Finite Volume Methods for Hyperbolic Problems PDF
Preview Finite Volume Methods for Hyperbolic Problems
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 Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi Cambridge University Press 32 Avenue of the Americas, New York, NY 10013-2473, USA www.cambridge.org Information on this title: www.cambridge.org/9780521810876 ©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
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