Non oscillatory central differencing for hyperbolic conservation laws - PDF document

ReprintedfromJournalofComputationalPhysicsAllRightsReservedbyAcademicPress,NewYorkandLondonVol.87,No2.,April1990(pp.408-463)Non-oscillatoryCentralDierencingforHyperbolicConservationLawsHaimNessyahuandEitanTadmorSchoolofMathematicalSciences,RaymondandBeverleySacklerFacultyofExactSciences,TelAvivUniversity,TelAviv,IsraelandInstituteforComputerApplicationsinScienceandEngineering,NASALangleyResearchCenter,Hampton,VirginiaReceivedAugust29,1988;revisedFebruary6,1989Manyoftherecentlydevelopedhigh-resolutionschemesforhyperbolicconservationlawsarebasedonupwinddierencing.ThebuildingblockoftheseschemesistheaveragingofanapproximateGodunovsolver;itstimeconsumingpartinvolvestheeld-by-elddecompositionwhichisrequiredinordertoidentifythe\directionofthewind."Instead,weproposetouseasabuildingblockthemorerobustLax-Friedrichs(LxF)solver.Themainadvantageissimplicity:noRiemannproblemsaresolvedandhenceeld-by-elddecompositionsareavoided.ThemaindisadvantageistheexcessivenumericalviscositytypicaltotheLxFsolver.Wecompensateforitbyusinghigh-resolutionMUSCL-typeinterpolants.Numericalexperimentsshowthatthequalityoftheresultsobtainedbysuchconvenientcentraldierencingiscomparablewiththoseoftheupwindschemes.AcademicPress,Inc.INTRODUCTIONInthispaperwepresentafamilyofnon-oscillatory,secondorder,centraldierenceapproximationstonon-linearsystemsofhyperbolicconservationlaws.Theseapproximationscanbeviewedasnaturalextensionsoftherst-orderLax-Friedrichs(LxF)scheme.Inparticular,total-variationandentropyestimatesareprovidedinthescalarcase,andunliketheupwindframework,noRiemannproblemsneedtobesolvedinthecaseofsystemsofconservationlaws.Theuseofsecond-orderpiecewise-linearapproximantsinsteadoftherst-orderpiecewise-constantones,compensatesfortheexcessiveLxFviscosity,andresultsinsecond-orderresolutionRiemann-solver-freefamilyofcentraldierenceschemes.Thepaperisorganizedasfollows.InSection2,wederiveourfamilyofhighresolutioncentraldier-encingschemes,usingtheLxFsolvertogetherwithMUSCL-typeinterpolants.Thus,ateachtime-levelwereconstructfromthepiecewiseconstantnumericaldata,anon-oscillatorypiecewiselinearapproximationofsecondorderaccuracy.Wethenfollowtheevolvingsolutiontothenexttimelevel,andendupbypro-jectingitbacktoapiecewiseconstantsolution.Theresultisafamilyofschemeswhichtakesaneasilyimplementedpredictor-correctorform.Theresolutionofourmethodhingesuponthechoiceofcertainlocalnumericalderivativeswithwhichonereconstructsthepiecewise-linearMUSCL-typeinterpolantsfromthepiecewise-constantdata.InSection3,weconcentrateonthescalarconservationlaw.Wediscussavarietyofchoicesfornumericalderivatives,andprovethattheresultingscalarfamilyofschemes,undertheappropriateCFLlimitation,satisesboththeTotalVariationDiminishing(TVD)propertyandacellentropyinequality.Thesepropertiesguaranteetheconvergencetotheuniqueentropysolution,atleastinthegenuinelynon-linearscalarcase. ThisresearchwassupportedbytheNationalAeronauticsandSpaceAdministrationunderNASAContractNos.NAS1-18107andNAS1-18605whilethesecondauthorwasinresidenceattheInstituteforComputerApplicationsinScienceandEngineering(ICASE),NASALangleyResearchCenter,Hampton,VA23665.AdditionalsupportwasprovidedbyU.S.-IsraelBSFGrantNo.85-00346,andbyNSFGrantNo.DMS85-03294whileinresidenceatUCLA,LosAngeles,CA90024.Copyright1990byAcademicPress,Inc. non-oscillatorycentraldifferencingInSection4,wedescribeseveralwaystoextendourscalarfamilyofcentraldierencingschemestosystemsofconservationlaws.Themainissueliesagaininthechoiceofvectorsofnumericalderivatives.First,wedescribeacomponent-wiseextensionforthedenitionofthesevectors,whichsharethesimplicityofthescalarfamilyofschemes.Next,wedemonstratethe exibilityofourcentraldierencingframework,whichenablesustoincorporatecharacteristicinformation{wheneveravailable,intothedenitionofnumericalderivatives.Wecontinue,byusingthischaracteristic-wiseframeworktoisolatethecontactwavewheretheArticialCompressionMethod(ACM)isemployed,whiletreatingthemorerobustsoundwavesusingthelessexpensivecomponent-wiseapproach.WeendupbypresentingacorrectivetypeACM,whichisimplementedinacomponent-wisemanner.Thisbothimprovesthecontactresolution,andretainsthesimplicityoftheRiemann-solver-freescalarapproach.Finally,inSection5wepresentnumericalexperimentswithourhigh-resolutionnon-oscillatorycentraldierenceschemes,andcomparetheresultswiththecorrespondingupwind-basedones.Boththequantitativeandqualitativeresultsforarepresentativesampleofcompressible owproblemsgovernedbytheEulerequations,arefoundtobeincompleteagreementwiththeresolutionexpectedbythescalaranalysis.Takingintoaccounttheeaseofimplementation,robustnessandtimeperformance,theseresultscomparefavorablywiththeresultsobtainedbythecorrespondingupwind-basedschemes.2.AFAMILYOFHIGH-RESOLUTIONCENTRALDIFFERENCINGMETHODSManyoftherecentlydevelopedhigh-resolutionschemes,whichapproximatetheonedimensionalsystemofconservationlaws @t+@ ))=0arebasedonupwinddierencing.TheprototypeofsuchupwindapproximationsistheGodunovscheme[4];itcomputesapiecewiseconstantapproximatesolutionovercellsofwidth
2�xj�1 ,whichisoftheform, x;t 2xxj+1 Toproceedintime,theGodunovschemerstevolvesthepiecewiseconstantsolution, x;t),forasucientlysmalltimestep
.Initiatedwith x;t),equation(2.1)consistsofasuccessivesequenceofnon-interactingRiemannproblems.Theirresultingsolutionattimelevel,canbeexpressedintermsoftheRiemannsolver, x;t 2 Thissolutionisthenprojectedbackintothespaceofpiecewiseconstantgridfunctions,seeFig.2.1, x;t
