Numerical Approximation of Hyperbolic Systems of Conservation Laws

519 Pages·1996·14.022 MB·English

by  Edwige Godlewski, Pierre-Arnaud Raviart (auth.)

Applied Mathematical Sciences
Volume 118
Editors
J.E. Marsden L. Sirovich F. John (deceased)
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Pattern Theory, Vol. Ill. (continued following index)
Edwige Godlewski Pierre-Arnaud Raviart
Numerical Approximation of
Hyperbolic Systems of
Conservation Laws
With 75 Illustrations
~Springer
Edwige Godlewski Pierre-Arnaud Raviart
Laboratoire d'Analyse Numerique Ecole Polytechnique
Universite Pierre et Marie Curie Centre de Mathematiques Appliquees
4 Place Jussieu 91128 Palaiseau Cedex
75252 Paris Cedex 05 France
France
Editors
J.E. Marsden L. Sirovich
Control and Dynamical Systems, 104-44 Division of Applied Mathematics
California Institute of Technology Brown University
Pasadena, CA 91125 Providence, RI 02912
USA USA
Mathematics Subject Classification (1991): 76N15, 76N20, 70005
Library of Congress Cataloging-in-Publication Data
Godlewski, Edwige.
Numerical approximation of hyperbolic systems of conservation laws/
Edwige Godlewski, Pierre-Arnaud Raviart.
p. em. - (Applied mathematical sciences; 118)
Includes bibliographical references and index.
I. Gas dynamics. 2. Conservation laws (Mathematics).
3. Differential equations, Hyperbolic-Numerical solutions.
I. Raviart, Pierre-Arnaud, 1939- . II. Title. III. Series: Applied
mathematical sciences (Springer Science+Business Media, LLC); v. 118.
QAI.A647 vol. 118
[QA930]
510 s-dc20
[533.2'01515353] 96-13585
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987654321
SPIN 10501626
ISBN 978-1-4612-6878-9 ISBN 978-1-4612-0713-9 (eBook)
DOI 10.1007/978-1-4612-0713-9
Preface
This work is devoted to the theory and approximation of nonlinear hyper
bolic systems of conservation laws in one or two space variables. It follows
directly a previous publication on hyperbolic systems of conservation laws
by the same authors, and we shall make frequent references to Godlewski
and Raviart (1991) (hereafter noted G.R.), though the present volume can
be read independently. This earlier publication, apart from a first chap
ter, especially covered the scalar case. Thus, we shall detail here neither
the mathematical theory of multidimensional scalar conservation laws nor
their approximation in the one-dimensional case by finite-difference con
servative schemes, both of which were treated in G.R., but we shall mostly
consider systems. The theory for systems is in fact much more difficult
and not at all completed. This explains why we shall mainly concentrate
on some theoretical aspects that are needed in the applications, such as
the solution of the Riemann problem, with occasional insights into more
sophisticated problems.
The present book is divided into six chapters, including an introductory
chapter. For the reader's convenience, we shall resume in this Introduction
the notions that are necessary for a self-sufficient understanding of this
book -the main definitions of hyperbolicity, weak solutions, and entropy
present the practical examples that will be thoroughly developed in the
following chapters, and recall the main results concerning the scalar case.
Chapter I is devoted to the resolution of the Riemann problem for a
general hyperbolic system in one space dimension, introducing the classi
cal notions of Riemann invariants and simple waves, the rarefaction and
shock curves, and characteristics and entropy conditions. The theory is
then applied to the p-system.
In Chapter II, we make a closer study of the one-dimensional system of
gas dynamics. We solve the Riemann problem in detail and then present
the simplest models of reacting flow, first the Chapman-Jouguet theory
and then the Z.N.D. model for detonation.
After this theoretical approach, we go into the numerical approximation
of hyperbolic systems by conservative finite-difference methods. The most
v
vi Preface
