Geometric space-time integration of ferromagnetic

GeometricSpace-TimeIntegrationofFerromagneticMaterialsJasonFrank1CWI,P.O.Box94079,1090GBAmsterdam,theNetherlandse-mail:[e-mail]-Lifshitzequation(LLE)governingtheflowofmagneticspininafer-romagneticmaterialisaPDEwithanoncanonicalHamiltonianstructure.InthispaperwederiveanumberofnewformulationsoftheLLEasapartialdifferentialequationonamultisymplecticstructure.Usingthisformweshowthatthestan-dardcentralspatialdiscretizationoftheLLEgivesasemi-discretemultisymplecticPDE,andsuggestanefficientsymplecticsplittingmethodfortimeintegration.Fur-thermoreweintroduceanewspace-timeboxschemediscretizationwhichsatisfiesadiscretelocalconservationlawforenergyflow,implicitintheLLE,andmadetransparentbythemultisymplecticframework.Keywords:ferromagneticmaterials,Landau-Lifshitzequation,multisymplecticstructure,geometricintegration1HamiltonianstructureoftheLandau-LifshitzequationThispaperaddressestheLandau-Lifshitzequation(LLE)asanonlinearwaveequationsupportingsolitonsandstablemagneticvortices,asconsiderede.g.in[5,20,24].TheLLEgovernstheflowofmagneticspininaferromagneticmaterial.Atapointx∈Rdthespinm(x,t)=(m1,m2,m3)TinCartesiancoordinatessatisfiesmt=m×[Δm+Dm+Ω],(1)whereΔistheLaplacianoperatorinRd,D=diag(d1,d2,d3)modelsanisotropyinthematerial,andΩisanexternalmagneticfield.1FundingfromanNWOInnovativeResearchGrantisgratefullyacknowledged.PreprintsubmittedtoAppliedNumericalMathematics20October2003Inapplicationsinmicromagnetics,theLLEmayadditionallyincludeanon-localterm,aspinmagnitude-preservingGilbertdampingterm,aswellasacouplingtermstoadynamicexternalfieldgovernedbyMaxwell’sequations,see[6].TheLLEcanbewrittenintheformofaHamiltonianPDEwithanonlinearLie-Poissonstructure(seee.g.[23,8]).ThegeneralformofaHamiltonianPDEisyt=B(y)δHδy,(2)wherey(x,t)∈Rp,Hisafunctional,δHδyisthevectorofvariationalderivativesofHwithrespecttoy,andB(y)isaPoissonstructurematrix,i.e.askew-symmetricmatrixoperatorsatisfyingtheJacobiidentity(see[23]).IfB(y)isaPoissonstructurematrix,continuouswithrespecttoy,thereisalocalchangeofvariables¯y=¯y(y)suchthatthestructureassumesacanonicalformδ¯yδyB(y)δ¯yδyT=J=00000Ip10−Ip10,(3)wherep=2p1+p2andIp1isthep1-dimensionalidentitymatrix.Expressedinthenewvariables,theHamiltoniansystem(2)becomes¯yt=JδH(¯y)δ¯y.ItisobviousfromthestructureofJthatthedependentvariables¯y1,...,¯yp2areconstantsofmotionforanyHamiltonianH.For(1)theHamiltonianfunctionalisthetotalenergyH=12Z|∇xm|2+m·Dm+2Ω·mdx.(4)andthePoissonstructureisB(m)=cm=0−m3m2m30−m1−m2m10,(5)whichisrelatedtothePoissonstructureofthefreerigidbody[17].Ifthespinisalternativelyrepresentedinthecoordinates¯m=(mℓ,mθ,mz)T,mℓ=qm21+m22+m23,mθ=tan−1m2m1,mz=m3,(6)2wheretan−1denotestheangle(m1,m2)makeswiththem1axis,thenthePoissonstructuretakesthecanonicalform(3)withp1≡p2≡1,whichshowsthatthespinlengthmℓ=|m|isaconservedquantity.Indeed,wehave∂∂t|m|2=2m·mt=2m·(m×δHδm)=0,(7)foranyH;thatis,|m|2isaCasimirof(5).Thepolarcoordinates(6)arewelldefinedexceptform1=m2=0,forwhichthespinisalignedwiththem3axis.Thedegeneratecasecanbetreatedbydefiningalocalchartwith,forexample,mℓ,my=m2andmφ=tan−1(m1/m3).Inthispaperwewillalwaysassumethatlocallyeitherm1orm2isnonzero.Althoughthisassumptioniscrucialfortheanalysis,thenu-mericalmethodsdevelopedherearegloballydefined,makingnouseoflocalcharts.AssumingDandΩareindependentoftandx,(1)istime-andspace-translationinvariant,implyingtheconservationofthetotalenergy(4)andtotalmomentum(givenhereformℓ≡1):P=Z11+m3(m1∇xm2−m2∇xm1)dx.(8)Bothglobalinvariantsareconsequencesofrelatedlocalconservationlaws.Forexample,inthesimplifiedcase:{D=I,Ω=0,d=1},theenergyandmomentumconservationlawsbecome,et+fx=0,e=12m·mxx,f=12(mx·mt−m·mxt),(9)at+bx=0,a=12(m3mθx−mθm3x),b=12(mθm3t−m3mθt−|mx|2).(10)Theseconservationlawscanbeintegratedoverthedomainofinterestandunderappropriate(forexample,periodic)boundaryconditions,implythein-varianceofthetotalintegral.ForΩ=0,(1)isalsotime-reversible.InnumericalsimulationsoftheLandau-Lifshitzandrelatedequations,itiscrucialtopreservetherelation(7).Anumberofstrategiesfordoingsoareen-counteredintheliterature.Ageneralnumericalintegratorcannotbeexpectedtodothisautomatically,makingitnecessarytoeitherimposetheconditionasaconstraint,ortorepeatedlyprojectthesolutionontotheconstraintmanifold[4].However,anumberofresultsundertheheadingof“geometricintegration”techniques(see[9])canbeusedtoconstructintegratorsthatautomaticallypreservethespinmagnitude.First,itiswellknownthattheclassofGauss-LegendreRunge-Kuttamethodspreservesanyquadraticinvariantsuchasthespinmagnitude(andthetotalenergy!).Theimplicitmidpointmethodisquite3commoninthiscontext;seetheworkofMonkandVacuswhouseafiniteelementdiscretizationofmicromagnetics[21,22].Second,giventhatm(x,t)evolvesonthesurfaceofasphere,onecanderiveanequivalentformulationof(1)intheLie-GroupSO(3)andapplyLieGroupintegrators,asin[11,14].Third,sincethespinmagnitudeisaCasimirofthePoissonmatrix(5),anyPoissonintegratorwillconserveitbydefinition.In[7]time-reversible,energyconserving,andPoissonintegratorswerecomparedagainststandardmethodsforthelattic