Finite Volume Methods for Hyperbolic Problems

579 Pages·2002·10.545 MB·English

by  LeVeque R.J.

FiniteVolumeMethodsforHyperbolicProblems
Thisbookcontainsanintroductiontohyperbolicpartialdifferentialequationsandapow-
erful class of numerical methods for approximating their solution, including both linear
problemsandnonlinearconservationlaws.Theseequationsdescribeawiderangeofwave-
propagation and transport phenomena arising in nearly every scientific and engineering
discipline.Severalapplicationsaredescribedinaself-containedmanner,alongwithmuch
ofthemathematicaltheoryofhyperbolicproblems.High-resolutionversionsofGodunov’s
methodaredeveloped,inwhichRiemannproblemsaresolvedtodeterminethelocalwave
structure and limiters are then applied to eliminate numerical oscillations. These meth-
ods were originally designed to capture shock waves accurately, but are also useful tools
forstudyinglinearwave-propagationproblems,particularlyinheterogenousmaterial.The
methodsstudiedareimplementedintheCLAWPACKsoftwarepackage.Sourcecodeforall
the examples presented can be found on the web, along with animations of many time-
dependent solutions. This provides an excellent learning environment for understanding
wave-propagationphenomenaandfinitevolumemethods.
RandallJ.LeVequeisaProfessorofAppliedMathematicsattheUniversityofWashington.

Cambridge Texts in Applied Mathematics
MaximumandMinimumPrinciples
M.J.SEWELL
Solitons
P.G.DRAZINANDR.S.JOHNSON
TheKinematicsofMixing
J.M.OTTINO
IntroductiontoNumericalLinearAlgebraandOptimisation
PHILIPPEG.CIARLET
IntegralEquations
DAVIDPORTERANDDAVIDS.G.STIRLING
PerturbationMethods
E.J.HINCH
TheThermomechanicsofPlasticityandFracture
GERARDA.MAUGIN
BoundaryIntegralandSingularityMethodsforLinearizedViscousFlow
C.POZRIKIDIS
NonlinearWaveProcessesinAcoustics
K.NAUGOLNYKHANDL.OSTROVSKY
NonlinearSystems
P.G.DRAZIN
Stability,InstabilityandChaos
PAULGLENDINNING
AppliedAnalysisoftheNavier–StokesEquations
C.R.DOERINGANDJ.D.GIBBON
ViscousFlow
H.OCKENDONANDJ.R.OCKENDON
Scaling,Self-SimilarityandIntermediateAsymptotics
G.I.BARENBLATT
AFirstCourseintheNumericalAnalysisofDifferentialEquations
ARIEHISERLES
ComplexVariables:IntroductionandApplications
MARKJ.ABLOWITZANDATHANASSIOSS.FOKAS
MathematicalModelsintheAppliedSciences
A.C.FOWLER
ThinkingAboutOrdinaryDifferentialEquations
ROBERTE.O’MALLEY
AModernIntroductiontotheMathematicalTheoryofWaterWaves
R.S.JOHNSON
RarefiedGasDynamics
CARLOCERCIGNANI
SymmetryMethodsforDifferentialEquations
PETERE.HYDON
HighSpeedFlow
C.J.CHAPMAN
WaveMotion
J.BILLINGHAMANDA.C.KING
AnIntroductiontoMagnetohydrodynamics
P.A.DAVIDSON
LinearElasticWaves
JOHNG.HARRIS
AnIntroductiontoSymmetryAnalysis
BRIANJ.CANTWELL
IntroductiontoHydrodynamicStability
P.G.DRAZIN
FiniteVolumeMethodsforHyperbolicProblems
RANDALLJ.LEVEQUE

Finite Volume Methods for Hyperbolic Problems
RANDALLJ.LEVEQUE
UniversityofWashington
CAMBRIDGEUNIVERSITYPRESS
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Cambridge University Press
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©Randall J. LeVeque 2002
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 2002
Reprinted 2003, 2005 (twice), 2006, 2007
Printed in the United States of America
Acatalogrecordfor this publication is available from the British Library.
Library of Congress Cataloging in Publication Data
LaVeque, Randall J., 1995–
Finite-volume methods for hyperbolic problems / Randall J. LeVeque.
p. cm. – (Cambridge texts in applied mathematics)
Includes bibliographical references and index.
ISBN 0-521-81087-6 – ISBN 0-521-00924-3 (pbk.)
1. Differential equations, Hyperbolic – Numerical solutions. 2. Finite volumes method
3. Conservation laws (Mathematics) I. Title. II. Series.
QA377.L41566 2002
515’.353-dc21 2001052642
ISBN 978-0-521-81087-6 hardback
ISBN 978-0-521-00924-9 paperback
Cambridge University Press has no responsibility for
the persistence or accuracyof URLs for external or
third-party Internet Web sites referred to in this publication
and does not guarantee that any content on such
Web sites is, or will remain, accurate or appropriate.
ToLoyceandBenjamin

Contents
Preface page xvii
1 Introduction 1
1.1 ConservationLaws 3
1.2 FiniteVolumeMethods 5
1.3 MultidimensionalProblems 6
1.4 LinearWavesandDiscontinuousMedia 7
1.5 CLAWPACKSoftware 8
1.6 References 9
1.7 Notation 10
PartI LinearEquations
2 ConservationLawsandDifferentialEquations 15
2.1 TheAdvectionEquation 17
2.2 DiffusionandtheAdvection–DiffusionEquation 20
2.3 TheHeatEquation 21
2.4 CapacityFunctions 22
2.5 SourceTerms 22
2.6 NonlinearEquationsinFluidDynamics 23
2.7 LinearAcoustics 26
2.8 SoundWaves 29
2.9 HyperbolicityofLinearSystems 31
2.10 Variable-CoefficientHyperbolicSystems 33
2.11 HyperbolicityofQuasilinearandNonlinearSystems 34
2.12 SolidMechanicsandElasticWaves 35
2.13 LagrangianGasDynamicsandthe p-System 41
2.14 ElectromagneticWaves 43
Exercises 46
3 CharacteristicsandRiemannProblemsforLinear
HyperbolicEquations 47
3.1 SolutiontotheCauchyProblem 47
ix