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RELAXATION SOLVERS FOR IDEAL MHD EQUATIONS -A REVIEW

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RELAXATION SOLVERS FOR IDEAL MHD EQUATIONS -A REVIEW

    RELAXATION SOLVERS FOR IDEAL

    MHD EQUATIONS -A REVIEW

AvailableonlineatWWW.sciencedirect.com

    ?

    ,

    ScienceDirect

    ActaMathematicaScientia2010,30B(2):621-632

    数学物理

    http://actams.wipm.ac.ca

    RELAXATIoNSoI?ERSFoRIDEALMHD

    EQUATIONSAREVIEW

    DedicatedtoProfessorJamesGlimmontheoccasionofhis75thbirthday ChristianKlingenberg

    DepartementoyMathematics,WiirzburgUniversity,AmHubland970Wiirzburg,Germany E-mail:klingenberg@mathematik.uni-wuerzburg.de

    Knutapart

    TheUniversityo|Maryland,CSCAMM,4146CSICBuilding#4o6PaintBranchDrive CollegeParkMD202-3289,USA

    E-mail:]~waagan@cscamm.umd.edu

    AbstractWehavedevelopedapproximateRiemannsolversforidealMHDequations basedonarelaxationapproachin4,[5.Theseleadtoentropyconsistentsolutionswith goodpropertieslikeguaranteedpositivedensity.Wedescribetheextensiontohigherorder andmultiplespacedimensions.Finallyweshowourimplementationofallthisintothe astrophysicscodeFLASH.

    Keywordsconservationlaws;idealmaguetohydrodynamics;finitevolumeschemes;en. tropystableschemes;positiveschemes

2000MRSubjectClassification35L65;65M08

    1Introduction

    InthispaperwestudyfinitevolumemethodsfortheequationsofidealMHD.Wederive approximateRiemannsolversbasedonarelaxationapproach.FortheEulerequationsof compressiblefluidsthisrelaxationapproachiswelldescribedinthebookofFranc0isBouchut 3andreferencestherein.ThegeneralisationofthistoidealMHDisdescribedin41,

    51which

    wereviewbelow.一~

    TheMHDsystem(1etting/3denotethe3×3identitymatrix)

    Pt+V?(pu)=0,

    (pu)t+?(pu.u+(p+1lBI)

    BB):0

    Et+V?(E+p+~1BI.)u(B.u)B=0,

    Bt+?(B0UUB)=0,

    V?B=0.

    ReceivedMarch12,2010

    622ACTAMATHEMATICASCIENTIAVb1.

    30Ser.B

    withaninternalenergyegivenbyE=pe+~PU+B.,andthepressuregivenbytheequation ofstateP=p(p,e).Thesystemfitsthegenericformofaconservationlawut+V.F(U)=0, exceptfortherestrictiononV?B.However,ifthisrestrictionissatisfiedattheinitialtime t=0,itautomaticallyholdsat1atertimest>0fortheexactsolution. Thermodynamicalconsiderationsleadstotheassumptionofexistenceofaspecificphysical entropy8=s(p,e)thatsatisfies

    )

    )

    wherethesubscriptsmeansthatthepartialderivativeistakenwithSconstant.Weshallalso maketheclassicalassumptionthat

    siSaconvexfunctionof

    Thesystemisthenequippedwiththeentropyinequality

(1,e)

    (p(s))t+(p(s))0

    inaccordancewiththesecondlawofthermodynamics. Consideraone-dimensionalhyperbolicconservationlaw +F()=0

    Aconservativeschemeforthissystemcanbegivenby Un.Ax(,j

    (1.4)

    (1.5)

    (1.7)

    Forafirstorderscheme,weset-n

    ,j(%,u?.n,j),where(',')istypicallygivenbyan exactoranapproximateRiemannsolver. 2TheRelaxationApproachinOneDimension TheequationsforidealMHDinonedimensionare +(pu)=0,

    (p)t+(pu+P+1B上『一言B)=0,

    (pu~h+(puu~B)=0,

    +[(E+p+IB上『一1D2)B(B?)=0,

    (Bz)t+(BzuB)=0.

