UNIVERSITY OF CAMBRIDGE Numerical Analysis Reports

UNIVERSITYOFCAMBRIDGENumericalAnalysisReportsLie-groupMethodsforIsospectralFlowsAntonellaZannaDAMTP1997/NA02March1997DepartmentofAppliedMathematicsandTheoreticalPhysicsSilverStreetCambridgeEnglandCB39EW1IntroductionWeaddressourselftotheproblemofevaluatingthenumericalsolutionofaclassofordinarydierentialequationsonmatrices,knownunderthenameofisospectralows.Theseowsariseinavarietyofmathematicalapplications,rangingfromlinearalgebratosolidstatephysicsandbiology.Untilfewyearsago,itwasacomputationalobservationthatclassicalmethodforOrdi-naryDierentialEquations(ODEs)producedanumericaloutputthatwasnotisospectral,thusfailingtoretainthemainqualitativefeatureoftheassignedproblem.However,in1995,Calvo,IserlesandZannaestablishedatheoryaccordingtowhichclassicalmethodsforODEscouldnotbeisospectral,andtheypresentedothermethods(modiedGauss{LegendreRunge{Kuttamethods)toconstructisospectralsolutions.InthispaperweproposetosolveisospectralowsusinganapproachbasedonLie-groupnumericalmethods.Althoughthisideamightappearcontrivedatarstglance,itiseasytoconvinceourselvesthatitisthenaturalwaytoanalysemathematicallyandsolvenumericallythisclassofproblems.TheproblemofintegratingODEsonLiegroupsstartedasearlyasinthe1880’s,withtheworkofSophusLie.Themethodofiteratedcommutators,thatwillbeintroducedinsectionx3isineectintimatelyrelatedtothemethodofreductionforlinearODEsonLiegroupproposedbyLie[28].Asfarasweareaware,therstnumerical(opposedtoanalytical)methodstointegrateODEsonLiegroupsappearedonlyin1993inapaperofCrouchandGrossman.Brieylater,Munthe-KaasdevelopedLie-groupmethodsofaRunge{Kuttatype,whileOwrenandMarthinsendevelopedthetheoryofthemethodsproposedbyCrouchandGrossmaninamorecompletemanner.In1996,Zannaproposedthemethodofiteratedcommutators,basedonnumericalschemesforODEsintroducedbyIserlesin1984[12]andsubsequentlyIserlesandNrsettgeneralizedandrenderednumericallyecientanumericalapproach(theMagnusseries)introducedin1954byMagnus.Atpresent,manyissuesrelatedtothisclassofmethodsarebeingintenselyanalysed.SomemathematicianshavebeenskepticalwithregardstothepracticaluseofLie-groupmethods.Inasense,itiscounterintuitivetobelievethatLie-groupmethodscancompetewithclassicalexplicitschemesorwithprojectionmethods.However,numericalexperimentsindicateclearlythatLie-groupmethodscanbecheaperthanclassicalexplicitschemes:Lie-groupequationspossessalotoffeaturesandsymmetriesthat,intelligentlyexploited,leadtosuperiornumericalmethods.WithregardtoLie-groupmethodsforisospectralows,itisfairtomentionthatin1994Helmke,MahonyandMooreintroducedanisospectralscheme(whichtheycalledaLiemethod)forthesolutionofdouble-bracketows[17].However,theirmethodhasorderoneonly,whichmeansthatitisnotverycompetitiveifwewantagoodaccuracy.Theapproachthatwepresentinthispaperismuchmoregeneralandallowsustointroduce,analyseandimplementLie-groupmethodsofarbitrarilyhighorder.2FromisospectralowstoLie-groupequations2.1IsospectralityandconservedintegralsIsospectralowsarecharacterizedbythematrixdierentialequationL0=[B;L];L(0)=L0;(2.1)whereL;B2RddandL0isagivenddinitialmatrix.ThematrixfunctionBB(t;L)dependsonLand,possibly,onthetimet.ThesquarebracketsdenotethecommutatorLiebracketonmatrices,[B;L]=BLLB.EachowischaracterizedbythematrixB,whichisusuallyskew-symmetric,whileLisgenerallysymmetric.Forinstance,ifLandL+denotetheloweranduppertriangularpartofthematrixL,whenB(L)=LL+,thenwehavetheTodaow,associatedwiththeTodalatticeequationsgoverningthemotionoftheparticlesonaone-dimensionallatticeunderexponentialnearest-neighbourinteraction[24].WhenB(L)=f(L)f(L)+,fbeingananalyticfunctionofthespectrumofL,thenwehavetheQRow(see[6]or[27]foralistofreferences).WhenB(L)=[N;L]foraxeddiagonalmatrixN,thenwehavethedouble-bracketow[2],andsoon.Itiswellknownthatthesolutionof(2:1)isisospectral,namelythethespectrumoftheintegralcurveL(t)of(2:1)doesnotchangewithtime[24].ThedeigenvaluesofLarerelatedtothedconservedintegralsthathadbeendiscoveredbyHenonandFlaschkafortheTodalatticeequations.Theintegralsareininvolution,thereforeHenonandFlaschkaconcludedthattheTodalatticeequations,andconsequentlyalsotheotherisospectralows,areintegrablesystems[24].Theconservedintegralscanbeintroducedbyconsideringthed-degreecharacteristicpolynomialassociatedwiththematrixL(t),p()=det(IL)=dpd1d1++(1)dp0;whosezerosaretheeigenvaluesi;i=1;:::;d;ofL.Sincetheeigenvaluesdonotdependontime,thecoecientspd1=1++d;pd2=dXi=1dXj=i+1ij;...p0=12d:(symmetricpolynomials)areconstantandconstitutethedintegralsassociatedwiththeow.Obviously,everyfunctionoftheisisanintegraloftheow.However,inordertocharacterizeunivocallytheeigenvalues,oneneedstoindicatedindependentconditions,thatcanbechosenastr(Lr)=dXi=1ri;r=1;:::;d;(2.2)2analternativetothesymmetricpolynomials[22,24].SuchconditionsareatthebasisoftheCalvo{Iserles{Zannaanalysisofnumericalmethodsandtheirretentionofisospectrality[6].Theyprovedthatclassicalnumericalmethods,suchasmultistepandRunge{Kuttaschemescannotbeisospectralford3,inthesensethat,givenaclassicalnumericalmethod,itisalwayspos