xZxj+1 2xj�1 y;tdy;x 2xxj+1 2: nessyahuandtadmor txvj(t)vj�1(t)vjt)vj(t)xj�1 2xj+1 Fig.2.1Integrationof(2.1)overatypicalcell[ 2+1 t;t]yields ThisshowstheupwindpropertyoftheGodunovscheme.Namely,ifthecharacteristicspeedsthroughouttherelevantneighboringcells,[],areallpositive(respectively,negative),then(2.5)issimpliedintotof(vj(t))�f(vj�1(t))](respectively,,f(vj+1(t))�f(vj(t))]).However,amorecomplexsituationoccurswhenthereisamixtureofbothrightgoingandleftgoingwaves.Inthiscase,thecomputationofGodunov'snumerical uxin(2.5)requiresustoidentifythe\directionofthewind,"i.e.,todistinguishbetweentheleft-andrightgoingwavesinsidetheRiemannfan.Theexact(orapproximate)solutionoftheRiemannfanmaybeanintricatetask,andinthiscontext,wementiontheeld-by-elddecompositionproposedbyRoe[19],whichintendstosimplifythistask.Instead,inthissectionweproposeahighresolutionapproximationof(2.1),whichisbasedonthestaggeredformoftheLax-Friedrichs(LxF)scheme, 2(t vj+vj+1]�[f(vj+1(t))�f(vj(t))]:(2:6)TheLxFscheme,[13],isaprototypeofacentraldierenceapproximation,whichoersagreatsimplicityovertheupwindGodunovscheme(2.5).Weobservethat(2.6)canalsobeinterpretedasapiecewiseconstantprojectionofsuccessivenon-interactingRiemannproblems,whichareintegratedoverastaggeredgrid,seeFig.2.2, 2(t) x;t
xZxjjR(x�xj+1 2 dx;x non-oscillatorycentraldifferencing txxjxj+1 2xjj+1 Fig.2.2TherobustnessoftheLxFscheme,(2.7),stemsfromthefactthatunliketheGodunovcase,hereweintegrateovertheentireRiemannfan,takingintoaccountboththeleft-andrightgoingwaves.Thisenablesustoignoreanydetailedknowledgeabouttheexact(orapproximate)Riemannsolver).Unfortunately,theLxFstaggeredsolver,(2.7),whichresultsinthesimplerecipe(2.6),suersfromexcessivenumericalviscosity,whichisevidentfromtheviscousform[23] 2[f(vjt)�f(vj�1(t1 2[Qj+1 2
vj+1 2(t)�Qj�1 2
vj�1 2(t
vj+1 Indeed,theclassofupwindschemesischaracterizedbyanumericalviscositycoecientmatrix 2j Aj+1 ,(here j+1 referstoanapproximateaverageoftheJacobianofx;t))overthecell[[t;t],e.g.,[22]).BytheCFLlimitation,thisamountofnumericalviscosityisalwayslessthantheamountofnumericalviscositypresentinthecentralLxFscheme,whosenon-staggeredformcorrespondsto.Consequently,theupwindGodunov-likeapproximationshavebetterresolutionthanthecentralLxFapproximation,thoughtheybothbelongtothesameclassofrst-orderaccurateschemes.Thisisoneofthemainmotivationsforusingupwindschemesasbuildingblocksforthemodernshockcapturingmethodsofhigher(thanrst-order)resolution,e.g.[7],[17],[24].Alternatively,ourproposedmethodwillusethesimplercentralLxFsolverasthebuildingblockforafamilyofhigh-resolutionschemes.InthismannerweshallretaintheLxFmainadvantageofsimplicity:noRiemannproblemsaresolvedandhenceeld-by-elddecompositionsareavoided.Themaindisadvantageofexcessivenumericalviscositywillbecompensatedbyusinghigh-resolutionMUSCLinterpolants,[24],insteadoftherst-orderpiecewiseconstantonesin(2.2).Tothisend,ateachtimelevelwerstreconstructfrom(2.2)apiecewiselinearapproximationoftheformx;t
xv0j�1 2xxj+1 Thisformretainsconservation,i.e.,(heretheoverbardenotesthe[]-cellaverage), x;t x;tsecond-orderaccuracyisguaranteediftheso-calledvectorofnumericalderivative, ,whichisyettobedetermined,satises
xv0j=@ (x=xj):9b) nessyahuandtadmorNext,wecontinuewithasecondstage,similartotheconstructionofthecentralLxFrecipe:weevolvethepiecewiselinearinterpolant,(2.9),whichisgovernedbythesolutionofsuccessivesequencesofnoninteractingGeneralizedRiemann(GR)problems,[1],seeFig.2.3,x;tx;tx;tx;t txvj+1 x;tx;t 2xj Fig.2.3Finally,theresultingsolutionisprojectedbackintothespaceofstaggeredpiecewise-constantgridfunc-tions 2(t x;t y;tdy;xInviewoftheconservationlaw(2.1),thelastintegralequals 2(t
x24Zxj+1 x;t x;t Thersttwolinearintegrandsontherightof(2.11),x;t)andx;t),aregivenby(2.9a)andcanbeintegratedexactly.Moreover,iftheCFLconditionmaxx;t ismet,thenthelasttwointegrandsontherightof(2.11),))and)),aresmoothfunctions;hencetheycanbeintegratedapproximatelybythemidpointruleattheexpenseoflocaltruncationerror.Thuswearriveat 2(t vj(t)+vj+1(t)]+ 8[v0j�v0j�[f(v(xj
t 2f(v(xj
t ByTaylorexpansionandtheconservationlaw(2.1), 2j(t)�1 2j; non-oscillatorycentraldifferencingmayserveasourapproximatemidvalue, withinthepermissiblesecond-orderaccuracyrequire-ment.Here, standsforanapproximatenumericalderivativeofthe ux
xf0j=@ ))+whichisyettobespecied.WeshouldemphasizethatwhileusingthecentraltypeLxFsolver,weintegratedovertheentireRiemannfan,seex;t)in(2.10),whichconsistsofboththeleft-andrightgoingwaves.Ontheonehand,thisenabledustoignoreanydetailedknowledgeabouttheexact(orapproximate)generalizedRiemannsolver);ontheotherhand,thisenablesustoaccuratelycomputethenumerical ux,x;whosevaluesareextractedfromthesmoothinterfaceoftwonon-interactingRiemannproblems.Insummary,ourfamilyofcentraldierencingschemestakestheeasilyimplementedpredictor-correctorform, 2j(t)�1 2j;)vj+1 2(t vj(t)+vj+1(t)]+ 8[v0j�v0j�[f(vjt+
t 2)�f(vj(t+
t Herethenumericalderivativesofbothgridfunctions,and,shouldobeytheaccuracyconstraints(2.9b)and(2.15).Inthismannerthesecond-orderaccuratecorrectorstep(2.16b),augmentstherst-orderaccuratepredictorstep(2.16a),andresultsinahigh-resolutionsecond-ordercentraldierenceapproximationof(2.1).Remarks.1.Thechoice
xv0j1 0in(2.16),recoverstheoriginalrst-orderaccurateLxFscheme(2.6).2.Ifinsteadof(2.6)weusethenon-staggeredversionoftheLxFscheme, 2[vjtj�1(t f(vj+1(t))�f(vj�1(t))];(2:17)andrepeatthereconstruction,evolutionandprojectionstepsdescribedabove,thentheresultinghighreso-lutioncentraldierencingapproximationamountsto 2j(t)�1 2j;)vj(t vj+1(t)+vj�1(t)]+ 4[v0j�1�v0j� 2[f(vjt+
t 2)�f(vj�1(t+
t Toguaranteethedesirednonoscillatorypropertyoftheseapproximations,thetwofreeingredientsatourdisposal{thenumericalderivatives and ,shouldbecarefullychosen.Thisissuewillbediscussedinthenexttwosections.3.THESCALARPROBLEMInthissection,weareconcernedwithnon-oscillatoryhigh-resolutioncentraldierencingapproximationsofthescalarconservationlaw @t@ ))=0Ourfamilyofhigh-resolutioncentraldierencingschemes(2.16)canberewrittenintheform 2(t vj(t)+vj+1(t)]�[gj+1�gj];(3:2a)wheretheso-calledmodiednumerical ux,,[18],isgivenby ))+ 8v0j(t+
t 2j(t)�1 2j:2b) nessyahuandtadmorHere, isanapproximateslopeatthegridpoint
xv0j=@ and isthenumericalderivativeofthegridfunction