    (2.

    (2.

    (2.

    (2.

    (2.

    ThevelocityissplitintoitslongitudinalandtransversecomponentsUand

    ,andthemagnetic

    dimensionalvectors.Sincethe fieldsimilarlyintoBnandB.HenceandBj_aretwo

    NO.2C.Klingenberg&K.waagan:RELAXATIONSOLVERSFORIDEALMHDEQ

    UATIONS623

divergence0fthemagneticfieldiszeroatalltimes,wetakeBnconstantforonedimensional

    data,butwewilleventuallyneedtorelaxthatrestriction? 2.1RelaxationsystemandapproximateRiemannsolver In[4]weintroducedtherelaxationsystem

    +(pu):0,

    (pu)+(pu.+)=0,

    )t+(puu~+7r):0, (p

    +[(E+7r)+"IRA_?Jz=0,

    (B1)t+(BluBnJ_)z=0

    (2.6)

    withE:pe+pu.+B.,

    andwheretherelaxationpressures7rand不上evolveacordingt0

    (P7r)t+(p~ru)+(I6I+c;)ucab?()z=0

    (p7rJ_)t+(pT:)cbu+c()=0.

    (2.7)

    Theparameters0,c60,andb?Rplaytheroleofapproximationsofv~lBnl,p,, andsign(Bn)Brespectively.Indeed,c.,Cb,barenottakenconstant,butareeVolVedwith

    (c.)t+u(c.):0,(cb)t+fl(cb)=0,bt+u=0(2.8)

    Wecanreferto

    =p+IBI.一去口:and7r=BB.(2.9)

    asanequilibriumstate.TheapproximateRiemannsolverresultsfromprojectingthesolution

    totheequilibriumstateatdiscretetimes,whilealsocomputingcellaverages?

    Theeigenvaluesofthesystem(2.6),(2.7)and(2.8)are,uT,uTand,where

    c=

    c}=

    V/(4+c~+Ibl2)2-4c14).

    (2.10)

    havingmultiplicity8.Allarelinearlydegenerate,hencetheRiemannproblemiseasyto

    solve.NotethatCC.c,,cs?Cbcf,andthattheeigenvaluesof(2.6),(2.7)and(2.8)

equaltheeigenvaluesof(2.1)(2.5)wheneverc=jl,Cb=p,/andb=sign(Bn)B.

    However,inordertosimplify,weshallmakeheredifferentchoices,leadingtoasolverwith3 wa?

    esor5wgveginsteadof7WgVe9.Thefullmotivationandanalysisoftherelaxationsystem isgiveninf41.WegetconditionstoensurethatforaRiemannproblemstartingatequilibrium f2.91,thesolutiontotherelaxationsystemsatisfies:(i)themassdensitystayspositive,and(ii) Theentropyinequality(1.5)holds.Asaconsequence,wegetaschemeofform(1.7)suchthat massdensityandinternalenergystayspositive,andthefollowingdiscreteentropyinequality holds

    (+)一叩()+At.G.(,1)G.(1,)]0

    with()=p(s(p,e)),andGanappropriatenumericalentropyflux

    +

    22

    66

    +

    ++

    1212

    624ACTAMATHEMATICASCIENTIAV_01.30Ser.B

    ThestabilityconditionsfortheapproximateRiemannsolveraregivenasinequalitiesin- volvingCb,Candb,andthesolutionoftherelaxationsystem.SincetheRiemannproblem solutionisexplicit,wecanobtainexplicitvaluesofCb,Candbthatensuregoodproperties.

    Thesimplestchoiceistotakeb=0andc=c0:Cb.TheresultingRiemannsolverhas threedistinctwavespeeds.Fastwaves,materialcontactwavesandtangentialdiscontinuities areresolvedsharplybythissolver.Wealsoobservegoodresolutionofotherwavesinpractice. Inordertoallowoptimalresolutionofvelocityshearwaveswhenvanishes,wealsoprovide a5wavesolverwherestillb=0,andcnisindependentofCbinsuchawaythatitvanishes withB.

