FINITE DIFFERENCE, FINITE ELEMENT,FINITE VOLUME ME

8.2FINITEDIFFERENCE,FINITEELEMENTANDFINITEVOLUMEMETHODSFORPARTIALDIFFERENTIALEQUATIONSJoaquimPeir´oandSpencerSherwinDepartmentofAeronautics,ImperialCollege,London,UKTherearethreeimportantstepsinthecomputationalmodellingofanyphysicalprocess:(i)problemdefinition,(ii)mathematicalmodel,and(iii)computersimulation.Thefirstnaturalstepistodefineanidealizationofourproblemofinterestintermsofasetofrelevantquantitieswhichwewouldliketomea-sure.Indefiningthisidealizationweexpecttoobtainawell-posedproblem,thisisonethathasauniquesolutionforagivensetofparameters.Itmightnotalwaysbepossibletoguaranteethefidelityoftheidealizationsince,insomeinstances,thephysicalprocessisnottotallyunderstood.Anexampleisthecomplexenvironmentwithinanuclearreactorwhereobtainingmeasurementsisdifficult.Thesecondstepofthemodelingprocessistorepresentouridealizationofthephysicalrealitybyamathematicalmodel:thegoverningequationsoftheproblem.Theseareavailableformanyphysicalphenomena.Forexample,influiddynamicstheNavier–Stokesequationsareconsideredtobeanaccuraterepresentationofthefluidmotion.Analogously,theequationsofelasticityinstructuralmechanicsgovernthedeformationofasolidobjectduetoappliedexternalforces.Thesearecomplexgeneralequationsthatareverydifficulttosolvebothanalyticallyandcomputationally.Therefore,weneedtointroducesimplifyingassumptionstoreducethecomplexityofthemathematicalmodelandmakeitamenabletoeitherexactornumericalsolution.Forexample,theirrotational(withoutvorticity)flowofanincompressiblefluidisaccuratelyrepresentedbytheNavier–Stokesequationsbut,iftheeffectsoffluidviscos-ityaresmall,thenLaplace’sequationofpotentialflowisafarmoreefficientdescriptionoftheproblem.1S.Yip(ed.),HandbookofMaterialsModeling.VolumeI:MethodsandModels,1–32.c
2005Springer.PrintedintheNetherlands.2J.Peir´oandS.SherwinAftertheselectionofanappropriatemathematicalmodel,togetherwithsuitableboundaryandinitialconditions,wecanproceedtoitssolution.Inthischapterwewillconsiderthenumericalsolutionofmathematicalproblemswhicharedescribedbypartialdifferentialequations(PDEs).ThethreeclassicalchoicesforthenumericalsolutionofPDEsarethefinitedifferencemethod(FDM),thefiniteelementmethod(FEM)andthefinitevolumemethod(FVM).TheFDMistheoldestandisbasedupontheapplicationofalocalTaylorexpansiontoapproximatethedifferentialequations.TheFDMusesatopo-logicallysquarenetworkoflinestoconstructthediscretizationofthePDE.Thisisapotentialbottleneckofthemethodwhenhandlingcomplexgeome-triesinmultipledimensions.ThisissuemotivatedtheuseofanintegralformofthePDEsandsubsequentlythedevelopmentofthefiniteelementandfinitevolumetechniques.Toprovideashortintroductiontothesetechniquesweshallconsidereachtypeofdiscretizationasappliedtoone-dimensionalPDEs.ThiswillnotallowustoillustratethegeometricflexibilityoftheFEMandtheFVMtotheirfullextent,butwewillbeabletodemonstratesomeofthesimilaritiesbetweenthemethodsandtherebyhighlightsomeoftherelativeadvantagesanddisadvan-tagesofeachapproach.Foramoredetailedunderstandingoftheapproacheswereferthereadertothesectiononsuggestedreadingattheendofthechapter.Thesectionisstructuredasfollows.WestartbyintroducingtheconceptofconservationlawsandtheirdifferentialrepresentationasPDEsandthealter-nativeintegralforms.Wenextdiscussestheclassificationofpartialdifferentialequations:elliptic,parabolic,andhyperbolic.ThisclassificationisimportantsincethetypeofPDEdictatestheformofboundaryandinitialconditionsrequiredfortheproblemtobewell-posed.Italso,permitsinsomecases,e.g.,inhyperbolicequations,toidentifysuitableschemestodiscretisethedifferen-tialoperators.Thethreetypesofdiscretisation:FDM,FEMandFVMarethendiscussedandappliedtodifferenttypesofPDEs.WethenendouroverviewbydiscussingthenumericaldifficultieswhichcanariseinthenumericalsolutionofthedifferenttypesofPDEsusingtheFDMandprovidesanintroductiontotheassessmentofthestabilityofnumericalschemesusingaFourierorVonNeumannanalysis.Finallywenotethat,giventhescientificbackgroundoftheauthors,thepresentationhasabiastowardsfluiddynamics.However,westressthatthefundamentalconceptspresentedinthischapteraregenerallyapplicabletocontinuummechanics,bothsolidsandfluids.1.ConservationLaws:IntegralandDifferentialFormsThegoverningequationsofcontinuummechanicsrepresentingthekine-maticandmechanicalbehaviourofgeneralbodiesarecommonlyreferredFinitemethodsforpartialdifferentialequations3toasconservationlaws.Thesearederivedbyinvokingtheconservationofmassandenergyandthemomentumequation(Newton’slaw).Whilsttheyareequallyapplicabletosolidsandfluids,theirdifferingbehaviourisaccountedforthroughtheuseofadifferentconstitutiveequation.Thegeneralprinciplebehindthederivationofconservationlawsisthattherateofchangeofu(x,t)withinavolumeVplusthefluxofuthroughtheboundaryAisequaltotherateofproductionofudenotedbyS(u,x,t).Thiscanbewrittenas∂∂t
Vu(x,t)dV+
Af(u)·ndA−
VS(u,x,t)dV=0(1)whichisreferredtoastheintegralformoftheconservationlaw.Forafixed(independentoft)volumeand,undersuitableconditionsofsmoothnessoftheinterveningquantities,wecanapplyGauss’theorem
V∇·fdV=
Af·ndAtoobtain
V∂u∂t+∇·f(u)−SdV=0.(2)Fortheintegralexpressiontobezero