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Precise

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PrecisePrecis

    Precise

    App1.Math.Mech.Eng1.Ed.31(11),14631472(2010)

    DOI10.1007/s10483010-1376-x

    (~)ShanghaiUniversityandSpringer-Verlag

    BerlinHeidelberg2010

    AppliedMathematics

    andMechanics

    (EnglishEdition)

    Preciseintegrationmethodforsolvingsingularperturbation

    problems

    Ming-huiFU(富明慧),Man-chiCHEUNG(张文志),S.V.SHESHENIN

    (1.DepartmentofAppliedMechanicsandEngineering,SunYatsenUniversity,

    Guangzhou510275,P.R.China;

    2.FacultyofMechanicsandMathematics,LomonosovMoscowStateUniversity, Moscow119992,Russia)

    (CommunicatedbyXing-mingGUO)

    AbstractThispaperpresentsaprecisemethodforsolvingsingularlyperturbed boundaryvalueproblemswiththeboundarylayeratoneend.Themethoddividesthe intervalevenlyandvesasetofalgebraicequationsinamatrixformbytheprecise integrationrelationshipofeachsegment.Substitutingtheboundaryconditionsintothe algebraicequations.thecoemcientmatrixcanbetransformedtotheblocktridiagonal matrix.Consideringthenatureoftheproblem.anecientreductionmethodiSgiven

    forsolvingsingularperturbationproblems.Sincethepreciseintegrationrelationshipin- troducesnodiscreteerrorinthediscreteprocess,thepresentmethodhashighprecision. Numericalexamplesshowthevalidityofthepresentmethod.

    Keywordssingularperturbationproblem,firstorderordinarydifferentialequation,

    two-pointboundaryvalueproblem,preciseintegrationmethod,reductionmethod

ChineseLibraryClassification0175.8.O241.8l

    2000MathematicsSubjectClassification65L10,76M45

    1Introduction

    Singularperturbationproblemsariseveryfrequentlyinfluidmechanics,fluiddynamics,aero dynamics,plasmadynamics,magneto

    hydrodynamics,oceanography,optimalcontrol,chemical

    reactions,etc.VariousmethodsweredevelopedduringthelastfeWyears.Thenotablemeth_ odsareasymptoticexpansionapproximations[1-2].

    finitedifierencemethods[3一引.finiteelement

    methods[6],

    boundary-valuetechniques[7],

    initialvaluetechniquesIs-9],

    splinetechniques[1011],

    andSOon.

    Theprecisetimeintegration(PTI)methodwasfirstproposedbyZhongandWilliamsfor thelinearinit[alvalueproblemofstructuraldynamics[12J.ThePTImethodhasattracted muchinterest.anditsapplicationhasbeenbroadenedtoinitialvalueproblemslikeheatcon

    ductproblems.randomresponseproblems[13l.andsometwopointboundary

    valueproblems

    }ReceivedJu1.8,2010/RevisedSept.15,2010

    ProjectsupportedbytheNationalNaturalScienceFoundationofChina(No.10672194)andthe

    China-RussiaCooperativeProject(theNationalNaturalScienceFoundationofChinaandtheRus-

    sianFoundationforBasicResearch)(No.10811120012)

    CorrespondingauthorMing-huiFU,Professor,Ph.D.,E-mail:stsfmh@mail.sysu.edu.cn 1464Ming-huiFU,ManchiCHEUNG,andS.V.SHESHENIN

    (TPBVPs)[6一引becauseofitsprominentnumericaladvantages,suchashighprecisionand highefficiency.

    Inthispaper,ahighefficientmethodbasedonthePTImethodispresentedforsolving

    singularlyperturbedtwo-pointboundary-valueproblemswiththeboundarylayeratoneend. Sincenodiscreteerrorisintroducedbyutilizingthepreciseintegrationrelationshipofstatus parametersbetweenadjacentnodestoestablishalgebraicequationsinthe"discrete"process (i.e.,thealgebraicequivalentprocess)ofordinarydifferentialequations(ODEs),theprecision

    ofthepresentmethodisalmostnotrelatedtothediscretesteplength.Hence.thereisnoneed tousedensegridsevenintheboundarylayerregion.Comparedwithothernumericalmethods, thepresentmethodhashigherprecisionandalargerscope.

