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OnsymplecticandmultisymplecticschemesfortheKdVequationU.M.Ascher∗R.I.McLachlan†January24,2004AbstractWeexaminesomesymplecticandmultisymplecticmethodsforthenotoriousKorteweg–deVriesequation,withthequestionwhethertheaddedstructurepreservationthatthesemethodsofferiskeyinprovidinghighqualityschemesforthelongtimeintegrationofnon-linear,conservativepartialdifferentialequations.Concentratingon2ndorderdiscretizations,severalinterestingschemesareconstructedandstudied.Ouressentialconclusionsarethatitispossibletodesignverysta-ble,conservativedifferenceschemesforthenonlinear,conservativeKdVequation.Amongthebestofsuchschemesaremethodswhicharesymplecticormultisymplectic.Semi-explicit,symplecticschemescanbeveryeffectiveinmanysituations.Compactboxschemesareeffectiveinensuringthatnoartificialwigglesappearintheapprox-imatesolution.Afamilyofboxschemesisconstructed,ofwhichthemultisymplecticboxschemeisaprominentmember,whichareparticularlystableoncoarsespace-timegrids.Keywords:Symplecticmethod,Multisymplecticmethod,Korteweg-deVriesequation,Boxscheme,Hamiltoniansystem1IntroductionThedesignanddevelopmentofsymplecticmethodsforHamiltonianor-dinarydifferentialequations(ODEs)hasyieldedverypowerfulnumerical∗DepartmentofComputerScience,UniversityofBritishColumbia,Vancouver,BC,V6T1Z4,Canada.(ascher@cs.ubc.ca).SupportedinpartunderNSERCResearchGrant84306.†IFS,MasseyUniversity,11-222,PalmerstonNorth,NewZealand.(R.McLachlan@massey.ac.nz)1OnsymplecticandmultisymplecticschemesforKdV2schemeswithbeautifulgeometricproperties.Symplecticandothersym-metricmethodshavebeennotedfortheirsuperiorperformance,especiallyforlongtimeintegration.See,e.g.,thebooksandmonographs[20,4,9]andthemanyreferencestherein.RecallthatHamiltoniansystemsdescribe,forinstance,themotionoffrictionless,energyconservingmechanicalsystems.Thus,theypossessmarginalstability,whichcorrespondingsymplecticnu-mericalschemesmimic.This“livingattheedgeofstability”isenabled,atleastforsufficientlysmall(andpossiblymany!)timesteps,bytheimpliedgeometricstructurethatsuchdiscreteschemesconserve.Ontheotherhand,ithaslongbeenknownthatconservativediscretiza-tionschemesfornonlinear,nondissipativepartialdifferentialequations(PDEs)governingwavephenomenatendtobecomenumericallyunstable,anddissipationhassubsequentlybeenroutinelyintroducedintosuchnu-mericalschemes.See,e.g.,thebooks[19,8],whichdescribetheseminalworkofKreiss[12]andmuchmore.Fornonlinearproblemsofthistype,inparticular,conservativedifferenceschemesareknowntooccasionallyyieldnumericalsolutionswhichatfirstlookfine,butatalatertimemaysuddenlyexplode–seeExample2.1below.Consequently,non-dissipativeschemeswerediscouraged,especiallyforlongtimeintegration.Typicalworkonpseudospectra,e.g.[21],whenappliedtostabilitystudiesofODEs,alsomustassumethateigenvaluesareplacedofftheimaginaryaxisandintothelefthalfplane,sothatsufficientlysmallcirclesofstabilitycanbedrawnaroundthem:InthecontextofHamiltoniansystemsthiscorrespondstousingaslightlydissipativediscretizationscheme.Thus,thecommonbeliefsoftwoestablishedcommunitiesseemheadedtoaclashuponconsideringsymplectictimediscretizationsforHamiltoniansemi-discretizationsandmultisymplecticdiscretizationsofcertainHamil-tonianPDEs.Thepurposeofthispaperistoexaminethissituationnu-merically,attemptingtoseewhethercarefullydesigned,conservativefinitedifferenceandfinitevolumediscretizationscanremainstableanddeliversharpsolutionprofilesforalongtime,andifyesthentowhatextentsym-plecticitystructureisessentialinsuchmethods.WethereforeconsidersomesymplecticandmultisymplecticmethodsforthenotoriousKorteweg-deVries(KdV)equation.Following[1]westudysemi-explicitsymplecticandfullyimplicitmultisymplectic,2ndorderdif-ferenceschemes.Wedemonstratetheiraccuracyandqualitativepropertiesaftermanytimesteps.Somesuchschemes,especiallythosebasedoncom-pactdiscretizationsinbothspaceandtime,remainremarkablystableoverawiderangeofproblemandgridparameters(includingcoarsespace-timegrids),andweinvestigatethisfurtherusingsteadystateanalysisandadispersionanalysis.Thisarticlebuildsontheworkreportedin[1],butitconcentratesmoreonthebroaderquestionposedabove.Section4.2developsadifferent,OnsymplecticandmultisymplecticschemesforKdV3finitevolumeviewofthenarrowboxscheme(8)whichthenprovesusefulinSection6.Thenumericalexamplesarenew,asisSection6.1,andsoarethemoregeneralboxschemes(10)andresultingminorimprovementssuchas(11).Indeed,thereisanentirefamilyofboxschemeswithsimilarstabilityandaccuracypropertiestothoseofthemultisymplecticboxscheme(6).Wesummarizeourconclusionsinthelastsectionofthearticle.2TheKdVequationTheKdVequationisgivenbyut=α(u2)x+ρux+νuxxx(1)=V0(u)x+νuxxx,V(u)=α3u3+ρ2u2.Weassumegiveninitialconditionsu(x,0)=u0(x)andperiodicboundaryconditions.See,e.g.,[6]forananalyticaltreatment.ForthisfamousequationweconsiderfinitevolumeandfinitedifferencediscretizationsonafixedgridwithstepsizesΔx,Δtinspaceandtime,respectively.Beforewestartthis,though,wemustexplainthechoiceoftheKdVequationasourtestbed.2.1WhyKdV?ThereareseveralgoodreasonstousetheKdVequationasaprototypeforourcomparativestudy.•Itisamodelnonlinearhype
本文标题:On symplectic and multisymplectic schemes for the
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