DIFFERENTIAL ALGEBRAIC EQUATIONS PART II NUMERICAL

Martin{Luther{UniversitatHalle{WittenbergFachbereichMathematikundInformatikLINEARPARTIALDIFFERENTIALALGEBRAICEQUATIONSPARTII:NUMERICALSOLUTIONW.Lucht,K.StrehmelandC.Eichler-LiebenowReportNo.18(1997)ReportsoftheInstituteofNumericalMathematicsEditors:ProfessorsandPrivateDocentsoftheDepartmentofMathematicsandComputerScience,Martin-Luther-UniversityHalle-Wittenberg.LINEARPARTIALDIFFERENTIALALGEBRAICEQUATIONSPARTII:NUMERICALSOLUTIONW.Lucht,K.StrehmelandC.Eichler-LiebenowReportNo.18(1997)Prof.Dr.WenfriedLuchtProf.Dr.KarlStrehmelDipl.-Math.ClaudiaEichler-LiebenowFachbereichMathematikundInformatikInstitutfurNumerischeMathematikUniversitatHalleTheodor-Lieser-Str.506120Halle,GermanyTel.:[phone],5524650,5524667email:[e-mail]@[e-mail]LALGEBRAICEQUATIONSPARTII:NUMERICALSOLUTIONW.LUCHT,K.STREHMEL,C.EICHLER-LIEBENOWMartin-Luther-UniversitatHalle{WittenbergFachbereichMathematikundInformatikInstitutfurNumerischeMathematikPostfach,D{06099Halle,Germanyemail:[e-mail]lutionoflinearpartialdierentialalgebraicequa-tions(PDAEs)ofthegeneraltypeAut(t;x)+Buxx(t;x)+Cu(t;x)=f(t;x)(whereA;B;Careconstant(nn)matrices)bymeansofthemethodoflines(MOL),thebackwardintime,centeredinspace(BTCS)andtheCrank-Nicolsonscheme.Usinganalgebraicquality(theproperquality)tocharacterizecertainmatrixpencilsresultingfromthePDAE,thecon-vergenceinnormofthenumericalsolutions(determinedbythedierenceschemes)towardstheexactsolutionisdescribedintermsoftwoindexesofthePDAE.IntwoTheoremsitisshownthatthereisastrongdependenceoftheorderofconvergenceontheseindexes.WepresentexamplesforthecalculationoftheorderofconvergenceandgiveresultsofnumericalcalculationsforseveralaspectsencounteredinthenumericalsolutionofPDAEs.AMSsubjectclassication:65M06,65M15,65M20Keywords:Partialdierentialalgebraicequations,coupledsystems,indexes,convergenceofdierenceschemes,methodoflines1.IntroductionAlongtheideasofthewell-knownmethodoflines(MOL)approachweconsidertwodis-cretizationmethodsforthenumericalsolutionoflinearpartialdierentialalgebraicequationswithconstantcoecientsoftheformAut(t;x)+Buxx(t;x)+Cu(t;x)=f(t;x)(1.1)witht2(0;te)andx2(l;l),A;B;C2Rnn,u;f:[0;te][l;l]!Rn.TheinterestisincaseswhereatleastoneofthematricesAandBissingular.ThetwospecialcasesA=0orB=0leadtoordinarydierentialequations(ODEs)ordierentialalgebraicequations(DAEs)whicharenotconsideredhere.Therefore,inthispaperweassumethatnoneofthematricesAandBisthezeromatrix.Eq.(1.1)istobesupplementedforallt2[0;te];te0;byboundaryconditions(BCs)forthecomponentsujofuforallj2MBC(thissetisdenedinthefollowingsection),andforsimplicityweassumeDirichletBCsRBuj(t;x):=uj(t;l)=0;j2MBC:(1.2)12W.LUCHT,K.STREHMEL,C.EICHLER{LIEBENOWFurthermore,initialconditions(ICs)ui(0;x)=gi(x)forx2[l;l];i2MIC;(1.3)areassumedtobegiven.Anycomponentsgiofgforalli2MICf1;2;::;ngcanbechosen.Forthecomponentsuk;k=2MBC;BCsofonlyDirichlettypeareinorder.WealsorequirethecompatibilityconditionsRBg(x)=RBu(0;x)(1.4)betweentheICsandtheBCs.IncontrasttoproblemswithregularmatricesAandB(e.g.parabolicproblems)theIC(1.3)and/ortheBC(1.2)forproblemswithsingularmatricesAand/orBhavetofulllcertainsupplementaryconditions(so-calledconsistencyconditions).Equationsofthetype(1.1)withBCsandICsasgivenareinvestigatedextensivelyin[7]whichisthebasisofthispaper.SomefactsnecessaryforanumericalsolutionofPDAEsarerecapitulatedinsection2.InthethirdsectionasemidiscretizationofthePDAE(1.1)withrespecttothespacecoor-dinate(withspacestepsizeh),i.e.themethodoflines(MOL)isconsidered.Ifthisspacediscretizationisofsecondorder,then(undersomeassumptions)theorderofconvergenceforh!0isshowntobealsoofsecondorder.ThefulldiscretizationofthePDAEisconsideredinsection4.Insubsections4.1and4.2theBTCSschemeisinvestigatedindetail.Ofspecialinterestistheinuenceofthedierentialspatialandtheuniformdierentialtimeindexontheconvergenceofthenumericalsolutiontotheexactsolution.Wederiveorderrelationsfortheconvergencebymeansofanalgebraiccharacterizationoftheindexes.TheCrank{Nicolsonschemeisthesubjectofsubsection4.3.ThisinvestigationisshortenedbecausetheideasarethesameasfortheBTCSscheme.NumericalexamplesusingtheBTCSschemearepresentedinsubsection4.4.Firstwedeter-minenumericallyorderrelationsforconvergence.Herebythetheoreticalresultsofsubsections4.1and4.2areconrmed.Inasecondexamplewestudynumericallytheinuenceofincon-sistentBCs.Athirdexampledealswiththeproblemofanindexjumpwhichisconsideredalsoin[7].AlthoughthePDAEdoesnothaveanindexjumptheMOL-DAEmayhavesuchajump.Thismaybetheoriginofaninstability(asdiscussedinsection4.4)andoflargeerrorsinthenumericalsolution.2.PreliminariesInthissectionwecollectforconveniencefrom[7]someassumptions,denitionsandfactsnecessaryforthenumericalsolutionoftheinitialboundaryvalueproblem(IBVP)(1.1){(1.4).Thegeneralassumptionsare(I):TheIBVP(1.1){(1.4)hasoneandonlyonesolution.(II):Eachcomponentofthesoluti