    2Precisemethodforsolvingsingularlyperturbedboundary-valueprob-

    lemsofODEs

    Consideralinearsingularlyperturbedtwo-pointboundaryvalueproblemintheformof

    ?()+A()+By(x)=,(),

    y(a)=oL,y(b)=,

    where?isasmallpositiveparameterf0<

    Wetransform(1)toasystemoffirstorder

    sothat

    where

    (1)

    (2)

    ?《1),andoL,,A,andBareknownconstants.

    ODEsbyintroducinganauxiliaryvariableP=Y

    =

    Hv+.x?[a,hi

    =

    (),=(一一?),r=.

    0)

    Obviously,thesolutiontothenonhomogeneousequation(3)canbegivenas ()=exp()t,.+J0xexp[(s)](s)dsl

    (3)

    (4)

First,wedividetheintervala,b]evenlyintosegmentsusingaconstantstepT=(ba)/m

    Then.thesolutiontothenonhomogeneousequationf31canbeexpressedas[718

    fxi+1

    =

    Tvi+exp(1s))

    =Tt+

    r

    exp[H(7-s)r(xi+s)ds,(5)

    wherexi=iT,Vi=v(xd(i=0,1,?,m),andT=exp(HT)isthematrixexponential,which

    canbegivenaccuratelybythepreciseintegrationmethodt12J.

    Theintegrationitemin(51correspondstotheparticularsolutiontothenonhomogeneous equation(3),whichcanbecomputedbyusingthePTImethodappliedtotheparticularsolution tothenonhomogeneoussystemsIls

    19]andnumericalintegrationmethods[20-221.Ontheother

    hand,thenonhomogeneoussystemofODEscanbetransformedtothehomogeneousoneby meansofthemethodofdimensiona1expansiontl?.

    Equation(5)canbethoughtasafinitedifferenceequationof(3).Equation(5)gives nodiscreteerror,whichisdirentfromconventionalfinitedifferencemethods.Moreover. thetransfermatrixTandtheparticularsolutiontothenonhomogeneousequationareboth calculatedaccuratelywiththepreciseintegrationmethod.Hence,theprecisionofthepresent Preciseintegrationmethodforsolvingsingularperturbationproblems1465 methodhasalmostnorelationtothesteplength.

    Thisisthereasonwhythepresentmethod

    isofhighprecision.

    Werewrite(5)inamatrixvectorformasfollows:

    whereVi:exp[H(7s)(+s)ds=1,2,,m).Wedenote

    :

    fj

    Substitutingtheboundaryconditions(2)into(6),weobtain

    where

,,1um

    llPi

    BL-~B,

    (-T222),c=(

    .

    (i=l,2,,m1),

    dtct:2,3,,m,d=t,(),

    (6)

    (7)

    (8)

    inwhichBListhefirstmatrix,andBRisthelastmatrixonthemaindiagona1. ThevalueoftheelementsinHmaybeverylargeifmistoosmallforavervsmaUE. Consequently,thevalueoftheelementsinTwillbeverylargeevenapproachinginfinity.That is,B=(?-1/27/222)cann.tbegivenaccurately.Toavoidthispr.blemIitrequiresan mthatislargeenough.

    Equation(8)isasystemofblocktridiagonalequations.Itisknownthat,ifthemattiX isdiagonallydominant,thentheLUeliminationisstable.However,theconditioncannotbe satisfiedfrequently,andBi,andBmaynotexist.Thatis,theLUeliminationmaybe unavailable.

    Inthefollowingpart,wegiveanefficientmethodtosolve(8).Here,weemployareduction procedureform=2M+1(M=0,1,2,).Theprocedureisexplainedbelow.Weaddthe

    ,l???????J

    /,,.,...............

    一一/

    =

    ,,l??I?Il?/

    :

    /,.,...............

    一一/

,l?????/

    B

    .

    .

    BA

    .

    .

    A

    CB...