    2.2Secondorderaccuracy

    LetU=~DU,whereDUisasecondorderaccurateapproximationto.Thenthe

    MUSCLscheme

+1

    =At((,

    )(,))(2.12)

    issecondorderaccurateinspace.SecondorderaccuracyintimecanbeachievedbyHeun's

    method.Onecanalsouseapredictorcorrectortypeschemetorestoretimeaccuracy,for exampleintheMUSCLHancockscheme,whereonetakes:UD+丢?F(U)DU.

    AstableMUSCLHancockschemeispresentedin181.F0rbrevity,westicktotheMUSCL schemewithU=U士去Dhere,thestabilityresultsofwhichareaspecialcaseinf18I. Thefirststepinmaking(2.12)stableistofollow[17andlimitthegradientD.This approachisinspiredbythetotalvariationdiminishingpropertyofscalarconservationlaws,

    suggestingthatthereconstructedstate,representedbyU,shouldnothavealargertotalvari- ationthattheoriginalstateU.OnemayalsousetheENO([12)orWENO([16)approaches,

    basedonsimilarprinciples,tofindU.Itisnotnecessarytobasethereconstructionona piecewiselinearapproximationtotheconserved

    linearapproximationstotheprimitivevariables

    weemploythemonotonizedcentrallimiter

    Dminf2I+

    quantities.Instead,wewillconsiderpiecewise

    W=(p,u,B,p).Forournumericalexamples,

    

    I,1

    .

    W.?一一Ij21一一I)

    

    >0,一一1>0

    <0,一一1<0

    (2.13)

    (2.14)

    Intheaboveformulas,minimizationandabsolutevaluearetobeunderstoodascomponent

    wise

operationsoneachscalarquantityof.

    Next,wewouldlikethepositivityandentropystabilityofthefirstorderschemetohold alsoforf2.12).itturnsouttonotbeverypracticaltouseaprovablyentropystablescheme forhigherorder.Instead,werelyonthegradientlimitingtoforcetheschemetowardsitsfirst orderversionnearshocks,henceensuringsufficientdissipation.Positivity,ontheotherhand, canonlybeensuredifsomeadditionallimitingisperformed.Werelyonthefollowingresult (See18],Prop.3.2):

    Proposition2.1Thescheme(2.12)ispositiveif

    DpI<2p,DpI<2p(2.15)

    No.2C.Klingenberg&K.Waagan:RELAXATIONSOLVERSFORIDEALMHDEQ

    UATIONS625

    and

    (p+)+<4pe(2.16)

    Notethattherelations(2.15)followfrom(2.13),andmoregenerallyholdforanyrecon- structionsuchthat(foreachcomponentofW)

    Thelastinequality

    replacingitwith

    WminWi.

    Z

    (2.16)canbeimposedbytakingagradientDWfrome.g.(2.13),and

    Numericaltestsin[18validatestheuseofProp2.1

    3Multidimensions

    D

    TheidealMHDequationsf1.1)consistsofasystemofconservation1aws,andthediver

    genceconstraint.B=0.TakingthedivergenceoftheevolutionequationforB,yieldsthat (?B)t=0.Hence,thedivergenceconstraintonlyneedstobeimposedontheinitialdata. Itisanontrivialissuehowtoaccountforthedivergenceconstraintinanumericalsimulation, particularlyinpresenceofthe1OWregularityexhibitedbysolutionstoidealMHD. Infinitevolumeschemesforequationslike(1.1),exactorapproximateRiemannsolversfor

one

    dimensionalsystemsarecrucialcomponentsalsoinmultipledimensions.Asanexample, considerintwospacedimensionsaCartesiangridwithcellcenters(Xi,),andthesystem

    +F()+G()=0

    Afinitevolumeschemeforthissystemcanbegivenby

    =%Ax(,j一一At.n'J+一一)

    (3.1)

    (3.2)

