Convergence of the finite volume method for multid

CONVERGENCEOFTHEFINITEVOLUMEMETHODFORMULTIDIMENSIONALCONSERVATIONLAWSB.Cockburn1,F.Coquel2andP.G.LeFloch3Abstract.Weestablishtheconvergenceofthenitevolumemethodappliedtomultidimensionalhyperbolicconservationlaws,andbasedonmonotonenumericalux-functions.Ourtechniqueappliesunderafairlyunrestrictiveassumptiononthetriangulations(\atelementsareallowed),andtoLipschitzcontinuousux-functions.Wetreattheinitialandboundaryvalueproblem,andobtainthestrongconvergenceoftheschemetotheuniqueentropydiscontinuoussolutioninthesenseofKruzkov.TheproofofconvergenceisbasedonaconvergenceframeworkduetoCoquelandLeFloch(Math.ofComp.57(1991),169{210),&J.Numer.Anal.30(1993),675{700.Fromaconvexdecompositionofthescheme,wederiveanewestimatefortherateofentropydissipation,andanewformulationofthediscreteentropyinequalities.Theseestimatesareshowntobesucientforthepassagetothelimitinthediscreteequation.ConvergencefollowsfromDiPerna’suniquenessresultintheclassofentropymeasure-valuedsolutions.Content1.Introduction2.Mainresultofconvergence3.Entropydissipationestimateandentropyinequalities4.Convergenceviameasure-valuedsolutions5.ExamplesandremarksReferences1.Introduction.Thispaperconsidersthenitevolumemethodfortheapproximationoftheinitialandbound-aryvalueproblemassociatedwithamultidimensionalconservationlaw:(1:1)@tu+divf(u)=0;u(t;x)2RI;t0;x2RId;(1:2)u(0;x)=u0(x);x2;SubmittedtoSIAMJ.NumericalAnalysis.Firstversion:december1990.Revisedversion:october1993.1SchoolofMathematics,UniversityofMinnesota,127VincentHall,Minneapolis,MN55455.ResearchpartiallysupportedbyNSFgrantDMS-91-03997.2Onera,29avenuedelaDivisionLeclerc,BP72,92320Ch^atillon(France).3DepartmentofMathematics,UniversityofSouthernCalifornia,LosAngeles,California90089;andCMAP-CNRS,EcolePolytechnique,Palaiseau(France).ResearchpartiallysupportedbyNSFgrantsDMS-88-06731andDMS-92-09326.1991MathematicsSubjectClassication.Primary65M12;secondary35L65.Keywordsandphrases:conservationlaw,measure-valuedsolution,nitevolumemethod,entropydissipation.TypesetbyAMS-TEX12CONVERGENCEOFTHEFINITEVOLUMEMETHODand,forallconvexentropypair(U;F)andalmostall(t;x)in@(withrespecttothe(d1)-dimensionalHaussdorfmeasure),(1:3)N(t;x)F(u(t;x))F(u1(t;x))rU(u1(t;x))f(u(t;x))f(u1(t;x))0:Theux-functionf:RI!RIdisassumedtobelocallyLipschitzcontinuous,andisanopen(andnotnecessarilybounded)subsetofRIdhavingapolygonalboundary@.WedenotebyN(t;x)theoutwardunitnormalalong@.Theinitialdatau0isassumedtobelongtoL1()\L1(),andtheboundarydatau1belongstoL1RI+;L1(@)\L1RI+;L1(@).ALipschitzcontinuousfunction(U;F):RI!RIRIdissaidtobeaconvexentropypairifUisaconvexfunctionand(1:4)dFdu=rUdfdu:Theformulation(1.3)isactuallyaweakformofthestandardboundaryconditionu(t;x)=u1(t;x),(t;x)2@,whichwouldnotleadtoawell-posedproblem,since(1.1)isanonlinearhyperbolicequation.InviewoftheresultsbyKruzkovandothers[1,21,22,37,43],itisnotdiculttoseethatthereexistsauniqueweaksolutioninL1(RI+)totheproblemthatsatisestheentropyinequalities:(1:5)@tU(u)+divF(u)0forallconvexentropypairs.Cf.alsoLax[23,24]forbackgroundonhyperbolicequations.Thenitevolumemethodisbasedonthelocalconservationpropertysatisedbythesolutionstoaconservationlaw,andcanbedenedongeneraltriangulations.Themethodiswidelyusedincomputationaluiddynamics.Thequestionoftheconvergencehencearisesnaturally,andhasindeedreceivedimportantattentioninthelasttenyears.Thecaseofcartesiantriangulations(madeofacartesianproductofone-dimensionalpartitions)withmonotonenumericalux-functionswascompletelysolvedaftertheworksbyConway-Smoller,Crandall{Majda,Lucier,Osher,Sandersandothers[6,12,22,28,29,32,33].Thecaseofcartesiantriangulationsisspecialbecause,whenthemeshistranslationinvariant,auniformestimateintheboundedvariationnorm(BV)canbederivedfromtheL1contractionproperty.CompactnessfortheschemethenfollowsfromtheBVestimateandHelly’stheorem.Theseargumentsdonotworkwiththenitevolumemethodonarbitrarytriangulations,aswasobservedbySanders[34].FollowingtheworkbySzepessy[36],wherethefocuswasthestreamlinediusionniteelementmethod,atechniqueforprovingconvergenceofnumericalschemesformultidimensionalequationswasintroducedbyCoquelandLeFlochin[7,8,9,26].Inthiswork,convergenceisestablishedwithoutappealingtoauniformBVestimatebutbasedonaresultbyDiPerna[14]which,undersomecircumstances,ensurestheuniquenessofthemeasure-valuedsolutionto(1.1).WealsorecallthataYoungmeasurecanbeconstructedfromanysequenceofapproximatesolutions,sayuh,inordertorepresentallitscompositeweak-starlimitsa(uh)fora2C0(RI).AYoungmeasure,thatisconsistentwithalltheentropyinequalities(i.e.,bydenition,anentropymeasure-valuedsolution)andreducestoaDiracmassu0(x)att=0,isaDiracmassu(t;x)foralltimes.Moreover,uistheuniqueentropysolutiontotheproblemwithinitialdatau0,cf.alsoSzepessy[37].DiPernaappliedthistechniquetochecktheconvergenceofthevanishingviscositymethodusingtheL1stabilityonly.Asamatteroffact,themainstepinapplyingDiPerna’suniquenesstheoremisthepassagetothelimitinthesenseofYoungmeasuresintheapproximateentropyinequalities.ForapproximateB.COCKBURN,F.COQUEL,AN