    A

    

    \,_?\

    lIr

    k

    tI

    ,

    ._:,d

    一二

    22一一一一

    一一00/,/\

    lI=

    BA

    1466Ming-huiFU,ManchiCHEUNG,andS.V.SHESHENIN

    secondequationmultipliedbyCB-.tothefirstequation.Then,thefm1)thequation

    multipliedbyAB?isaddedtothemthequation.Finally,each2ithequationmultipliedby

    

    CBandthef2i+2)thequationmultipliedby-AB_1areaddedtothe(2i+1)thequation

    (i=1,2,?,(m3)/2),whichgives

    where

    /,B;c(1)f

A)B(1)c(1)

    l-.?.?.1'A1)B1)c(1) \A(1)

    ui'

    t5

    ,

    (1)

    rm1)/2

    .(1)

    "(m+1)/2

    di)

    d5)

    d(1)

    rm1)/2

    (1)

    t~(m+1)/2

    B':B

    CBA,B?=BRABC,B()=BABcCBA,

    (1):CB,

    A(1)一一AB-A,)----U2i--1(1,2,,re+i), di):dCB_ld.,drAB_ld2CB(_2j3,,), dmABdm.1-

    (9)

    Thenumberofequationsandunknownsin(9)is2M_.,whichislessthanthatin(8),and

    thesystemhasthesamestructure.Therefore,thereductionprocessisrepeatedresultingin

    the~llowingsystemafterktimesofiterations:

    ,,Bc()f

    )B(k)(?) A

    1..?.?.I'ABc?

    \A()B'

    '

    .

    ()

    2Mk

    ,

    ()

    2Mk+1 d'

    d

    ()

    2Mk

    ()

    "2M+1 (10)

    where B'=B一?一 c((B()A(一?,B=B?一?一A(k-1)(B(k-1))C'一?,

    A()=A('(B(?)A(一?,c()=((B())C一?,

    B():B()A()(B()()c()(B())A(一?,

    ):u

    2

    (k

    

    -

    l

    1)(=1,2,,2M+1),d=d一?一c((B()d一?,

    d)=d(k

    -

1

    

    A((B()d2(k

    -

    2

    1

    

    c((B()d'(:2,3,,2M-k),

    d

    +l=d2(k-

    

    1)

    ++1A((B()4~--2+(=2,3,,M).

    AfterMtimesofiterations,thesystemreads

    ((=((11)

    Preciseintegrationmethodforsolvingsingularperturbationproblems1467 Thesolutionto(11)canbeeasilydeterminedas

    u

    (c'()-,(12)

    Ii=(B)(dc(M)u1.

    Noticethatui'is1,i.e.,P0isgivensimultaneously.Then,using(5)orsubstituting1and U2backinthereversepathofthereductionprocedureyieldstherequiredsolutionsto(1)inthe desiredpoint.Sometimes,afterP0isgiven,theintervalisredivided,andthetransfermatrix isrecomputedinordertoderivetherequiredfunctionvaluesinthepointwhichmaynotbein thenodes.

    Sincethecomputationsofthematrixbymatrixmultiplicationandthematrixinversehave muchmoretimeconsumingoperationsthanthematrix-vectormultiplication,weconsideronly

    firsttwotypesofoperations.TheUfactorizationforasystemofequationswithatridiagonal matrixrequires0(21floatingpointoperations.Thematrixinversecomputationrequires

    0(22)operations,andthematrixmultiplicationcomputationis2+1?1times.However, theinversematrixcomputationinf9)(12)isonly+2times,andthematrixmultiplication

    computationis6M+2times.Thesemeanthatthepresentreductionalgorithmcangreatly

    improvethecomputationalemciencywhenMislarge. Particularly,when(1)iSahomogeneousODE,thereiSnoneedtocomputetherighthand

    sideof(10)duringthepreviousMtimesofelimination,whichalsoreducesthecomputation.

    3High-ordersingularperturbationproblems Intheprevioussection,onlythesecondorderODEsarediscussed.Infact,themethod canbeeasilyextendedtohigher-orderODEs.Inthefollowingpart,thefourthorderODEs

    aregivenasanexampletoillustratetheprocess.Considerafourth-ordersingularlyperturbed

    two-pointboundaryvalueproblemasfollows:

    ?()()+4"()+By(x)=.,(),

    y(a)=Q,y(b)=/3,()=,"(6)=0,

    whereO/,,7,and0areknownconstants.

    LetPY,q=P,andZ=q,wehave

    0

    1

    O

    E

    Letv=(ypqz)T,=o

    g

    1/

    T

    ,

    andH=

    ?+

    0

    0

    0

    B

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