xf0j=@ ))+Theconstraints(3.3)withsmooth(=Lipschitzcontinuous)rstorderperturbationsontheirright,guaranteethesecond-orderaccuracyofthecentraldierencingschemes(3.2).Inordertoensurethattheseschemesarealsonon-oscillatoryinthesensetobedescribedbelow,ournumericalderivatives, ,shouldsatisfyforeverygridfunctionsgn Const:MinMod 2;
wj�1 Here,theMinModstandsfortheusuallimiter,x;yMinModx;y sgnsgnMinandcanbesimilarlyextendedtoincludemore(thantwo)variables.Theconstraint(3.4)isrequiredinordertoguaranteetheTotalVariationDiminishing(TVD)propertyforthefamilyofcentraldierencingschemes(3.2).WerecallthatTVDisadesirablepropertyinthecurrentsetup,foritimpliesnospuriousoscillationsinourapproximatesolutionx;t),[7].However,itiswellknown,e.g.[7],[18],thatonecannotsatisfyboththeaccuracyrequirement,(3.3),andtheTVDrequirement,(3.4),atthenon-soniccriticalgridvalues,,where
2

vj�1 Therefore,thesecond-orderaccuracyrequirement,(3.3),mustbegivenupatthesecriticalgridvalues.Dierenceschemeswith(formal)secondorderofaccuracyatallbutthesecriticalgridvaluesmaybeclassiedashavingsecondorderresolutioninthesensethatthelocaltruncationerrorisalmosteverywhereandtheoverallsecond-orderaccuracydoesnotseemtobedegradedinsuchcases,atleastinthe-norm.WeshallverifytheTVDpropertyofthecentraldierencingschemes,(3.2),withthehelpofLemma3.1Thescheme(3.2a)isTVD,ifitsmodiednumerical ux,,satisesthefollowinggeneralizedCFLcondition 2
vj+1 2 2;
gj+1 Indeed,by(3.2a),thedierence 2(t)�vj�1 )equals 2(t)�vj�1 )=
2(1 2�
gj+1 2
vj+1 )+
2(1 2+
gj�1 2
vj�1 Condition(3.5)tellsusthatthetermsinsidetheparenthesisarepositiveandTVDfollowsalongthelinesof[7], 2(t)�vj�1 Equippedwithlemma3.1weturntoTheorem3.2Letthenumericalderivatives and in(3.3)bechosensuchthattheTVDrequirement(3.4)holds,say,sgn Const 2;
vj�1 ;Constsgn Const 2;
vj�1 27b) non-oscillatorycentraldifferencingAssumethatthefollowingCFLconditionissatisedmax;Const Const 2(p Thenthefamilyofhigh-resolutioncentraldierencingschemes(3.2),(3.3)isTVD.Proof:By(3.2b)wehave: 2
vj+1 2jf(vjt+
t 2f(vj(t+
t 2 vj+1 2j+1 8v0j+1 2
vj+1 2jf(vjt+
t 2f(vj(t+
t 2 jt+
t 2)�vj(t+
t j
j 2
t)�vj(t+
t 2)
vj+1 2j+1 8j
v0j+1 2
vj+1 OurCFLcondition(3.8)impliesthatthersttermontherightof(3.9)doesnotexceed 2f(vj(t+
t 2 jt+
t 2)�vj(t+
t Usingthemidvalue )in(3.2b),wecanestimatethesecondtermontherightof(3.9), 2)�vj(t+
t 2)
vj+1 2 2j
f0j+1 2
vj+1 2j;
f0j+1 whereinviewof(3.7b)and(3.8), 2
vj+1 max
vj+1 2j;jf0j vj+1 Const :Finally,theTVDrequirement,(3.7a),givesusanupperboundforthethirdtermontherightof(3.9), 2
vj+1 max(
vj+1 2j;jv0j vj+1 Using(3.10),(3.11)and(3.12),wendthat(3.9)boilsdowntothequadraticinequality(1+ 2 81 whosesolutionyieldstheCFLlimitation(3.8).Remarks.1.Thevalueswhichpermitapositivesolutionof(3.8),0,are02.TheTVDconstraints(3.7)with=0yields0,whichrecoversthestaggeredLxFscheme(2.6)withthecorrespondingCFLcondition 3.TheCFLrestriction(3.5)isasucientbutnotnecessaryconditionfortheTVDproperty.Inpracticeonemayusehighervaluesof,upto 4.Asimilaranalysiscarriedoutforthenon-staggeredform,(2.18),yields (p insteadof(3.8).Inpracticeonemayuse1inthiscase.Weshallnowdiscussafewexamplesofnumericalderivatives,whichretainboththesecondorderreso-lutionconstraint,(3.3),andtheTVDconstraints,(3.7).Asourrstexampleforthenumericalderivative,,wechoose 2;
vj�1 2g:) nessyahuandtadmorThischoicemayoversmearastrong\discontinuity",wheretheorderofaccuracyislesssignicant.Apreferablesecondchoice,whichallowsforasteeperslopenearsuchdiscontinuitiesandyetretainshigheraccuracyinsmoothregions,isgivenby 2;1 2(vjvj�1)vj�1 Thelimitingparametercanrangebetweenthevalues=1,whichcorrespondstothebasicMinModlimiterin(3.13a),andupto4,whichispermittedbytheCFLcondition(3.8).Similarly,the uxnumericalderivativemaybechosenas 2;
fj�1 whichisaspecialcaseof 2;1 2(fjfj�1)fj�1 Asimpleralternativefor(3.14)isgivenbywhereisalreadycomputedby(3.13).WeobservethatthischoicesaveshalfthecomputationtimeoftheMinModoperation;yet,itrequiresthecomputationoftheJacobian,),whendealingwithsystemsofconservationlaws.Thenumericalderivativechosenin(3.13a),(3.14a)satises(3.7)with=1,whichimpliestheTVDpropertyundertheCFLlimitation(3.8)with 2(p Thenumericalderivativechosenin(3.13b),(3.14b)clearlysatises(3.7)andconsequentlytheTVDproperty,foreverypermissible4.WesummarizetheabovebystatingCorollary3.3Letthenumericalderivative bechosenby 2;
vj�1 ;(3letthe uxnumericalderivativebechoseneitherby 2;
fj�1 Thenthefamilyofhighresolutioncentraldierencingschemes(3.2),(3.16)isTVDundertheCFLcondition; 2(p SimilarilywehaveCorollary3.4Letthenumericalderivative bechosenby 2;1 2(vjvj�1);j�1 ;(3letthe uxnumericalderivativebechoseneitherby 2;1 2(fjfj�1);j�1 2g:) non-oscillatorycentraldifferencingThenthefamilyofhighresolutioncentraldierencingschemes(3.2),(3.17)isTVD,undertheCFLcondi-tion; 2(p Remarks.1.WenotethattheCFLlimitationsinCorollaries3.3and3.4arenotsharp.Intherstcase,(3.16),wherealimiterparameter=1wasused,thereconstructionstepisaTVDoperation;replacingtheexactTVDevolutionoperatorbythemidpointrulein(2.11)togetherwiththenalaveragingstepisalsoTVD,undertheCFLlimitation .Similarly,onecanarguethatinthesecondcase,(3.17),wherealimiterparameter=2wasused,theaveragingstepretainstheTVDproperty(thoughnotnecessarilytheentropycondition),aslongastheCFLcondition ismet.Indeed,thisCFLconditionwasveriedasthestabilitiylimitation,bythenumericalexperimentsreportedinSection5.2.Recently,non-oscillatoryschemeswereconstructed,suchthatbysacricingtheTVDproperty,theyachievehigher(thansecond-order)resolutionincludingthecriticalgridvalues,e.g.,theUNOschemein[12]andtheENOclassofapproximationsin[9].Toimplementsuchideaswithinourframework,onecanborrowtheirdenitionofnumericalderivative.Forexample,insteadoftheTVDchoices(3.4),ourcentraldierencingscheme(3.2)maybeaugmentedbytheUNOchoice(here
2+1 2vj�1;
2vj);
