Finite-volume-method

Finitevolumemethod1FinitevolumemethodThefinitevolumemethodisamethodforrepresentingandevaluatingpartialdifferentialequationsintheformofalgebraicequations[LeVeque,2002;Toro,1999].Similartothefinitedifferencemethodorfiniteelementmethod,valuesarecalculatedatdiscreteplacesonameshedgeometry.Finitevolumereferstothesmallvolumesurroundingeachnodepointonamesh.Inthefinitevolumemethod,volumeintegralsinapartialdifferentialequationthatcontainadivergencetermareconvertedtosurfaceintegrals,usingthedivergencetheorem.Thesetermsarethenevaluatedasfluxesatthesurfacesofeachfinitevolume.Becausethefluxenteringagivenvolumeisidenticaltothatleavingtheadjacentvolume,thesemethodsareconservative.Anotheradvantageofthefinitevolumemethodisthatitiseasilyformulatedtoallowforunstructuredmeshes.Themethodisusedinmanycomputationalfluiddynamicspackages.1DexampleConsiderasimple1DadvectionproblemdefinedbythefollowingpartialdifferentialequationHere,representsthestatevariableandrepresentsthefluxorflowof.Conventionally,positiverepresentsflowtotherightwhilenegativerepresentsflowtotheleft.Ifweassumethatequation(1)representsaflowingmediumofconstantarea,wecansub-dividethespatialdomain,,intofinitevolumesorcellswithcellcentresindexedas.Foraparticularcell,,wecandefinethevolumeaveragevalueofattimeand,asandattimeas,whereandrepresentlocationsoftheupstreamanddownstreamfacesoredgesrespectivelyofthecell.Integratingequation(1)intime,wehave:where.Toobtainthevolumeaverageofattime,weintegrateoverthecellvolume,anddividetheresultby,i.e.Weassumethatiswellbehavedandthatwecanreversetheorderofintegration.Also,recallthatflowisnormaltotheunitareaofthecell.Now,sinceinonedimension,wecanapplythedivergencetheorem,i.e.,andsubstituteforthevolumeintegralofthedivergencewiththevaluesofevaluatedatthecellsurface(edgesand)ofthefinitevolumeasfollows:Finitevolumemethod2where.Wecanthereforederiveasemi-discretenumericalschemefortheaboveproblemwithcellcentresindexedas,andwithcelledgefluxesindexedas,bydifferentiating(6)withrespecttotimetoobtain:wherevaluesfortheedgefluxes,,canbereconstructedbyinterpolationorextrapolationofthecellaverages.Equation(7)isexactforthevolumeaverages;i.e.,noapproximationshavebeenmadeduringitsderivation.GeneralconservationlawWecanalsoconsiderthegeneralconservationlawproblem,representedbythefollowingPDE,Here,representsavectorofstatesandrepresentsthecorrespondingfluxtensor.Againwecansub-dividethespatialdomainintofinitevolumesorcells.Foraparticularcell,,wetakethevolumeintegraloverthetotalvolumeofthecell,,whichgives,Onintegratingthefirsttermtogetthevolumeaverageandapplyingthedivergencetheoremtothesecond,thisyieldswhererepresentsthetotalsurfaceareaofthecellandisaunitvectornormaltothesurfaceandpointingoutward.So,finally,weareabletopresentthegeneralresultequivalentto(7),i.e.Again,valuesfortheedgefluxescanbereconstructedbyinterpolationorextrapolationofthecellaverages.Theactualnumericalschemewilldependuponproblemgeometryandmeshconstruction.MUSCLreconstructionisoftenusedinhighresolutionschemeswhereshocksordiscontinuitiesarepresentinthesolution.Finitevolumeschemesareconservativeascellaverageschangethroughtheedgefluxes.Inotherwords,onecell'slossisanothercell'sgain!Finitevolumemethod3References•Eymard,R.Gallouët,T.R.Herbin,R.(2000)ThefinitevolumemethodHandbookofNumericalAnalysis,Vol.VII,2000,p.713-1020.Editors:P.G.CiarletandJ.L.Lions.•LeVeque,Randall(2002),FiniteVolumeMethodsforHyperbolicProblems,CambridgeUniversityPress.•Toro,E.F.(1999),RiemannSolversandNumericalMethodsforFluidDynamics,Springer-Verlag.Furtherreading•Hirsch,C.(1990),NumericalComputationofInternalandExternalFlows,Volume2:ComputationalMethodsforInviscidandViscousFlows,Wiley.•Laney,CulbertB.(1998),ComputationalGasDynamics,CambridgeUniversityPress.•LeVeque,Randall(1990),NumericalMethodsforConservationLaws,ETHLecturesinMathematicsSeries,Birkhauser-Verlag.•Patankar,SuhasV.(1980),NumericalHeatTransferandFluidFlow,Hemisphere.•Tannehill,JohnC.,etal.,(1997),ComputationalFluidmechanicsandHeatTransfer,2ndEd.,TaylorandFrancis.•Wesseling,Pieter(2001),PrinciplesofComputationalFluidDynamics,Springer-Verlag.Externallinks•Thefinitevolumemethod[1]byR.Eymard,TGallouëtandR.Herbin,updateofthearticlepublishedinHandbookofNumericalAnalysis,2000•TheFiniteVolumeMethod(FVM)-Anintroduction[2]byOliverRübenkönigofAlbertLudwigsUniversityofFreiburg,availableundertheGFDL.(Archivecopy[3]attheWaybackMachine.)•FiPy:AFiniteVolumePDESolverUsingPython[4]fromNIST.•CLAWPACK[5]:asoftwarepackagedesignedtocomputenumericalsolutionstohyperbolicpartialdifferentialequationsusingawavepropagationapproachReferences[1]~herbin/PUBLI/bookevol.pdf[2][3]*/[4][5]~claw/ArticleSourcesandContributors4ArticleSourcesandCo