Coquel, and E. Godlewski. . . . . . . . . . . . . . . . . . . . . . . . . . . 567

Nonlocal Sources in Hyperbolic Balance Lawswith ApplicationsR.M. Colombo and G. Guerra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577

Comparison of Several Finite Difference Methods forMagnetohydrodynamics in 1D and 2DP. Havlk and R. Liska . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585

On Global Large Solutions to 1-D Gas Dynamics

E.E. Endres and H.K. Jenssen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .593

A Carbuncle Free Roe-Type Solverfor the Euler EquationsF. Kemm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601

WENOCLAW: A Higher Order Wave Propagation MethodD.I. Ketcheson and R.J. LeVeque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609

Unsteady Transonic Airfoil Flow Simulations usingHigh-Order WENO SchemesI. Klioutchnikov, J. Ballmann, H. Oliver, V. Hermes, and A. Alshabu 617

The PredictorCorrector Method for Solving

of Magnetohydrodynamic ProblemsT. Kozlinskaya and V. Kovenya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625

A Central-Upwind Scheme for Nonlinear Water WavesGenerated by Submarine LandslidesA. Kurganov and G. Petrova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635

An A Posteriori Error Estimate for Glimms SchemeM. Laforest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643

Multiphase Flows in Mass Transfer in Porous MediaW. Lambert and D. Marchesin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653

Nonlinear HyperbolicElliptic Coupled Systems Arising in

Radiation DynamicsC. Lattanzio, C. Mascia, and D. Serre . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661

The Lagrangian Coordinates Applied to the LWR ModelL. Ludovic, C. Estelle, and L. Jorge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671


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Contents XIII

Hyperbolic Conservation Laws and Spacetimeswith Limited RegularityP.G. LeFloch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679

Arbitrary LagrangianEulerian (ALE) Method in CylindricalCoordinates for Laser Plasma SimulationsM. Kucharik, R. Liska, R. Loubere, and M. Shashkov. . . . . . . . . . . . . . . 687

Numerical Aspects of Parabolic Regularization for ResonantHyperbolic Balance LawsM. Kraft and M. Lukacov a-Medvidov a . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695

Three-Dimensional Adaptive Central Schemes on

Unstructured Staggered GridsA. Madrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703

High Amplitude Solutions for Small Data in Pairs ofConservation Laws that Change TypeV. Matos and D. Marchesin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711

Asymptotic Behavior of Riemann Problem with Structurefor Hyperbolic Dissipative SystemsA. Mentrelli and T. Ruggeri. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721

Maximal Entropy Solutions for a Scalar Conservation Lawwith Discontinuous FluxS. Mishra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731

Semidiscrete Entropy Satisfying Approximate RiemannSolvers and Application to the Suliciu RelaxationApproximationT. Morales and F. Bouchut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739

On the L2-Well Posedness of an Initial Boundary ValueProblem for the Linear Elasticity in Two and ThreeSpace DimensionsA. Morando and D. Serre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747

Intersections Modeling with a Class of Second-OrderModels for Vehicular Traffic Flow

M. Herty, S. Moutari, and M. Rascle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .755

Some Contributions About an Implicit Discretization of a 1DInviscid Model for River FlowsA. Bermudez de Castro, R. Munoz-Sola, C. Rodrguez,

and M. Angel Vilar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765


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XIV Co ntents

Remarks on the Nonhomogeneous Oseen Problem Arisingfrom Modeling of the Fluid Around a Rotating BodyS. Kracmar, S. Necasov a, and P. Penel. . . . . . . . . . . . . . . . . . . . . . . . . . . 775

Multi-D Bony Type Potentialfor the BoltzmannEnskog EquationS.-Y. Ha and S.E. Noh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783

Convergence of Well-Balanced Schemes for the InitialBoundary Value Problem for Scalar Conservation Laws in 1DM. Nolte and D. Kr oner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791

Stability for Multidimensional Periodic Waves

Near Zero FrequencyM. Oh and K. Zumbrun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799

Existence of Strong Traces for Quasisolutions of ScalarConservation LawsE.Y. Panov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807

Path-Conservative Numerical Schemes for NonconservativeHyperbolic SystemsC. Pares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817

Numerical Modeling of Two-Phase Gravitational GranularFlows with Bottom TopographyM. Pelanti, F. Bouchut, A. Mangeney, and J.-P. Vilotte . . . . . . . . . . . . 825

Linear Lagrangian Systems of Conservation LawsY.-J. Peng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833

Normal Modes Analysis of Subsonic Phase Boundariesin Elastic MaterialsH. Freist uhler and R.G. Plaza . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841

