Numerical Simulation for Nonlinear Partial Differe

NumericalSimulationforNonlinearPartialDifferentialEquationwithVariableCoefficientsbyMeansofTheDiscreteVariationalMethod∗TakanoriIdeProductEngineeringDepartmentNo.1,AISINAWCO.LTD.Anjo,Japan.andDepartmentofMathematics,TokyoMetropolitanUniversityHachioji,Japan.E-mail:[e-mail]fMathematics,TokyoMetropolitanUniversityHachioji,Japan.E-mail:[e-mail]fferentialequationswithvariablecoefficientsinvolvingdiscontinuouscaseplayanimportantpartinengineering,physicsandecology.Inthispaper,wewillstudynonlinearpartialdifferentialequationswithvariablecoefficientsarisedfrompopulationmodels.Generallyspeaking,itishardtoanalyzethebehaviorofnonlinearpartialdifferentialequations,thereforeweusuallyrelyonthenumericalapproximation.Currently,thereisanincreasinginterestindesigningnumericalschemesthatpreserveinvariantsfordifferentialequations.Wewilldesignthenumericalschemesthatpreserveenergypropertyandgiveconjecturesforourtargetequation.1IntroductionThemainpurposeofthispaperistoinvestigatethenumericalsolutionsofnonlinearpartialdifferentialequationwithvariablecoefficientsarisedfrompopulationmodel[1].Thevariablecoefficientsofourequationinvolvethediscontinuouscase.Therefore,wethinkthatthebehaviorofsolutionrapidlychangesaroundtheneighborhoodofthediscontinuouspoint.Rannacherhasalreadystudiedthefiniteelementsolutionofdiffusionproblemwithirreguralitiesintheinitialorboundarydata[30].Thesedays,manyworksexistconcerningfordesigningofstructurepreservingnumericalschemesforordinarydifferentialequations[5],[13],[15],[19],[22],[23],[24],[31],[33],[34](andtheirreferences)andpartialdifferentialequations[6],[16],[21],[29],[32](andtheirreferences).Recently,FurihataandMoriproposedthediscretevariationalmethodwhichisanexactdiscretizationofthecontinuousvariationalmethod[9],[10],[11].Thismethodisoneprocedurefordesigningfinitedifferenceschemepreservingenergypropertiessuchasenergydissipation,energyconservationandsoon[17].Matsuoandothersexpandedthediscretevariationalmethodtocomplexvaluedproblem[25].Theyalsoexpandedthismethodtodesigningspatiallyaccurateschemesunderperiodicconditions[26].Inthispaper,wewilldesignthefinitedifferenceschemefornonlinearpartialdifferentialequationwithvariablecoefficientsthatFurihataandMatsuodidnotconsiderbymeansofthediscretevariationalmethod.Thecontentsofthispaperareasfollows:Insection2,wewilldefinethenotationofthediscretesymbolsandtheircalculusemployedthroughoutthispaper.Insection3,wewilldescribetherelationshipbetweenthecontinuousvariationalderivativeandthediscretevariationalderivative[9],[10][11].Insection4,we∗AMSSubjectClassification:35-04,35K55,37K05,65M061willshowthepartialdifferentialequationthatweanalyzeinthispaperanditsenergyproperty.Wewillalsoderivethefinitedifferenceschemefornonlinearpartialdifferentialequationwithvariablecoefficientsthatpreserveenergyproperty.Insection5,wewillshowourtargetequation.Insection6,wewillprovethestabilityanduniqueexistenceofnumericalsolutionthatweproposeinsection5.Finally,wewilldemonstratethenumericalsimulation.Wewillalsogiveconjecturesofourtargetequation.2NotationsandpreliminariesInthissection,wedefinetheone-dimensionaldiscreteoperators.LetΩ=[a,b]andthemeshsizewithrespecttoxisdefinedby∆xdef=b−aN.(1)Wedenotebyfkanumericalvaluesupposedtobeanapproximationoff(a+k∆x),k=0,1,···,Nfkf(a+k∆x),f={fk}Nk=0∈RN+1.(2)DifferenceoperatorsWedefinethedifferenceoperatorsasfollows:δ+kfkdef=fk+1−fk∆x,(3)δ−kfkdef=fk−fk−1∆x,(4)δ(1)kfkdef=fk+1−fk−12∆x,(5)δ(2)kfkdef=fk+1−2fk+fk−1(∆x)2.(6)2.1OtherdiscreteoperatorsWedefinetheshiftoperatorsandaveragingoperatorsinthissubsection.ShiftoperatorsTheshiftoperatorsaredefinedasfollows:s+kfkdef=fk+1,(7)s−kfkdef=fk−1,(8)s(1)kfkdef=s+k+s−k2.(9)AveragingoperatorsWedefinetheaveragingoperatorsbyusingshiftoperatorsasfollows:µ+kdef=12(s+k+1),(10)µ−kdef=12(s−k+1),(11)µ(1)kdef=µ+k+µ−k2=1+s(1)k2.(12)2.2SummationAsthediscretizationoftheintegral,weadopttheperiozoidalrule
Nk=0

thatisdefinedbyNk=0

fk∆xdef=12f0+N−1k=0fk+12fN∆x.(13)2SummationbypartswithperiozoidalruleThefollowingisthediscretizationoftheintegralbypartsformulainusualcalculusfortwosequencesNk=0

fk(δ+kgk)∆x+Nk=0

(δ−kfk)gk∆x=fk(s+kgk)+(s−kfk)gk2Nk=0(14)wheref={fk}Nk=0,g={gk}Nk=0∈RN+1andgkisanapproximationofg(a+k∆x).Remark:Thediscreteoperatorswithrespecttotaredefinedinthesamemannerwithfncorrespondingtof(n∆t),n=0,1,2,···.3ThediscretevariationalmethodInthissection,wegiveabriefdescriptionofourmethodasitappliestothepartialdifferentialequationsthatconcernsushere.Thediscretevariationalmethodisanexactdiscretizationoftheusualcontinuousvariationalmethod.Wewillsummarizethedefinitionofthediscretevariationalderivative[10].Wewillrecallthecontinuousvariationalmethodbriefly.Firstly,wedefinetheenergyasfollowsJ[u]def=baG(u,ux)dx,(15)wherewecallG(u,ux)theenergyfunctioninthispaper.Bytheusualvariationalargument,wehaveJ[u+δu]−J[u]baδGδuδudx+∂G∂uxδuba,(16)whereδGδuistheEuler-LagrangevariationalderivativedefinedbyδGδudef=∂G∂u−∂∂x∂G∂ux.(17)Fromnowon,wewilldiscretizethevariationalderivativeinanexactmanner.Firstly,thediscreteenergyisde