ERROR ESTIMATES FOR THE RUNGE-KUTTA DISCONTINUOUS

ERRORESTIMATESFORTHERUNGE-KUTTADISCONTINUOUSGALERKINMETHODFORTHETRANSPORTEQUATIONWITHDISCONTINUOUSINITIALDATABERNARDOCOCKBURN∗ANDJOHNNYGUZM´AN†Abstract.Westudytheapproximationofnon-smoothsolutionsofthetransportequationinone-spacedimensionbyapproximationsgivenbyaRunge-KuttadiscontinuousGalerkinmethodofordertwo.Wetakeaninitialdatawhichhascompactsupportandissmoothexceptatadiscontinuity,andshowthat,iftheratioofthetimestepsizetothegridsizeislessthan1/3,theerroratthetimeTintheL2(R\RT)−normistheoptimalordertwowhenRTisaregionofsizeO(T1/2h1/2log1/h)totherightofthediscontinuityandofsizeO(T1/3h2/3log1/h)totheleft.Numericalexperimentsvalidatingtheseresultsarepresented.Keywords.discontinuousGalerkinmethods,errorestimates,hyperbolicproblemsAMSsubjectclassifications.65N301.Introduction.Inthispaper,wepresentthefirsterrorestimatesfortheRunge-KuttadiscontinuousGalerkin(RKDG)methodforthetransportequationwithdiscontinuousinitialdata.Theseresultsareobtainedforaformallysecond-orderaccurateRKDGmethodappliedtothemodelproblemUt+Ux=0inR×(0,T),(1.1a)U(·,0)=U0(·)onR,(1.1b)wheretheinitialconditionU0hascompactsupport;ithasadiscontinuityatx=0andissmootheverywhereelse.Roughlyspeaking,weshowthatthequalityoftheapproximationattimeTisofsecondorderinthesizeofthemesh,h,outsidearegionofsizeO(T1/2h1/2log1/h)totherightofthediscontinuityandofsizeO(T1/3h2/3log1/h)totheleft.AnillustrationofthisresultcanbeseeninFig.1.1.TheRKDGmethodwasintroducedbyCockburnandShuetal.inaseriesofpapers[10,9,8,6,11];seealsothemonographs[4,5]andthereview[12].MostaprioriandaposteriorierroranalyzesofdiscontinuousGalerkin(DG)methodsforhyperbolicproblemshavebeencarriedoutforeitherthesemidiscreteversionofthemethodorforDGmethodsusingspace-timeelements;see[7],wherethedevelopmentoftheDGmethodsuptotheendoflastcenturyisdescribed.Totheknowledgeoftheauthors,theonlyapriorierroranalysisfortheRKDGmethodisduetoShuandZhang[19]whoproved,amongotherthings,thatthesamemethodconsideredhere(butappliedtononlinearscalarconservationlawsinseveralspacedimensions)convergeswithordertwointheL∞(0,T;L2(Rd))-normprovidedthatthesolutionissmooth.Inthispaper,wecontinuethisefforttounderstandtheRKDGmethodandanalyzeitincaseofsolutionsthathavediscontinuities.Asasteppingstonetowardsthegoalofsolvingthemuchmorecomplicatedcaseofnonlinearscalarconservationlawsinseveralspacedimensions,weconsiderherethesimplermodelproblem(1.1)andfind,foreachtime∗SchoolofMathematics,UniversityofMinnesota,VincentHall,Minneapolis,MN55455,USA,email:[e-mail]onalScienceFoundation(GrantDMS-0411254).†SchoolofMathematics,UniversityofMinnesota,VincentHall,Minneapolis,MN55455,USA,email:guzma033@.umn.edu.SupportedbyaNationalScienceFoundationMathematicalSciencePostDoctoralResearchFellowship(DMS-0503050).12B.CockburnandJ.Guzm´anx00.20.40.60.81-0.6-0.4-0.200.20.40.60.811.21.41.6uhuh2/3log(1/h)h1/2log(1/h)O()O()T1/3T1/2Fig.1.1.ThetwopartsoftheregionRTcontainingthenumericallayeroftheRKDGmethod:Totheleft(top)andtotheright(bottom)ofthediscontinuityoftheexactsolutionu.Theapprox-imatesolutionuhwasobtainedbyusingh=1/100andk/h=0.3.ThetimeTis0.5.T,theregionRTaroundthediscontinuityoftheexactsolutionU(·,T)suchthattheapproximatesolutionuh(·,T)givenbytheRKDGmethodconvergeswiththeoptimalorderoftwointheL2(R\RT)-norm.Todothat,weuseanapproachwhichisamodificationoftheclassicalL2-argumenttoobtainerrorestimatesintheL∞(0,T;L2(Rd))-norm,see,forexample,subsection2.7of[5],wherethesemidiscretecaseistreated.Themodificationhasthreemainfeatures.ThefirstistheuseofthedecompositionoftheerroroftheapproximationgivenbytheRKDGmethodproposedbyShuandZhang[19].Thesecond,theuseofspecialprojectionsthatallowustoobtainthefullorderofcon-vergenceoftheapproximation;see[3].Inourtechnique,withouttheseprojections,theorderofconvergenceisreducedby1/2.Thethird,theintroductionofsuitablychosenweightsthankstowhichwecanlocalizetheestimatesandmakethedifferencebetweentheregiontotheleftofthediscontinuityandthattotherightofit.SimilarweightswereoriginallyusedbyJohnsonetal.[16]toprovelocalL2errorestimatesforasingularlyperturbedreaction-convection-diffusionproblemapproxi-matedbythestreamlinediffusion(SD)method;see[17]forL∞results.Recently,Guzm´an[13]provedsimilarresultsforaDGmethod.Moreover,ifoneapproximates(1.1)witheitherthestandardSDorDGmethodandlinearspace-timeelements,onecanshowusingtechniquesin[16],[13]thatthenumericallayerresultingfromdis-continuousinitialdataiscontainedinaregionwhosesizeisatmostO(log(1/h)h1/2)fromeithersideofthediscontinuity.Inthisarticleweshowthatinonesideofthediscontinuity,thesizeofthenumericallayercanbereducedtoO(log(1/h)h2/3)fortheRKDGmethod.Weaccomplishthisbytakingadvantageofthemonotonicityofoneofthetwoweightfunctionsthatweuse;seeTheorem4.11below.Resultsofthistypewereobtainedmanyyearsagoforfinitedifferencemethodsforthemodelproblem(1.1)withdiscontinuousinitialdata;see[1],[14],[2]and[15].Indeed,byusingofFouriertechniques,thesizeofthenumericallayertotheleftofthediscontinuitywasshowntobedifferenttothesizeofthenumericallayertoitsErrorestimatesforRKDGmethods3rightfo