Large Time Step Positivity-Preserving Methodfor Multiphase FlowsF. Coquel, Q.-L. Nguyen, M. Postel, and Q.-H. Tran . . . . . . . . . . . . . . . 849

Velocity Discretization in Numerical Schemesfor BGK EquationsA. Alaia, S. Pieraccini, and G. Puppo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857

A SpaceTime Conservative Method for Hyperbolic Systemsof Relaxation TypeS. Qamar and G. Warnecke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865


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Contents XV

A Numerical Scheme Based on Multipeakons for ConservativeSolutions of the CamassaHolm EquationH. Holden and X. Raynaud. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873

Consistency of the Explicit Roe Scheme for Low MachNumber Flows in Exterior DomainsF. Rieper and G. Bader. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883

Weak and Classical Solutions for a Model Problem inRadiation HydrodynamicsC. Rohde, N. Tiemann, and W.-A. Yong. . . . . . . . . . . . . . . . . . . . . . . . . . 891

Spectral Analysis of Coupled HyperbolicParabolic Systems

on Finite and Infinite IntervalsJ. Rottmann-Matthes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 901

Toward an Improved Capture of Stiff Detonation WavesO. Rouch, M.-O. St-Hilaire, and P. Arminjon . . . . . . . . . . . . . . . . . . . . . 911

Generalized Momenta of Mass and Their Applications to theFlow of Compressible FluidO. Rozanova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 919

ADERRungeKutta Schemes for Conservation Lawsin One Space DimensionG. Russo, E.F. Toro, and V.A. Titarev . . . . . . . . . . . . . . . . . . . . . . . . . . . 929

Strong Boundary Traces and Well-Posedness for ScalarConservation Laws with Dissipative Boundary ConditionsB. Andreianov and K. Sbihi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937

A Relaxation Method for the Couplingof Systems of Conservation LawsA. Ambroso, C. Chalons, F. Coquel, E. Godlewski, F. Lagoutiere,

P.-A. Raviart, and N. Seguin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947

Increasing Efficiency Through Optimal RK Time Integrationof Diffusion EquationsF. Cavalli, G. Naldi , G. Puppo, and M. Semplice . . . . . . . . . . . . . . . . . . 955

Numerical Simulation of Relativistic Flows Described by aGeneral Equation of StateS. Serna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963

On Delta-Shocks and Singular ShocksV.M. Shelkovich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 971


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XVI Co ntents

Finite Dimensional Representation of Solutions of ViscousConservation LawsW. Shen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 981

A Moving-Boundary Tracking Algorithm for InviscidCompressible FlowK.-M. Shyue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989

Transparent Boundary Conditions for the Elastic Waves inAnisotropic MediaI.L. Sofronov and N.A. Zaitsev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997

Counterflow Combustion in a Porous Medium

A.J. de Souza . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1005

Global Attractor and its Dimension for aKleinGordonSchrodinger SystemM.N. Poulou and N.M. Stavrakakis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013

A Few Remarks About a Theoremby J. RauchF. Sueur. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1021

A Riemann Solver Approach for Conservation Laws withDiscontinuous FluxM. Garavello, R. Natalini, B. Piccoli, and A. Terracina. . . . . . . . . . . . . 1029

The Strong Shock Wave in the Problem on Flow AroundInfinite Plane WedgeD.L. Tkachev, A.M. Blokhin, and Y.Y. Pashinin. . . . . . . . . . . . . . . . . . . 1037

The Derivative Riemann Problemfor the BaerNunziato EquationsE.F. Toro and C.E. Castro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045

Stability of Contact Discontinuities for the NonisentropicEuler Equations in Two-Space DimensionsA. Morando and P. Trebeschi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1053

Three-Dimensional Numerical MHD Simulationsof Solar ConvectionS.D. Ustyugov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061

Large-Time Behavior of Entropy Solutions to ScalarConservation Laws on Bounded DomainJ. Vovelle and S. Martin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069


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Contents XVII

A Second-Order Improved Front Tracking Method for theNumerical Treatment of the Hyperbolic Euler EquationsJ.A.S. Witteveen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077

Simulation of Field-Aligned Ideal MHD Flows AroundPerfectly Conducting Cylinders Using an ArtificialCompressibility ApproachM.S. Yalim, D.V. Abeele, and A. Lani . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085

Vanishing at Most Seventh-Order Terms of ScalarConservation LawsN. Fujino and M. Yamazaki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093

Large-Time Behavior for a Compressible EnergyTransport ModelL. Hsiao and Y. Li. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101

Novel Entropy Stable Schemes for 1D and 2D Fluid EquationsE. Tadmor and W. Zhong. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1111

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1121

Documents

J. Vovelle and S. Martin- Large-Time Behavior of Entropy Solutions to Scalar Conservation Laws on Bounded Domain