Riemann Solvers and Numerical Methods for Fluid Dynamics A Practical Introdu.pdf
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Riemann Solvers and Numerical Methods for Fluid Dynamics A Practical Introdu.pdf
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RiemannSolversandNumericalMethodsforFluidDynamicsThirdEditionEleuterioF.ToroRiemannSolversandNumericalMethodsforFluidDynamicsAPracticalIntroductionThirdEdition123ISBN978-3-540-25202-3e-ISBN978-3-540-49834-6DOI10.1007/978-3-540-49834-6SpringerDordrechtHeidelbergLondonNewYorkLibraryofCongressControlNumber:
2009921818c?
Springer-VerlagBerlinHeidelberg2009Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,reproductiononmicrofilmorinanyotherway,andstorageindatabanks.DuplicationofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.ViolationsareliabletoprosecutionundertheGermanCopyrightLaw.Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.Coverdesign:
eStudioCalamarS.L.Printedonacid-freepaperSpringerispartofSpringerScience+BusinessMedia()ProfessorEleuterioF.ToroUniversityofTrentoItalyeleuteriofrancisco.torounitn.itInmemoryoftworemarkablepersons:
mymotherOlgaMunozdeToro(Chile,19302003)andmyfatherinlawFritzFreund(Germany,19262008).PrefacetotheFirstEditionIn1917,theBritishscientistL.F.Richardsonmadethefirstreportedattempttopredicttheweatherbysolvingpartialdifferentialequationsnu-merically,byhand!
ItisgenerallyacceptedthatRichardsonswork,thoughunsuccessful,markedthebeginningofComputationalFluidDynamics(CFD),alargebranchofscientificcomputingtoday.Hisworkhadthefourdistinguish-ingcharacteristicsofCFD:
apracticalproblemtosolve,amathematicalmodeltorepresenttheproblemintheformofasetofpartialdifferentialequa-tions,anumericalmethodandacomputer,humanbeingsinRichardsonscase.EightyyearsonandthesefourelementsremainthepillarsofmodernCFD.ItisthereforenotsurprisingthatthegenerallyaccepteddefinitionofCFDasthescienceofcomputingnumericalsolutionstopartialdifferentialorintegralequationsthataremodelsforfluidflowphenomena,closelyembodiesRichardsonswork.Computershave,sinceRichardsonsera,developedtounprecedentedlev-elsandataneverdecreasingcost.Therangeofapplicationareasgivingrisetopracticalproblemstobesolvednumericallyhasincreaseddramati-cally.Inadditiontothetraditionaldemandsfrommeteorology,oceanogra-phy,somebranchesofphysicsandfromarangeofengineeringdisciplines,thereareatpresentfreshdemandsfromadynamicandfastmovingman-ufacturingindustry,whosetraditionalbuildtestfixapproachisrapidlybe-ingreplacedbytheuseofquantitativemethods,atalllevels.Theneedfornewmaterialsandfordecisionmakingunderenvironmentalconstraintsareincreasingsourcesofdemandsformathematicalmodelling,numericalalgo-rithmsandhighperformancecomputing.mathematicalmodelshaveim-proved,thoughthebasicequationsofcontinuummechanics,alreadyavailablemorethanacenturybeforeRichardsonsfirstattemptsatCFD,arestillthebasesformodellingfluidflowprocesses.Progressisrequiredatthelevelofthermodynamics,equationsofstate,andadvancesintothemodellingofnonequilibriumandmultiphaseflowphenomena.numericalmethodsareper-hapsthesuccessstoryofthelasteightyyears,thelasttwentybeingperhapsthemostproductive.ThissuccessisfirmlybasedonthepioneeringworksofVIIIPrefacetotheFirstEditionscientistssuchasvonNeumann,whoseresearchonstabilityexplainedandresolvedthedifficultiesexperiencedbyRichardson.ThissuccesswouldhavebeenimpossiblewithoutthecontributionsfromCourant,Friedrichs,Richt-myer,Lax,Oleinik,Wendroff,Godunov,Rusanov,vanLeer,Harten,Roe,Osher,Colella,Yee,andmanyothers.Thenetresultis:
moreaccurate,moreefficient,morerobustandmoresophisticatednumericalmethodsareavailableforambitiouspracticalapplicationstoday.DuetothemassivedemandsonCFDandthelevelofsophisticationofnumericalmethods,newdemandsoneducationandtrainingofthescientistsandengineersofthepresentandthefuturehavearisen.Thisbookisanattempttocontributetothetrainingandeducationinnumericalmethodsforfluiddynamicsandrelateddisciplines.Thecontentsofthisbookweredevelopedoveraperiodofmanyyearsofinvolvementinresearchonnumericalmethods,applicationofthemethodstosolvepracticalproblemsandteachingscientistandengineersatthepostgraduatelevel.ThestartingpointwasamoduleforaMastersCourseinCom-putationalFluidDynamicsattheCollegeofAeronautics,Cranfield,UK.ThematerialwasalsopartofshortcoursesandlecturesgivenatCranfield,UK;theErnstMachInstitute,Freiburg,Germany;theShockWaveResearchCentre,TohokuUniversity,Sendai,Japan;theDepartmentofMathematicsandtheDepartmentofCivilandEnvironmentalEngineering,UniversityofTrento,Italy;theDepartmentofMathematics,TechnicalUniversityFedericoSantaMaria,Chile;theDepartmentofMechanics,TechnicalUniversityofAachen,Germany;andtheManchesterMetropolitanUniversity(MMU),Manchester,UK.Thisbookisaboutmodernshockcapturingnumericalmethodsforsolv-ingtimedependenthyperbolicconservationlaws,withsmoothanddiscon-tinuoussolutions,ingeneralmultidimensionalgeometries.Theapproachiscomprehensive,practicaland,inthemain,informal.Allnecessaryitemsofinformationforthepracticalimplementationofallmethodsstudiedhere,areprovidedindetail.Allmethodsstudiedareillustratedthroughpracticalnu-mericalexamples;numericalresultsarecomparedwithexactsolutionsandinsomecaseswithreliableexperimentaldata.MostofthebookisdevotedtoacoherentpresentationofGodunovmeth-ods.ThedevelopmentsofGodunovsapproachoverthelasttwentyyearshaveledtoamaturenumericaltechnology,thatcanbeutilisedwithconfidencetosolvepracticalproblemsinestablishedaswellasnewareasofapplication.GodunovmethodsrelyonthesolutionoftheRiemannproblem.Theexactsolutionispresentedindetail,soastoaidthereaderinapplyingthesolutionmethodologytootherhyperbolicsystems.Wealsopresentavarietyofapprox-imateRiemannsolvers;again,theamountofdetailsuppliedwillhopefullyaidthereaderwhomightwanttoapplythemethodologiestosolveotherprob-lems.OtherrelatedmethodssuchastheRandomChoiceMethodandtheFluxVectorSplittingMethodarealsoincluded.Inaddition,westudycentred(nonupwind)shockcapturingmethods.TheseschemesaremuchlesssophisticatedPrefacetotheFirstEditionIXthanGodunovmethods,andofferacheapandsimplealternative.HighorderextensionsofthesemethodsareconstructedforscalarPDEs,alongwiththeirTotalVariationDiminishing(TVD)versions.MostoftheseTVDmethodsarethenextendedtoonedimensionalnonlinearsystems.TechniquestodealwithPDEswithsourcetermsarealsostudied,asaretechniquesformultidi-mensionalsystemsingeneralgeometries.ThepresentationoftheschemesfornonlinearsystemsiscarriedoutthroughthetimedependentEulerequationsofgasdynamics.Havingreadtherelevantchapters/sections,thereaderwillbesufficientlywellequippedtoextendthetechniquestootherhyperbolicsystems,andtoadvectionreactiondiffusionPDEs.Thereareatleasttwowaysofutilisingthisbook.First,itcanbeusedasameansforselfstudy.Inthepresentationoftheconcepts,theemphasishasbeenplacedonclarity,sometimessacrificingmathematicalrigour.Thetypicalreaderinmindisagraduatestudentinadepartmentofengineering,physics,appliedmathematicsorcomputerscience,embarkingonaresearchtopicthatinvolvestheimplementationofnumericalmethods,fromfirstprinciples,tosolveadvectionreactiondiffusionproblems.Thecontentsofthisbookmayalsobeusefultonumericalanalystsbeginningtheirresearchonalgorithms,aselementarybackgroundreading.Suchusersmaybenefitfromacompre-hensiveselfstudyofallthecontentsofthebook,inaperiodofabouttwomonths,perhapsincludingthepracticalimplementationandtestingofmostnumericalmethodspresented.Anotherclassofreaderswhomaybenefitfromselfstudyingthisbookarescientistsandengineersinindustryandresearchlaboratories.Atthecostofsomerepetitiveness,eachchapterisalmostselfcontainedandhasplentyofcrossreferencing,sothatthereadermaydecidetostartreadingthisbookinthemiddleorjumptothelastchapter.Thisbookcanalsobeusedasateachingaid.Academicsinvolvedintheteachingofnumericalmethodsmayfindthisworkausefulreferencebook.Selectedchaptersorsectionsmaywellformthebasesforafinalyearunder-graduatecourseonnumericalmethodsforPDEs.Inamathematicsorcom-putersciencedepartment,thecontentsmayinclude:
somesectionsofchapter1,chapters2,5,13,somesectionsofchapter14,chapter15andsomesectionsofchapter16.Inadepartmentofengineeringorphysics,onemayincludechapters3,4,6,7,8,9,10,11,12,15,16and17.Apostgraduatecoursemayinvolvemostofthecontentsofthisbook,assumingperhapsaworkingknowl-edgeofcompressiblefluiddynamics.Shortcoursesfortrainingengineersandscientistsinindustryandresearchlaboratoriescanalsobebasedonmostofthecontentsofthisbook.EleuterioToroManchester,UKMarch1997.XPrefacetotheFirstEditionACKNOWLEDGEMENTSFORFIRSTEDITIONTheauthorisindebtedtomanycolleagueswhoovertheyearshavekindlyarrangedshortandextendedvisitstotheirestablishments,haveorganisedseminars,workshopsandshortcourses.Theseeventshavestronglyinflu-encedthedevelopmentofthisbook.Specialtha
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