    Forafirstorderscheme,thenumericalflux

    ,

    issetas(,Un1,),asintheone

    dimensionalscheme(1.7).Theresultingschemecanbeprovedtopreservethestabilityprop- ertiesofthebasiconedimensionalschemeattheexpenseofloweringtheCFLnumberbya factorof2.Typically,suchareductionintheCFLnumberisnotfoundnecessaryinpractice. IndevelopingtheapproximateRiemannsolverforonedimension,weassumedthatBnWas constantduetothedivergenceconstraint.Ontheotherhand,whenevaluatingthenumerical fluxesin(3.2),thisisnolongertrue.Asimplewaytogetaroundtheproblemistonsealocal averageofBntoevaluatetheflux.Thistechniquecannotbeexpectedtopreservethestability propertiesoftheRiemannsolver.Instead,wewillbasetheschemeonthePowellsystem[15,

    whichmeansthatthemagneticinductionequationiSrewrittenas

    Bt+?(BUUB)=uV?B.(3.3)

    NotethatanonconservativetermproportionaltoV?Bhasbeenadded.In[15],similarsource termswereaddedtothemomentumandenergyequations,resultinginasymmetrizablesystem.

    626ACTAMATHEMATICASCIENTIAV01.30Ser.B

    Forourpurposeitissufficienttomodifytheinductionequation.Takingthedivergenceof(3.3) yields

    (V?B)t+?(uV?B)=0(3.4)

    Thismeansthatif?B=0initially,itremainsSO,andconsequentlythestandardform(1.1) andthePowellsystemareequivalent.Wealsonotethattheentropyinequality(1.5)holdsalso

    forPowell'Ssystem.However,inthepresenceofsmallerrorsinthedivergenceconstraint,the Powel1systemisstable,whilethestandardformisnot.Furthermore,weareabletoextend ourstabilityresultsinonedimensiontomultidimensionsviathePowelltermsapproach.

    3.1Firstorder

    Thenextstepistodescribehowtherelaxationsystem,andcorrespondingRiemannsolver, canbeextendedtothePowell'ssystem.TherelaxationsystemforPowell'Ssystemonlydiffers fromthestandardcase(2.6)~(2.7)intheform(3.3)oftheinductionequation.Thesolution oftheRiemannproblemfortherelaxationsystemwith(3.3)isthesameasbeforewiththe additionthatBnlumpsfromBtoBacrossthemiddlewave.Theresultingapproximate Riemannsolversatisfiesthesameentropystabilityandpositivityconditionsasarevalidfor constantBn.

    ThenumericalfluxesforP,puandEarecalculatedasbefore(withBnevaluatedlocally), whilethenumericalfluxesforBbecomenon-conservativeandarefoundinthefollowingway: Theapproximatesolutionsatisfies

    Bt+(BuBu)+u(Bn)z=0

    Denotebythevalueofthroughthematerialcontact.Then,

    (B)=()(tu),u(B).:U(BB)),

    (3.5)

    (3.6)

    whereUisthevalueofuthroughthematerialcontact.Wealsohavethejumpcondition Bu]:0+u(BB)I0

    where[...Ix=0denotesthejumpthroughtheline=0.

    Now,integrate(3.5)over(0,At)×(一?,0).Weget

    )dx-Bg+At((Bu-

    B~u

    +u

    Next,integrate(3.5)over(0,At)×(0,?).Weget

    Denote

    )dx-Br+At

+uB:.

    =

    =

    (BuBu)f)

    ((BuBu)(Buju)0+)

    (BuBu)0+u(BrB)I<0, (BuBnu)0+u(B二一)I>0. (3.7)

    (3.8)

    (3.9)

    (3.10)

    N0.2C.Klingenberg&K.Waagan:RELAXATIONSOLVERSFORIDEALMHDEQ

    UATIONS627

    Weendupwith

    B+B+

    Accordingto(3.7),aformulaforthe

    Ifu*>0then

    {

    (()i+1/2-()?/)=0.

    numericalfluxesis (BuBu)0,

    (BuBu)0一一u(~),

    r.then

    ;兰二:;::.+u(B~B)

    (3.11)

    (3.12)

    (3.13)

    Thiscompletesthedescriptionofthefirstordermultidimensionalschemeofform(3.2).Note

    thatwhenBnisallowedtovary,thefirstorderscheme(1.7)generalisestotheform

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