vj+1 2�1 Theorem3.2anditscorollaries3.3and3.4demonstratehigh-resolutioncentraldierencingmethodswhichsatisfythenon-oscillatoryTVDproperty,andhenceareconvergenttoalimitsolutionx;t).Toguaranteethatthislimitsolutionistheuniqueentropysolutionofthescalarconservationlaw(3.1),weshallappealtothefollowingcellentropyinequality,see[10], 2(t1 U(vj)+U(vj+1)]�[Gj+1�Gj]:(3:19)Here)isaconvexentropyfunctionand)isthenumericalentropy uxwhichisconsistentwiththecorrespondingdierentialoneu;u;uWerecallthatLaxhasveriedsuchcellentropyinequalityfortheLxFscheme,[14].FollowingLax,wewillcontinuouslydeforminto+(1(0)=(1)=andinasimilarmanner,wewillfurtherdeform)intor;s)+(1IntheAppendixweproveLemma3.5Letbeapiecewisedierentiableinterpolantofthegridfunction.Thenthefollowingidentityholds 2(t1 2[U(vj(vjZvjjU0(v)g0(v)RUj+1 Heretheresidualterm, 2(g)RUj+1 isgivenby, ))=(
r;s r;s dsdr:Addingandsubtracting nessyahuandtadmorthenafterintegrationbyparts,therighthandsideof(3.21)willamountto: ))= 2[U(vj(vj[F(vj�F(vj(v)
(g(v)�f(vvjj+ZvjjUv)
(g(v)�f(vdvRUj+1 Consequently,theinequality providesuswithasucientconditionforthefamilyofcentraldierencingschemes(3.2)tosatisfythecellentropyinequality,(3.19),withnumericalentropy ux)).ThisbringsusLemma3.6Letbethepiecewiselinearinterpolantofthemodied uxgridfunction 2
vj+1 min(max(Assumethatthecentraldierencingschemes(3.2),satisfytheTVDconstraint(consult(3.7)),sgn Const 2;
v+j�1 ;Constwhere 2
vj+1 �
(max Theentropydissipativelimiterin(3.25b),isintroducedinordertopreventthenonexpansiveentropyviolatingrarefactions,consult[18,Section8].Moreover,assumethatthe uxnumericalderivativesatisestheTVDconstraint:sgn Const 2;
vj�1 Const ConstsothattheCFLcondition(3.8)holds.Thenthefollowinginequalityholds 2(g(v0u Remarks.1.WeobservethatintheGenuinelyNon-Linear(GNL)case,where,say,0,theentropyentropy(3.25b)becomeseectiveonlyinrarefactioncellswhere
0,inagreementwith[18].Itretainsthesecond-orderresolutionofthecentraldierencingschemes(3.2),exceptforanitenumberofcriticalcellswhichcontainstrongrarefactions,(
1,whereitreduces(3.2),(3.3)totheoriginalLxFscheme.2.Lemma3.6appliestochoicesofnumericalderivatives,,subjecttotheTVDconstraint(3.7a)with0Inpractice,highervalues,1,canbeused.Lemma3.6-whichisprovedintheappendix,showsthatourcentraldierencingTVDschemes(3.2),(3.7)fulllthesucientcondition(3.23)andconsequentlythecellentropyinequality(3.19),withrespecttothequadraticentropyfunction .Thus,thelimitsolutionofourcentralTVDschemes,x;tsatises 1 2u2 du:Thissinglesoutx;t)astheuniqueentropysolutionof(3.1),atleastintheGNLcase[2].WehaveshownTheorem3.7ConsidertheGNLscalarconservationlaw(3.1).Itisapproximatedbythefamilyofhighresolutioncentraldierencingschemes(3.2),(3.3)whichsatisfytheTVDandentropyconstraints,(3.25).Then,iftheCFLcondition(3.8)holds,wehave: non-oscillatorycentraldifferencing1.Second-orderresolution;2.TotalVariationDiminishingproperty;3.Aconsistentquadraticcell-entropyinequality;and,asaconsequenceof2.and3.:4.thecorrespondingcentraldierencingschemesconvergetotheuniquephysicallyrelevantsolutionoftheGNLconservationlaw(3.1).Weshallclosethissectionwithsomescalarnumericalexamples.WeconsidertheapproximatesolutionoftheinviscidBurger'sequation usingseveralofthepreviouslymentionedcentraldierencingschemes.Theyinclude:1.Therst-orderLxFschemeinitsnon-staggeredform(2.17).2.Thesecond-ordernon-oscillatorycentraldierencingscheme(2.18),(3.13a),(3.15).Thisistheordinarynon-staggeredversionofourcentraldierencingwhichwillbereferredtoasORD.3.Thesecond-ordernon-oscillatorycentraldierencingscheme(3.2),(3.13a),(3.15).ThisisthestaggeredversionofourcentraldierencingwhichwillbereferredtoasSTG.Equation(3.27)issolvedwithtwosetsofinitialconditions.Intherstcase,wehavethesmooth1-periodicinitialdata,0)=Thewellknownsolutionof(3.27),(3.28a),e.g.[15],developsashockdiscontinuityat31.TableIshowsusthenormoftheerrorsatthepre-shocktime15.ItindicatestherstorderaccuracyoftheLxFschemeincontrasttothesecondorderaccuracyofourcentraldierencing,ORDandSTG.InTableIwealsorecordedthesameerrorsatthepost-shocktime4.Thepresenceofashockdiscontinuityreducestheglobalerrortorstorder.However,thecentraldierencingSTGschemeperformssomewhatbetterthanthecentraldierencingORDschemeandtheybothhavebetterresolutionthantherst-orderLxFschemeinshock-freezones.TABLEI-NormoftheErrorsforNumericalSolutionsof0)=sin( t4 NLxFORDSTGLxFORDSTG 400.0237020.0026200.0008590.0444490.0036120.000849800.0122490.0006670.0002320.0234860.0012910.0002771600.0062460.0001690.0000610.0113830.0004980.0000983200.0031580.0000430.00000160.0052350.00002090.000038 nessyahuandtadmor 0.00.20.40.60.81.01.21.41.61.82.0 0.00.20.4U(X,T=0.4) 0.00.20.40.60.81.01.21.41.61.82.0 0.00.20.4U(X,T=0.4) 0.00.20.40.60.81.01.21.41.61.82.0 0.00.20.60.8U(X,T=0.4) 0.00.20.40.60.81.01.21.41.61.82.0 0.00.20.60.8U(X,T=0.4) 0.00.20.40.60.81.01.21.41.61.82.0 0.00.20.4U(X,T=0.4) 0.00.20.40.60.81.01.21.41.61.82.0 0.00.20.4U(X,T=0.4) Figure3.1: non-oscillatorycentraldifferencingThisbehaviorisampliedinthesecondcaseofsolvingtheRiemanproblem(3.27),withRiemaninitialdata:0)=InthiscasethesteadyshocksolutionisresolvedbythenumericalschemesasaviscousproleshoeninFig.3.1.Figure3.1illustratestheover-smearingoftheLxFprole,whencomparedwiththoseoftheORDandSTGschemes.Oncemore,weobservethattheSTGschemehassomewhatbetterresolutionthenitsnon-staggeredcounterpartORD.Yet,theCFLlimitationinthenon-staggeredform,1,resultsinabettertimepreformancethantheSTGschemewhichissubjecttotheCFLlimitation .(WerecallthatthesucientTVDconstraintinTheorem3.2ismorerestrictivethantheusualCFLlimitation;indeed,wenotethatthenumericalsolutionbytheORDversionofourschemeisTVDunderCFLlimitationyetitsvariationslightlyincreaseswith95.)Ineithercase,theseeasilyimplementednon-oscillatorycentraldierencingoutperformtherst-orderLxFone.4.SYSTEMSOFCONSERVATIONLAWSInthissection,wedescribehowtoextendourscalarfamilyofcentraldierencingschemestotheone-dimensionalsystemofconservationlaws, @t@ ))=0Herex;t)istheunknown-vectoroftheformx;tx;tx;tand)isthe uxvector,)=(;:::;fwithanJacobianmatrix, p;q;:::;N:Ourapproximatesolutionatthegridpointisgivenbythe-vector;:::;vj;Nandthecorrespondingvectorofdierences,
,consistsof-componentsdenotedby j;kEquippedwiththesenotations,ourfamilyofhigh-resolutioncentraldierencingschemes(3.2),(3.3),takestheform, 2(t vj(t)+vj+1(t)]�[gj+1�gj];(4:2a)wherethemodiednumerical ux,,isgivenby ))+ 8v0j(t+
