Stochastic mode reduction for the immersed boundar

STOCHASTICMODEREDUCTIONFORTHEIMMERSEDBOUNDARYMETHOD∗PETERR.KRAMER†ANDANDREWJ.MAJDA‡SIAMJ.APPL.MATH.c2003SocietyforIndustrialandAppliedMathematicsVol.64,No.2,pp.369–400Abstract.Weapplytheformulationofastochasticmodereductionmethoddevelopedinare-centpaperofMajda,Timofeyev,andVanden-Eijnden[Comm.PureAppl.Math.,54(2001),pp.891–974](MTV)toobtainsimplifiedequationsforthedynamicsofstructuresimmersedinathermallyfluctuatingfluidatlowReynolds(orKubo)number,assimulatedbyarecentextensionoftheim-mersedboundary(IB)methodbyKramerandPeskin[ProceedingsoftheSecondMITConferenceonComputationalFluidandSolidMechanics,ElsevierScience,Oxford,UK,2003,pp.1755–1758].Theeffectivedynamicsoftheimmersedstructuresarenotobviousintheprimitiveequations,whichinvolvebothfluidandstructuredynamics,buttheprocedureofMTVallowstherigorousderivationofareducedstochasticsystemfortheimmersedstructuresalone.Wefind,inthelimitofsmallReynolds(orKubo)number,thattheLagrangianparticleconstituentsoftheimmersedstructuresundergoadrift-diffusivemotionwithseveralphysicallycorrectfeatures,includingthecouplingbe-tweendynamicsofdifferentparticles.TheMTVprocedureisalsoappliedtothespatiallydiscretizedformoftheIBequationswiththermalfluctuationstoassistinthedesignandassessmentofnumericalalgorithms.Keywords.stochasticmodereduction,immersedboundarymethod,BrownianmotionAMSsubjectclassifications.60H10,60H30,60J60,60J65,60J70,76R50,82C31,82C70,82C80DOI.10.1137/S00361399034221391.Introduction.Inseveralapplicationsofmoderninterest,thegoverningequa-tionscanbewrittenasacomplexsystemofstochasticdifferentialequations,withthevariables(modes)evolvingoverawiderangeofcharacteristictimescales.Sometimes,thevariablescanbegroupedintoa“fast”classofmodesanda“slow”classofmodes,withawideseparationbetweenthetimescalesofthetwoclasses.Insuchasituation,onecanexploitsingularperturbationtechniquesusingtheratioofthefasttoslowtimescalesasasmallparametertoreducethesystembyaveragingtheeffectsofthefastmodesonthesystem.ArigorousprocedureforaveragingoverfastfluctuationsinastochasticsystemwasfirstprovidedbyKhas’minskii[24,23]andthenlaterde-velopedintomorewidelyapplicabletheoremsbyKurtz[30](seealso[9]),EllisandPinsky[7],andPapanicolaou[39].(Seethetextbook[17]foranappliedexposition.)Recently,Majda,Timofeyev,andVanden-Eijnden[34,35,36]([36]hereafterreferredtoasMTV)havedevelopedthesemathematicaltechniquesintoamethodologicalframeworkforclimatemodeling,wherethegoverningequationsareoftenessentiallyquadraticallynonlinearandcontainbothslowlyvaryingclimateand“meanflow”modesandmorerapidlyfluctuatingmodes.Inthisworkandthecompanionpa-per[26],wedemonstratehowtheMTVframeworkcanbeappliedproductivelytoaquitedifferentclassofapplications,namelythesimulationofmicroscalefluidsys-temswithimmersedstructuresandthermalfluctuations,suchasmicrophysiological∗ReceivedbytheeditorsJanuary30,2002;acceptedforpublication(inrevisedform)June11,2003;publishedelectronicallyDecember31,2003.†DepartmentofMathematicalSciences,RensselaerPolytechnicInstitute,301AmosEatonHall,1108thStreet,Troy,NY12180([e-mail]).‡CourantInstituteofMathematicalSciences,NewYorkUniversity,251MercerStreet,NewYork,NY10012([e-mail]).Thisauthor’sresearchwassupportedinpartbyAROgrantDAAD19-01-10810,NSFgrantDMS9972865,andONRgrantN00014-96-1-0043.369370PETERR.KRAMERANDANDREWJ.MAJDAsystems,colloidsuspensions,andpolymersuspensions.Inthepresentpaper,wefo-cusontheimmersedboundary(IB)method[42]forsimulatingbiologicalsystems,asextendedrecentlybyKramerandPeskin[28]toincludethermalfluctuationsandtherebyextenditsapplicabilitytosmallscales(microns).TheIBmethodemphasizesthedynamicsofthefluidenvironmentinwhichthebiologicalstructures,suchaspolymers,membranes,andparticles,areimmersed.Theforcesgeneratedbythebiologicalstructuresastheyaredeformedorinteractwithexternalfieldsarecommunicatedlocallyasforcesonthefluid.ThefluidthenrespondsdynamicallytotheseforcesinawayrepresentedbytheNavier–Stokesequations.Thestructuresarethenadvected(andstrained)bytheirlocalfluidvelocity.Thermalfluctuationsareintroducedthroughforcesonthefluid.Theimmersedstructuresarenotdirectlythermallyforced,butratherundergothermalfluctuationsthroughadvectionbythethermallyfluctuatingfluid[28].TheIBmethod,then,isdescribedbyacoupledsystemofdifferentialequationswiththefluidmodesstochasticallyforced.Thedetailsofthisdynamicalsystemwillbepresentedinsection2.Inthemicroscopicsystemsforwhichthemethodisdesigned,viscosityplaysastrongrole.Moreprecisely,onecandefineathermalReynoldsnumber(productofparticlesizeandthermalvelocitydividedbykinematicviscosity)whichwilloftenbesmallintypicalsystemsofinterest.Insuchinstances,theMTVstochasticmodereductionframeworkcanbeappliedbasedonthissmallparametertodeduceasimplifiedsystemgoverningtheimmersedparticlesandstructures,withthefluidvariableseliminated(seesection3).Tounifythediscussionwiththeothersimulationmethodsin[26],itisusefultonotethatthethermalReynoldsnumberintheIBmethodcanbeidentifiedwitha“thermalKubonumber,”definedastheratiooftherateofdecorrelationofaparticle’s(Lagrangian)th