usual schemes for one-dimensional systems are developed in Chapter III,
with special emphasis on the application to gas dynamics. The last section
begins with a short account on the kinetic theory so as to introduce kinetic
schemes.
Chapter IV is devoted to the study of finite volume methods for
bidimensional systems, preceded by some theoretical considerations on
multidimensional systems.
For the sake of completeness, we could not avoid the problem of boundary
conditions. Chapter V is but an introduction to the complex theory and
presents some numerical boundary treatment.
The authors wish to thank R. Abgrall, F. Coquel, F. Dubois, and particu
larly T. Gallouet, B. Perthame, and D. Serre, from whom they learned
a great deal and who answered willingly and most amiably their many
questions.
They owe thanks to the SMAI reading committee and to the reviewers,
who made very valuable suggestions.
The first author is grateful to all her colleagues who encouraged her in
completing this huge work, especially to H. Le Dret and F. Murat for so
often giving her their time, and to L. Ruprecht for her kind and competent
assistance in the retyping of the final manuscript; such friendly help was
invaluable.
Paris, France E. Godlewski and P.-A. Raviart
September 1995
Contents
Preface v
Introduction 1
1. Definitions and examples 1
2. Weak solutions of systems of conservation laws 11
3. Entropy solutions 21
Notes 35
I. Nonlinear hyperbolic systems in one space dimension 37
1. Linear hyperbolic systems with constant coefficients 37
2. The nonlinear case. Definitions and examples 40
3. Simple waves and Riemann invariants 49
4. Shock waves and contact discontinuities 60
5. Characteristic curves and entropy conditions 70
6. Solution of the Riemann problem 83
7. The Riemann problem for the p-system 87
Notes 97
II. Gas dynamics and reacting flows 99
1. Preliminaries 99
2. Entropy satisfying shock conditions 108
3. Solution of the Riemann problem 126
4. Reacting flows. The Chapman-Jouguet theory 142
5. Reacting flows. The Z.N.D. model for detonations 160
Notes 166
III. Finite difference schemes for one-dimensional
systems 167
1. Generalities on finite difference methods for systems 167
2. Godunov's method 182
vii
viii Contents
3. Roe's method 196
4. The Osher scheme 229
5. Flux vector splitting methods 237
6. Van Leer's second-order method 245
7. Kinetic schemes for the Euler equations 269
Notes 301
IV. The case of multidimensional systems 303
1. Generalities on multidimensional hyperbolic systems 303
2. The gas dynamics equations in two space dimensions 316
3. Multidimensional finite difference schemes 343
4. Finite-volume methods 360
5. Second-order finite-volume schemes 403
Notes 415
V. An introduction to boundary conditions 417
1. The initial boundary value problem in the linear case 417
2. The nonlinear approach 435
3. Gas dynamics 442
4. Absorbing boundary conditions 446
5. Numerical treatment 453
Notes 460
Bibliography 461
References 461
Index 501
Introduction
1 Definitions and examples
In this section, we present the general form of systems of conservation laws
in several space variables and we give some important examples of such
systems that arise in continuum physics.
n
Let be an open subset of JR.P, and let fj, 1 ~ j ~ d, be d smooth
functions from n into JR.P; the general form of a system of conservation
laws in several space variables is
au a
d
(1.1) at +Lax- fj(u) = 0, X= (xl, ... ,xd) E JR.d, t > 0,
j=l J
where
CJ
u~
is a vector-valued function from JR_d X [0, +oo[ into n. The set n is called
the set of states and the functions
are called the flux-functions. One says that system (1.1) is written in
conservative form.
Formally, the system (1.1) expresses the conservation of the p quan-
tities Ut, ... ,Up· In fact, let D be an arbitrary domain of JR.d, and let
n = (n1, ... , nd)T be the outward unit normal to the boundary aD of
D. Then, it follows from (1.1) that
!! udx+t1
fj(u)njdS=O.
D J=l {)D
1