t 2j(t)�1 Asbefore,thecomputationofand )requiresthenumericalderivativesofthegridfunctionsand.ThistimewehavetochoosetwoN-vectorsofnumericalderivatives, j;N j;N nessyahuandtadmorIntherestofthissection,weshalldescribetheprosandconsofseveralchoicesforthesevectorsofnumericalderivatives.Ourrstchoiceisacomponent-wiseextensionofthescalardenitioninSection3.Tothisendwemayuseeither(4.4a),j;k 2
vj�1 ;:::;N;orthemoregeneral(4.4b),j;k 21 2(vjvj�1vj�1 ;:::;N;orinstead,usetheUNO-likenumericalderivativein(3.18),j;k 21 j;k 21 j;k;:::;N:Apossiblechoiceforthevectorofnumerical uxderivativemaybeThisapproachinvolvesmultiplicationoftheJacobianmatrixbythevectorofderivatives,.Thismultipli-cationmaybeavoidedifweuseacomponent-wisedenitionforthevectorofnumerical uxderivatives,inanalogyto(3.14).Forexample,wemayusej;k 2
fj�1 oralternatively,j;k 21 2(fjfj�1fj�1 WeobservethattheJacobianFreeForm(JFF),(4.4),(4.6)avoidstheuseoftheJacobianmatrixrequiredby(4.4),(4.5),attheexpenseofcarryingouttheMinModoperationtwice.Theresultingcentraldierencingschemes,(4.2),whicharebaseduponthecomponent-wisedenitionofthenumericalderivativesin(4.4)-(4.6),sharethesimplicityofthescalarframework.Namely,noRiemannproblemsaresolvedandconsequentlycharacteristicdecompositions{requiredinordertodistinguishbetweentheleftandrightgoingwavesinsidetheRiemannfan,areavoided.Atthesametime,ourcentraldierencingapproachis exibleenoughsothatitenablesustoincorporatecharacteristicinformation,wheneveravailable,inordertoachieveimprovedresolution.Ournextchoiceshowshowtoincorporatethecharacteristicinformationintothedenitionofthenu-mericalderivatives.TothisendweshallemployaRoeMatrix, ),namely,anaveragedJacobian, ,satisfying,e.g.[11],[19], andhavingcompleterealeigensystem 2^Rj+1 ;:::;N:Letusprojectthevectorofdierences onto ,i.e.weusethecharacteristicdecomposition 2=Xk^j+1 2^Rj+1 ;:::;N;where 2^Lj+1 2
vj+1 ;:::;N:Thenthecorrespondingprojectionofthe uxvectorofdierencesisgivenby 2=Xk^j+1 2aj+1 2Rj+1 2 non-oscillatorycentraldifferencingNow,apossiblecharacteristic-wisechoiceforthenumericalderivativesinanalogywith(4.4),maybe(herej;kisdenotestheaveragedeigenvectorcenteredatj;k 2^j�1 j;kandthenumerical uxderivativescanbecalculatedas OnceagainwecanusetheJFFandavoidthemultiplicationofRoe'smatrixbythevectorofnumericalderivatives,ifinsteadof(4.11)weemploy,consult(4.9),j;k 2aj+1 2^j�1 2aj�1 j;kAsanexample,letusconsidertheEulerequations, 4mE35+@ 4m+pu(E+p)35�(E�1 Here;u;mu;pandarerespectivelythedensity,velocity,momentum,pressureandtotalenergy.ThecorrespondingRoematrix,),isassociatedwiththeeigensystem 2^Rj+1 wheretheeigenvalues^ aregivenby 2;1j+1 2�^cj+1 2;^aj+1 2;2j+1 2;^aj+1 2;3j+1 2j+1 andtherighteigenvectorsaregivenby 2;1=241^u�^c^H�^u^c35j+1 2^Rj+1 2;2=241^u1 2^u235j+1 2^Rj+1 2;3=241^u^H^c35j+1 Theaveragequantitiesontherightof(4.14)-(4.15)givenin[19]are, u� p ^H=p H� p ^c=r 1)( 2^u2)=Ej+pj where&#xw-38; )denotestheusualarithmeticmean.Thisbringsustothecharacteristicdecomposition(4.8),wherethecharacteristicprojections, 2;1=1 2(1�2);^j+1 2;2j+1 2�1;^j+1 2;3=1 areexpressedintermsof,whicharegivenby 2=^c2j+1 =(
2�
j+1 2^uj+1 2)=^cj+1 Wenotethatthesecondcontacteldassociatedwith isindependentofthesquarerootwhichisrequiredonlyinthecomputationofthemeanvaluesoundspeed^ .Sincethiseldisalinearlydegenerate,itlacksthestrongentropyenforcementtypicaltotheothertwogenuinelynon-lineareld,andtherefore,isusuallysmearedbynumericalschemes.Inournextchoiceofnumericalderivatives,weincorporateonlypartialcharacteristicinformation.Namely,weisolatethelessexpensive(i.e.,squareroot-free)characteristicprojectiononthecontacteld,andusethecomponent-wiseapproachfortheothertwoelds. nessyahuandtadmorThus,werstseparatethecontacteld, 2j+1 2
~Ej+1 237524
j+1 2
mj+1 2
Ej+1 235�^j+1 2;2
241^u1 2^u235j+1 andthendenethevectorofnumericalderivativeas 2;2;^j�1 241^1 2^u2�35j+64j+1 2;j�1 2j+1 2;j�1 2
~Ej+1 2;
~Ej�1 Similarly,computingthenumerical uxderivativewithacharacteristicapproachappliedonlytotheisolatedcontactwave, 2;1
~fj+1 2;2
~fj+1 2;337524
fj+1 2;1
fj+1 2;2
fj+1 2;335�^j+1 2;2
^aj+1 2;2
241^u1 2^u235j+1 amountsto 2;2^aj+1 2;2;^j�1 2;2^aj�1 2;2)
241^1 2^u2�35j+64
~fj+1 2;1;
~fj�1 2;1
~fj+1 2;2;
~fj�1 2;2
~fj+1 2;3;
~fj�1 ThelatterapproachenablesustouseeectivelytheArticialCompressionMethod(ACM)ontheisolatedcontacteld,e.g.[6],[7].Tothisend,thecontactwaveisolatedin(4.19)ismodiedby 2;2;^j�1 2;2g+jrj]
241^u1 2^u235j+64j+1 2;j�1 2j+1 2;j�1 2
~Ej+1 2;
~Ej�1 whereandaregivenby 2;2�^j�1 2;2j j^j+1 2;2j+j^j�1 2;2j;)rj=1 2^uj+1 2)2
^j+1 2;2;1 2^uj�1 2)2
^j�1 Finally,weshallmentionanalternativeapproachtothecharacteristicimplementationoftheACMin(4.22).Tothisend,theArticialCompressionisimplementedasafurthercorrectorsteptothecomponent-wiseapproachpresentedin(4.2a)-(4.2b).ThiscorrectivetypeACMtakestheform, 2�Wj�1 Here,thecompressioncoecient,,andaregivenby 2=8�������:wj;
wj+1 2

vj+1 2�0;wj=
vj�1 2;
vj+1 2g;wj
wj+1 2

vj+1 whereisrelatedtosubcellresolutioninformation(Harten,privatecommunication,[8]), 2(jj�1)j=
vj�1 2;
