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1、PatchdynamicswithbuffersforhomogenizationproblemsGiovanniSamaey1,DirkRoose1,andIoannisG.Kevrekidis21Dept.ofComputerScience,K.U.Leuven,Celestijnenlaan200A,3001Leuven,Belgium2Dept.ofChemicalEngineeringandPACM,PrincetonUniversity,Princeton,NJ08544February2,2008AbstractAnimportantclassofproblemsexhibits
2、smoothbehaviouronmacroscopicspaceandtimescales,whileonlyamicroscopicevolutionlawisknown.Forsuchtime-dependentmulti-scaleproblems,an“equation-free”frameworkhasbeenproposed,ofwhichpatchdynamicsisanessentialcomponent.Patchdynamicsisdesignedtoperformnumericalsimulationsofanunavailablemacroscopicequatio
3、nonmacroscopictimeandlengthscales;itusesappropriatelyinitializedsimulationsoftheavailablemicroscopicmodelinanumberofsmallboxes(patches),whichcoveronlyafractionofthespace-timedomain.Toreducetheeffectoftheartificiallyintroducedboxboundaries,weusebufferregionsto“shield”theboundaryartefactsfromtheinterior
4、ofthedomainforshorttimeintervals.Weanalyzetheaccuracyofthisschemeforadiffusionhomogenizationproblemwithperiodicheterogeneity,andproposeasimpleheuristictodetermineasufficientbuffersize.Thealgorithmperformanceisillustratedthroughasetofnumericalexamples,whichincludeanon-linearreaction-diffusionequationandt
5、heKuramoto–Sivashinskyequation.arXiv:physics/0412005v1[physics.comp-ph]1Dec200411IntroductionForanimportantclassofmulti-scaleproblems,aseparationofscalesprevailsbetweenthe(microscopic,detailed)levelofdescriptionoftheavailablemodel,andthe(macroscopic,continuum)levelatwhichonewouldliketoobserveandana
6、lyzethesystem.Consider,forexample,akineticMonteCarlomodelofbacterialgrowth[31].Astochasticmodeldescribestheprobabilityofanindividualbacteriumtorunor“tumble”,basedontherotationofitsflagellae.Technically,itwouldbepossibletosimplyevolvethedetailedmodelandobservethemacroscopicvariablesofinterest(e.g.cel
7、ldensity),butthiscouldbeprohibitivelyexpensive.Itisknown,however,that,undercertainconditions,onecouldwriteadeterministicequationfortheevolutionofthemacroscopicobservable(herebacteriaconcentration,thezerothm
