Explicit Runge-Kutta Methods for the Numerical Sol

UNIVERSITYOFMANCHESTERExplicitRunge-KuttaMethodsfortheNumericalSolutionofSingularDelayDierentialEquationsC.A.H.Paul&C.T.H.BakerNumericalAnalysisReportNo.212(revised)April1992UniversityofManchester/UMISTManchesterCentreforComputationalMathematicsNumericalAnalysisReportsDEPARTMENTOFMATHEMATICSReportsavailablefrom:DepartmentofMathematicsUniversityofManchesterManchesterM139PLEnglandAndbyanonymousftpfrom:vtx.ma.man.ac.uk(130.88.16.2)inpub/narepsExplicitRunge-KuttaMethodsfortheNumericalSolutionofSingularDelayDierentialEquationsC.A.H.PaulandC.T.H.BakerApril27,1992AbstractInthispaperweareconcernedwiththedevelopmentofanexplicitRunge-Kuttaschemeforthenumericalsolutionofdelaydierentialequations(DDEs)whereoneormoredelayliesinthecurrentRunge-Kuttainterval.TheschemepresentedisalsoapplicabletothenumericalsolutionofVolterrafunctionalequations(VFEs),althoughthetheoryisnotcoveredinthispaper.WealsoderivethestabilityequationsoftheschemefortheODEy0(t)=y(t);andtheDDEy0(t)=y(t)+y(t);wherethedelayandtheRunge-KuttastepsizeHarebothconstant.InthecaseoftheDDE,weconsiderthetwodistinctcases:(i)H,and(ii)H.WeevaluatetheperformanceoftheschemebysolvingseveraltypesofsingularDDEandaVFE.Keywords.ExplicitRunge-Kutta,singulardelaydierentialequationsAMSsubjectclassications.primary65Q051IntroductionWeareconcernedwithextendingexplicitRunge-Kuttaordinarydierentialequation(ODE)solverstosolvedelaydierentialequations(DDEs).AnyRunge-KuttamethodmaybeexpressedintheformofaButchertableau.Thus,a-stageRunge-KuttamethodtosolveanODEfromaninitialpointt0withinitialvaluey(t0)andderivativefunctionf(t;y(t))maybeassociatedwiththeButchertableaucAbc1a11a1.........ca1a.b1b(1)MathematicsDepartment,UniversityofManchester1TheRunge-Kuttamethodrepresentedby(1)yieldsanapproximationey(t)tothetruesolutiony(t)ofthedierentialequation,whichisgivenbyey(tn+H)=ey(tn)+HXi=1bifni;whereey(t0)=y(t0);tni=tn+ciH;fni=f(tni;ey(tni));ey(tni)=ey(tn)+HPj=1aijfnj:(2)Forcoincidentfcig,theseformulaecontainanambiguitywhichisresolvedinanobviousway.ClearlytheRunge-Kuttaformulae(2)mayonlybesolveddirectlyifaij=0forallji.InthiscasetheRunge-Kuttamethodiscalledexplicit,andthecorrespondingmatrixAisstrictlylowertriangular.Theterm\orderofaRunge-Kuttamethodreferstotheglobalorder(Defn.1.3).TheorderofaRunge-Kuttamethodgovernshowclosetheapproximatesolutioney(t)istothetruesolutiony(t)forsmallstepsizesH,givensmoothness.TheorderequationsforaRunge-Kuttamethoduptoorderthreeare:Pibi=1order1;Pibici=12order2;Pibic2i=13Pi;jbiaijcj=169=;order3:Theattainableorderofamethodisdeterminedbytheorderequations,ofwhichtherearetwosub-typesthatareofinteresttoushere.Denition1.1Thep-thorderquadratureequationisXibicp1i=1p:(3)Thequadratureorderequations1areassociatedwiththeintegrationofpurequadratureterms{elementarydierentialsoftheform@[e-mail]lsodependsony(t),thenon-quadratureorderequationsmustbeconsidered.In[2]thetotalityoforderconditionsispresentedintermsofsimplifyingconditions.Denition1.2ARunge-Kuttamethodislocalorderp+1forsucientlysmoothderivativefunctionf(t;y(t))andsolutiony(t),ifalltheorderequationsuptoandincludingthep-thorderequationsaresatised.1IntheshorthandnotationofButcher[2],theorderequations(3)uptoordermmaybewrittenasB(m).2Denition1.3ARunge-Kuttamethodisofglobalorderp(orap-thorderRunge-Kuttamethod)ifforxedTt0jey(t)y(t)j=O(Hp)forallt2[t0;T],asH!0.Theorem1.4ARunge-Kuttamethodisglobalorderpifitislocalorderp+1.Denition1.5Acontinuousextensioney(tn+H)ofa-stageRunge-Kuttamethodisanapprox-imanttoy(t)overtheinterval[tn;tn+H],whichmaybewrittenasey(tn+H)=ey(tn)+HXi=1bi()fnifor01:(4)Thereexistothercontinuousextensionsintheliterature,suchashighlycontinuousextensions[4]andmultistageRunge-Kuttacontinuousextensions[5].Denition1.6Acontinuousextensioney(tn+H)hascontinuousquadratureorderqifXibi()csi=s+1s+1fors=0;...;q1:(5)Denition1.7Acontinuousextension(tn+H)ofasucientlysmoothfunction(t)haslocalorderpifmax01(tn+H)(tn)+HXibi()0(tn+ciH)!=O(Hp):TheorderofacontinuousRunge-Kuttamethod(thatdenesanextensioney(tn+H))isdeter-minedbyreplacingHbyH,fbigbyfbi()gandey(tn+H)byey(tn+H)intheorderequations[2].ThecontinuousorderequationsuptoorderthreearethusPibi()=order1;Pibi()ci=22order2;Pibi()c2i=33Pi;jbi()aijcj=369=;order3:(6)ARunge-KuttamethodcanbecombinedwithacontinuousextensiontoproduceacontinuousRunge-Kuttamethod,whichmaybeassociatedwiththecontinuousButchertableaucAb()c1a11a1.........ca1a.b1()b()(7)Generallythevaluesfbi()j=1gcoincidewiththevaluesfbig.Denition1.8Thefull-steporderofacontinuousRunge-Kuttamethodistheorderofthemethodat=1.32ClassesofDDEDenition2.1Adelaydierentialequation(DDE)isadierentialequationinwhichthederiva-tivefunctiondependsonanitenumberofpreviousvaluesofthesolutiony(t).ThesimplestDDEisapuredelaydierentialequationwithconstantdelay(t)=,forexampley0(t)=y(t)(tt0);(t)=f(t);(8)wheretheinitialfunction(t)isdenedontheinterval[t0;t0].Overtheinterval[t0;t0+],(8