vj+1 2g:) non-oscillatorycentraldifferencingTheresultisthecentraldierencingscheme(4.2),appendedbythecomponent-wisedenitionsofnumer-icalderivativesin(4.10)-(4.12),andcomplementedbytheACMcorrectorstep(4.23).Thisscheme,unlikethecharacteristic-wiseimplementationoftheACMin(4.22),enjoysthesimplicityofthecomponent-wiseapproach,andatthesametime,enablesustodealeectivelywiththedelicatecontactwave.WeremarkthatoneshouldbecarefulnottoovercompressdiscontinuitiesusingsuchcorrectivetypeArticialCom-pression:itshouldbeimplementedaftertherarefactionwaveshaveevolvedusinganappropriatelychosencompressioncoecient5.NUMERICALEXAMPLESInthissection,wewillpresentnumericalexampleswhichdemonstratetheperformanceofourfamilyofhighresolutioncentraldierencingschemesforsystemsofconservationlaws.WeconsidertheapproximatesolutionoftheEulerequationsofgasdynamics,seesection4, 4mE35+@ 4mu(E+p)35�(E�1 Weexperimentwiththefollowingmembersfromourfamilyofhigh-resolutioncentraldierencingschemes:1.Thecentraldierencingscheme(4.2),(4.4a),(4.5).Thisisthecomponent-wiseextensionofthescalarSTGschemepresentedinSection3andisthereforereferredtobythesameabbreviation.2.Thecentraldierencingscheme(4.2),(4.4b),(4.5)withalimitervalue=2.ThisschemeisreferredtoasSTG2.3.Thecomponent-wiseUNO-typeversionofourscheme,(4.2),(4.4c),(4.5).ItisreferredtoasSTGU.4.TheschemeSTGwiththeadditionofthecorrectivetypeACMdescribedby(4.23)isreferredtoasAlltheaboveexamplesusecomponent-wisedenitionsforthevectorsofnumericalderivatives,andarebasedonthestaggeredgridformulation.Ourlastexampleisbasedonnon-staggeredLxFscheme,namely,5.Thecentraldierencingscheme(2.18),(4.4a),(4.5).Thisisthecomponent-wiseextensionofthescalarORDschemepresentedinsection3andisthereforereferredtobythesameabbreviation.Forthepurposeofperformancecomparisonweincludeheretheresultsofseveralwellknownupwindandcentralschemesaswell.Theseschemesinclude:1.Therstordercentralnon-staggeredLxFscheme,(2.17),[13].2.TherstorderaccurateGodunov-typeschemeofRoe,e.g.[7].3.Harten'ssecondorderaccurateupwindULT1scheme,[7].4.Harten'ssecondorderaccurateupwindULT1Cscheme,[7],whereArticialCompressionisaddedtoULT1inthelinearilydegeneratecontacteld.ItisreferredtoasULTC.Wesolvethesystem(5.1)withthreesetsofinitialconditions.OurrstexampleistheRiemannproblemproposedbySod[21](abbreviatedhereafterasRIM1),whichconsistsofinitialdata0)=;x=(1;x�=(0TableIIshowsthenormoftheerrors.Thoughtheresultsareelddependent,the\quantitativepicture"isfavourablewiththecentraldierencingschemes.TableIIIshowsthetimeperformanceofthevariousschemes.Alltheschemeshavetimeperformancesoforder,whereisthenumberofspatialcells.Figures5.1-5.4includeacomparisonbetweenthenumericalsolutionandtheexactsolution(shownby nessyahuandtadmorthesolidline),e.g.[3],[20],at1644.Asexpected,theoverallresolutionoftherstorderschemesisoutperformedbythesecondorderschemes.Weobservethatoursecondorderstaggeredschemes,STG,STG2,andSTGU,andsimilarily,thesecondorderupwindULT1scheme,smeartheshockdiscontinuityovertwocells.Thecontactdiscontinuity,however,ismoredelicate:hereweobservesmearingofabout5-6cellsbythesecondorderschemes,bothinthecentralandupwindcases.Wecanalsoobservetheover-andundershootsgeneratedbyboththeupwindULT1andcentralORD.TheseunsatisfactoryresultssuggesttointroduceACMinthecontacteld.ForthispurposewepresenttheupwindULTCschemeandthecentralcomponent-wiseSTGCschemeinFig.5.4.WenotethattheACMisappliedinSTGConlyatthelast10%ofthetimestepswiththecompressioncoecientThisresultsin2cellsresolutionofthecontactwave,andsomewhatbetterresolutionintheotherwavesaswell.Yet,smallover-andundershootswhichareduetoovercompression,stillremain.OursecondRiemanntestproblem(abbreviatedhereafterasRIM2),istheoneproposedbyLax[5].Itisinitiatedwith,=(0=(0andtheresultsat16canbefoundinFig.5.5-5.8.ThedensityproleinRIM2lacksthemonotonicitywehadinRIM1,andtherefore,itismoredicultfor\non-oscillatory"numericalschemestorecoverthecontactwaveandtheintermediate\plateau"whichfollows.Consequently,theupwindschemesperformheresomewhatbetterthanthecentralschemes:ULTCresolutionisbetterthanSTGCwhichhasmoreover-andundershootsthanbefore.WenotethatSTG2hasbetterresolutionanderrorsthanSTGUinallelds.ThisisduetothefactthatSTG2hassteeperslopeneardiscontinuities,consultSection2.Finally,theresultsofthenonstaggeredcentraldierenceschemeORDforbothRIM1andRIM2problemsarepresentedinFigure5.9.WerecallthattheCFLlimitationinthestaggeredcase, isnowdoubledtobeconsultSection3.Moreover,acomponent-wisereconstructionofthevectorofnumericalderivatives,enabledustoavoidanyRiemannsolverinthisnonstaggeredcase.Consequently,theORDschemeistwicefasterthanthestaggeredcentralversionsbasedonSTG,aswellastheupwindschemeULT1whichnecessitatesthe(approximate)solutionofaRiemannproblemateachcell.However,theresolutionofthisnonstaggeredversion,ORD,deteriorates,whencomparedtothestaggeredversionsandtheupwindmethods.Ourthirdproblem,discussedbyWoodward-Collelain[25],consistsofinitial-data,0)=where=100=100.Asolidwallboundaryconditions(re ection)isappliedtobothends.TheresultsarecomparedwiththefourthorderENOscheme[9],inFig.5.10-5.12.ThecontinouslineistheresultoftheENOschemewith800cells.WepresenttheresultsofSTG2andULT1with400cellsinFig.5.13-5.15atand038respectively.WeobservethattheupgradefromtherstorderLxFschemetothesecondorderSTG2,resultsinasubstantialimprovementofresolution,seeFig.5.10-5.15;moreover,STG2comparesfavourablywiththesecondorderupwindULT1scheme.Insummary,wemayconcludethatwhenstrongdiscontinuitiesarepresent,STG2seemstooerthebestresults,STGCcanbetunedtoobtainsharpresolutionattheexpenseofovercompression,andORDversionwasfoundtobethemosteconomical.Furtherextensivenumericalexperimentsdonealongtheselinesarereportedin[16]. WethankA.HartenforallowingustousehisENOresultsin[9]. non-oscillatorycentraldifferencingTABLEIIComputationTimeofRiemannProblems,resultsat ULT1/CSTGROEORDLxFSTGCSTGUSTG2NX 1.231.230.740.690.221.431.471.37504.934.752.922.710.855.675.885.4310019.8119.3211.6810.713.3722.7423.4921.662002.872.741.721.550.483.243.353.075011.5410.936.836.161.9012.8813.3012.2210046.3443.5027.2724.407.5251.4653.2048.83200 Notes.1.Duetoourmethodofimplementation,ULT1andULTChavethesamecomputationtime.InfactULT1issomewhatfasterthenULTC.2.AlltheaboveschemesuseaCFLnumberof0.95,exceptfortheversions,STG*,whichuseaCFLnumberof0.475.TABLEIIIRiemannProblems,NormErrors DensityVelocityPressure Nx501002005010020050100200 SchemeRiemanProblem-RIM1,t=0.1644LxF.03121.02460.01769.06651.04583.02814.03602.02458.01582ROE.01918.01308.00836.03224.02090.01145.01762.01109.00666ORD.01868.01026.00578.03315.01807.00959.01630.00861.00460STG.01495.00741.00409.02812.01105.00550.01232.00581.00294ULT1.01338.00806.00437.02933.01177.00820.01285.00736.00362STG2.01241.00619.00297.02449.01132.00494.01019.00487.00228STGU.01146 .00544 .00291 .02300 .00816 .00403 .00961 .00432 .00216 STGC.00982 .00322 .00172 .01994 .00481 .00276 .00705 .00270 .00153 ULTC.01269.00715.00361.02923.01761.00804.01283.00735.00362RiemanProblem-RIM2,t=0.16LxF.12162.09044.06165.13523.09294.05557.15860.10767.06537ROE.06630.04334.02827.07397.04144.02192.08399.04826.02655ORD.06791.03824.02231.07158.03623.01709.07836.04056.01995STG.04972.02903.01776.04392.02416.01307.05118.02669.01426ULT1.04518.03572.01477.05570.02603.01094.06075.02841.01206STG2.03473.02129.01151.03369.01655 .00849 .03956.02037.00988 STGU.03668.02152.01302.03323 .01657.01046.03907 .02031 .01121STGC.02764 .01291 .00647 .02285 .01356 .00836 .02355 .01409 .00873 ULTC.03001 .01566 .00872 .05504.02545.01074.05997.02784.01183 Notes.1.AlltheaboveschemesuseCFLnumberof0.95,exceptforthestaggeredversions,STG*,whichuseaCFLnumberof0.475.2.Theunderlinedresultsindicatethesmallestnormerrorsineverycolumn. nessyahuandtadmor 0.00.20.40.60.81.0 0.00.20.4DENSITY 0.00.20.40.60.81.0 0.00.20.4DENSITY 0.00.20.40.60.81.0 0.00.20.4VELOCITY 0.00.20.40.60.81.0 0.00.20.4VELOCITY 0.00.20.40.60.81.0 0.00.20.4PRESSURE 0.00.20.40.60.81.0 0.00.20.4PRESSURE Figure5.1:RIM1-ROEvs.ULT1 non-oscillatorycentraldifferencing 0.00.20.40.60.81.0 0.00.20.4DENSITY 0.00.20.40.60.81.0 0.00.20.4DENSITY 0.00.20.40.60.81.0 0.00.20.4VELOCITY 0.00.20.40.60.81.0 0.00.20.4VELOCITY 0.00.20.40.60.81.0 0.00.20.4PRESSURE 0.00.20.40.60.81.0 0.00.20.4PRESSURE Figure5.2:RIM1-LXFvs.STG2 nessyahuandtadmor 0.00.20.40.60.81.0 0.00.20.4DENSITY 0.00.20.40.60.81.0 0.00.20.4DENSITY 0.00.20.40.60.81.0 0.00.20.4VELOCITY 0.00.20.40.60.81.0 0.00.20.4VELOCITY 0.00.20.40.60.81.0 0.00.20.4PRESSURE 0.00.20.40.60.81.0 0.00.20.4PRESSURE Figure5.3:RIM1-STGvs.STGU non-oscillatorycentraldifferencing 0.00.20.40.60.81.0 0.00.20.4DENSITY 0.00.20.40.60.81.0 0.00.20.4DENSITY 0.00.20.40.60.81.0 0.00.20.4VELOCITY 0.00.20.40.60.81.0 0.00.20.4VELOCITY 0.00.20.40.60.81.0 0.00.20.4PRESSURE 0.00.20.40.60.81.0 0.00.20.4PRESSURE Figure5.4:RIM1-ULTCvs.STGC nessyahuandtadmor 0.00.20.40.60.81.0 0.00.20.4DENSITY 0.00.20.40.60.81.0 0.00.20.4DENSITY 0.00.20.40.60.81.0 0.00.51.01.52.0VELOCITY 0.00.20.40.60.81.0 0.00.51.01.52.0VELOCITY 0.00.20.40.60.81.0 0.01.02.03.04.0PRESSURE 0.00.20.40.60.81.0 0.01.02.03.04.0PRESSURE Figure5.5:RIM2-ROEvs.ULT1 non-oscillatorycentraldifferencing 0.00.20.40.60.81.0 0.00.20.4DENSITY 0.00.20.40.60.81.0 0.00.20.4DENSITY 0.00.20.40.60.81.0 0.00.51.01.52.0VELOCITY 0.00.20.40.60.81.0 0.00.51.01.52.0VELOCITY 0.00.20.40.60.81.0 0.01.02.03.04.0PRESSURE 0.00.20.40.60.81.0 0.01.02.03.04.0PRESSURE Figure5.6:RIM2-LXFvs.STG2 nessyahuandtadmor 0.00.20.40.60.81.0 0.00.20.4DENSITY 0.00.20.40.60.81.0 0.00.20.4DENSITY 0.00.20.40.60.81.0 0.00.51.01.52.0VELOCITY 0.00.20.40.60.81.0 0.00.51.01.52.0VELOCITY 0.00.20.40.60.81.0 0.01.02.03.04.0PRESSURE 0.00.20.40.60.81.0 0.01.02.03.04.0PRESSURE Figure5.7:RIM2-STGvs.STGU non-oscillatorycentraldifferencing 0.00.20.40.60.81.0 0.00.20.4DENSITY 0.00.20.40.60.81.0 0.00.20.4DENSITY 0.00.20.40.60.81.0 0.00.51.01.52.0VELOCITY 0.00.20.40.60.81.0 0.00.51.01.52.0VELOCITY 0.00.20.40.60.81.0 0.01.02.03.04.0PRESSURE 0.00.20.40.60.81.0 0.01.02.03.04.0PRESSURE Figure5.8:RIM2-ULTCvs.STGC nessyahuandtadmor 0.00.20.40.60.81.0 0.00.20.4DENSITY 0.00.20.40.60.81.0 0.00.20.4DENSITY 0.00.20.40.60.81.0 0.00.20.4VELOCITY 0.00.20.40.60.81.0 0.00.51.01.52.0VELOCITY 0.00.20.40.60.81.0 0.00.20.4PRESSURE 0.00.20.40.60.81.0 0.01.02.03.04.0PRESSURE Figure5.9:RIM1andRIM2byORD non-oscillatorycentraldifferencing 0.00.51.0 0.00.5DENSITY AT T=0.01 0.00.20.40.60.81.0 0.01.05.06.0DENSITY AT T=0.01 0.00.51.0 0.00.5VELOCITY AT T=0.01 0.00.20.40.60.81.0 -10.0-5.00.010.015.0VELOCITY AT T=0.01 0.00.51.0 0.00.5PRESSURE AT T=0.01 0.00.20.40.60.81.0 0.0100.0200.0PRESSURE AT T=0.01 Figure5.10:WC-BANGT=0.01-ENOvs.LXF(theENOresultsarein[9]) nessyahuandtadmor 0.00.51.0 0.00.5DENSITY AT T=0.03 0.00.20.40.60.81.0 0.05.010.0DENSITY AT T=0.03 0.00.51.0 0.00.5VELOCITY AT T=0.03 0.00.20.40.60.81.0 -7.0-5.0-1.01.05.07.09.011.0VELOCITY AT T=0.03 0.00.51.0 0.00.5PRESSURE AT T=0.03 0.00.20.40.60.81.0 0.0100.0400.0500.0800.0PRESSURE AT T=0.03 Figure5.11:WC-BANGT=0.03-ENOvs.LXF(theENOresultsarein[9]) non-oscillatorycentraldifferencing 0.00.51.0 0.00.5DENSITY AT T=0.038 0.00.20.40.60.81.0 0.01.03.04.0DENSITY AT T=0.038 0.00.51.0 0.00.5VELOCITY AT T=0.038 0.00.20.40.60.81.0 -1.01.03.09.011.015.0VELOCITY AT T=0.038 0.00.51.0 0.00.5PRESSURE AT T=0.038 0.00.20.40.60.81.0 0.050.0100.0200.0250.0350.0400.0PRESSURE AT T=0.038 Figure5.12:WC-BANGT=0.038-ENOvs.LXF(theENOresultsarein[9]) nessyahuandtadmor 0.00.20.40.60.81.0 0.01.05.06.0DENSITY AT T=0.01 0.00.20.40.60.81.0 0.01.05.06.0DENSITY AT T=0.01 0.00.20.40.60.81.0 -10.0-5.00.010.015.0VELOCITY AT T=0.01 0.00.20.40.60.81.0 -10.0-5.00.010.015.0VELOCITY AT T=0.01 0.00.20.40.60.81.0 0.0100.0200.0PRESSURE AT T=0.01 0.00.20.40.60.81.0 0.0100.0200.0PRESSURE AT T=0.01 Figure5.13:WC-BANGT=0.01-STG2vs.ULT1 non-oscillatorycentraldifferencing 0.00.20.40.60.81.0 0.05.010.0DENSITY AT T=0.03 0.00.20.40.60.81.0 0.05.010.0DENSITY AT T=0.03 0.00.20.40.60.81.0 -7.0-5.0-1.01.05.07.09.011.0VELOCITY AT T=0.03 0.00.20.40.60.81.0 -7.0-5.0-1.01.05.07.09.011.0VELOCITY AT T=0.03 0.00.20.40.60.81.0 0.0100.0400.0500.0800.0PRESSURE AT T=0.03 0.00.20.40.60.81.0 0.0100.0400.0500.0800.0PRESSURE AT T=0.03 Figure5.14:WC-BANGT=0.03-STG2vs.ULT1 nessyahuandtadmor 0.00.20.40.60.81.0 0.01.03.04.0DENSITY AT T=0.038 0.00.20.40.60.81.0 0.01.03.04.0DENSITY AT T=0.038 0.00.20.40.60.81.0 -1.01.03.09.011.015.0VELOCITY AT T=0.038 0.00.20.40.60.81.0 -1.01.03.09.011.015.0VELOCITY AT T=0.038 0.00.20.40.60.81.0 0.050.0100.0200.0250.0350.0400.0PRESSURE AT T=0.038 0.00.20.40.60.81.0 0.050.0100.0200.0250.0350.0400.0PRESSURE AT T=0.038 Figure5.15:WC-BANGT=0.038-STG2vs.ULT1 non-oscillatorycentraldifferencingAPPENDIX:ONACELLENTROPYINEQUALITYInthissection,weprovidethepromisedproofsforLemmatta3.7and3.8,whichverifythecellentropyinequalityforourscalarfamilyofhigh-resolutioncentraldierencemethods.WebeginwithaproofofLemma3.7. )denotethedierence, 2(g 2[U(vj(vjZvjjU0(v)g0(v)U(vj+1 Wenowcontinuouslydeform+(1,between(0)and(1),see(3.20a).Withthisinmind, )mayberewrittenintheform 2(g10d RHS�From(3.2a)wemayndthedependenceof )onthecontinuationparameter(forsimplicityweomittheexplicitdependenceontime): 2(s v(s)+vj+1]�[gj+1�g(v(s))];(A:3)whichinviewof (s
vj+1 2;(d (vj+1 ))= 2(s 2+(v(s
vj+1 Inasimilarmanner,wehave ))= andLeibnitzrulegivesus �Zvj(s)U0(v)g0(v)(v(s0(v(s
vj+1 Substitutionof(A.5),(A.6),and(A.7)into(A.2)yields 2(g
vj+1 2Z10[1 2+(v(s[U0(v(sU0(vj+1 Next,weusethecontinuationr;s)+(1in(3.20b)inordertoexpressthelastdierenceontherightas ))= 0(vj+1 r;sdr:Thisequalitycomesaboutasfollows:inviewof(3.20b),(3.2a), r;s)isgivenby r;s v(s)+v(r;ssg(v((r;s))];( 2(s)+1 2j+1 )and(A.9)follows.Notingthat r;s thenbycarryingoutthedierentiatonontheRHSof(A.9)weobtain, 0(vj+1 r;s))= r;s r;s 2:( nessyahuandtadmorSubstituting(A.9),(A.11)and(A.12)into(A.8),wewillendupwiththedesiredidentity(3.22) WeclosethissectionwiththeproofofLemma3.8.Thepiecewiselinearinterpolantofthegridfunction,,chosenin(3.24), 2
vj+1 hasaxedslopeateachcell:r;s))=))= 2
vj+1 �From(A.14)and(3.22)weobtainthatinthecaseofquadraticentropyfunctionwhere 2(g 2j+1 2)2241 4� 
gj+1 2
vj+1 Moreover,thedierence)betweentwoneighboringvaluesand,coversanareaofsize, 2[gjgjj+1 dv:Thus,inviewof(A.15)and(A.16),thedesiredinequality,(3.26),boilsdownto 2[gjgj]

vj+1 2�Zvjjf(v)1 2(
gj+1 2
vj+1 2)2�1 8j+1 Toverifytheinequality(A.17),werecallthatby(3.2a),(3.2b)wehave ))+ 8v0m=f(vm(t)� 2f0m j;jandTaylor'sexpansionyields 8v0m42j+1 2)2vj+1 Thisenablesustowritethersttwotermsontheleftof(A.17)as 2[gjgj]

vj+1 2�Zvjjf(v)1 842)
(v0jv0j j+1 2)
j+1 2)2+Oj+1 Considernowthethirdtermontheleftof(A.17):by(A.19)wehave 2=
fj+1 2+1 842)
(
v0j+1 2
vj+1 2)

vj+1 inserting 2=
vj+1 2+Oj+1 into(A.21a),squaringtheresultandrearrangingweobtain 2(
gj+1 2)2=2 2j+1 2)2+ 842)
(
v0j+1 2
vj+1 2)
j+1 2)2++1 42)2
(
v0j+1 2
vj+1 2)2
j+1 2)2+Oj+1 2)3:() non-oscillatorycentraldifferencingWenotethatthecubictermontherightof(A.20),(A.21b),consistsoftheerrorinthetrapezodialrule 2[f(vj(vjj+1 2�Zvjjf(v) v( xj+1 aswellasadditionalcontributionswhichareofthesameorderofmagnitude 2)3[fv( xj+1 Inserting(A.20),(A.21b),and(A.22)intotheinequality(A.17)givesus 8j+1 2)2
"v0jv0j j+1 2�
v0j+1 2
vj+1 2+1�42
v0j+1 2
vj+1 max Finally,sinceandagreeinsignwith
,theexpressioninsidetheleftbracketscanbeupperboundedby
vj+1 2v0jv0j 20jv0j 2j��1 2)j
v0j+1 2
vj+1 2j+1�42
v0j+1 2
vj+1 BytheCFLlimitation, ,thesumofthelasttwotermsisnonpositive,andweareleftwiththeinequalitymax
vj+1 2;v0j vj+1 max whichismetbythechoiceofentropysatisfyinglimiterin(3.25a),(3.25b). nessyahuandtadmor[1]Ben-Arzi,M.andFalcovitz,J.,\AnUpwindSecond-OrderSchemeforCompressibleDuctFlows,"SIAMJ.Sci.Stat.Comput.,Vol.7,No.3,July1986.[2]DiPerna,R.J.,\ConvergenceofApproximateSolutionstoConservationLaws,"Arch.Rat.Mech.Anal.,Vol.82,1983,pp.27-70.[3]Dutt,P.,\ARiemannSolverBasedonaGlobalExistenceProoffortheRiemannProblem,"ICASEReportNo.86-3,January1986.[4]Godunov,S.K.,\AFiniteDierenceMethodfortheNumericalComputationofDiscontinuousSolutionsoftheEquationsofFluidDynamics,"Mat.Sb.,Vol.47,1959,pp.271-290,inRussian.[5]Harten,A.,\TheMethodofArticialCompresssion:I.ShocksandContactDiscontinuities,"ResearchDevelop.Report,COU-3077.50,CourantInstitute,N.Y.U.,June1974[6]Harten,A.,\TheArticialCompressionMethodforComputationofShocksandContactDiscontinu-ities,III.Self-AdjustingHybridSchemes,"Math.Comput.,Vol.32,NewYork,1978,pp.363-389.[7]Harten,A.,\HighResolutionSchemesforHyperbolicConservationLaws,"J.Comput.Physics,Vol.49,1983,pp.357-393.[8]Harten,A.,\ENOSchemeswithSubcellResolution,"JCPVol.83,1989,pp.148-.[9]Harten,A.,Engquist,B.,andOsher,S.,andChakravarthy,S.R.\UniformlyHighOrderAccurateEssentiallyNon-OscillatorySchemesIII,"JCPVol.71,1987,pp.231-.[10]Harten,A.,Hyman,J.M.,andLax,P.D.,\OnFiniteDierenceApproximationsandEntropyCondi-tionsforShocks,"Commun.PureAppl.Math.,Vol.29,1976,pp.297-322.[11]Harten,A.andLax,P.D.,\ARandomChoiceFiniteDierenceSchemeforHyperbolicConservationLaws,"SIAMJ.Num.Anal.,Vol.18,1981,pp.289-315.[12]Harten,A.andOsher,S.,\UniformlyHighOrderAccurateNon-OscillatorySchemeI,"SINUMVol.24,1987,pp.279-.[13]Lax,P.D.,\WeakSolutionsofNon-LinearHyperbolicEquationsandTheirNumericalComputations,"Commun.PureAppliedMath.,Vol.7,1954,pp.159-193.[14]Lax,P.D.,\ShockWavesandEntropyinContributionstoNonlinearFunctionalAnalysis,"EditorE.A.Zarantonello,AcademicPress,NewYork,Vol.31,1971,pp.611-613.[15]Lax,P.D.,\HyperbolicSystemsofConservationLawsandtheMathematicalTheoryofShockWaves,"RegionalConf.SeriesinAppl.Math.,1973,SIAM,Phila.[16]Nessyahu,H.,\Non-OscillatorySecondOrderCentralTypeSchemeforSystemsofNon-LinearHyper-bolicConservationLaws,"M.Sc.Thesis,Tel-AvivUniversity,1987.[17]Osher,S.,\RiemannSolvers,TheEntropyConditionandDierenceApproximating,"SIAMJ.Num.Anal.,Vol.21,1984,pp.217-235.[18]Osher,S.andTadmor,E.,\OntheConvergenceofDierenceApproximationstoScalarConservationLaws,"Math.Comp.,Vol.50,1988,pp.19-.[19]Roe,P.L.,\ApproximateRiemannSolvers,ParameterVectors,andDierenceSchemes,"J.Comput.Phys.,Vol.43,1981,pp.357-372.[20]Smoller,J.,\ShockWavesandReaction-DiusionEquations,"SpringerVerlag,N.Y.,1983. non-oscillatorycentraldifferencing[21]Sod,G.A.,\ASurveyofSeveralFiniteDierenceMethodsforSystemsofNonlinearHyperbolicCon-servationLaws,"J.Comp.Phys.,Vol.27,1978,pp.1-31.[22]Tadmor,E.,\TheLargeTimeBehaviouroftheScalar,GenuinelyNonlinearLax-FriedrichsScheme,"Math.Comp.,Vol.43,1984,pp.353-368.[23]Tadmor,E.,\NumericalViscosityandtheEntropyConditionforConservativeDierenceSchemes,"Math.Comp.,Vol.43,1984,pp.369-382.[24]VanLeer,B.,\TowardstheUltimateConservativeDierenceSchemeV.A.Second-OrderSequeltoGodunov'sMethod,"J.Comp.Phys.,Vol.32,1979,pp.101-136.[25]Woodward,P.andCollellaP.,\TheNumericalSimulationofTwo-DimensionalFluidFlowwithStrongShocks,"J.Comp.Phys.,Vol.54,1984,pp.115-173.