equations Galerkinscheme for solving extended magnetohydrodynamics A positivity-preserving semi-implicit discontinuous Journal of Computational Physics

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Journal of Computational Physics

www.elsevier.com/locate/jcp

A positivity-preserving semi-implicit discontinuous

Galerkin scheme for solving extended magnetohydrodynamics equations

Xuan Zhao

^a

, Yang Yang

^b

, Charles E. Seyler

^a,∗

aSchoolofElectricalandComputerEngineering,CornellUniversity,Ithaca,NY14853,USA bDepartmentofMathematicalScience,MichiganTechnologicalUniversity,Houghton,MI49931,USA

a rt i c l e i n f o a b s t ra c t

Articlehistory:

Received26February2014

Receivedinrevisedform20August2014 Accepted26August2014

Availableonline3September2014

Keywords:

Positivity-preserving DiscontinuousGalerkin Relaxationmethod Magnetohydrodynamics Two-fluid

HEDplasma

A positivity-preserving discontinuous Galerkin (DG) scheme [42] is used to solve the ExtendedMagnetohydrodynamics(XMHD)model,whichis atwo-fluid modelexpressed withacenter-of-massformulation.WeprovethatDGschemewithapositivity-preserving limiterisstableforthesystemgovernedbytheXMHDmodelortheresistiveMHDmodel.

Weusetherelaxationsystemformulation[28]fordescribingtheXMHDmodel,andsolve theequationsusingasplitlevelimplicit–explicittimeadvancescheme,steppingoverthe timestepconstraintimposedbythestiffsourceterms.Themagneticfieldisrepresented inanexactlocallydivergence-freeformofDG[23],whichgreatlyimprovestheaccuracy andstabilityofMHDsimulations.Aspresentlyconstructed,themethodisabletohandle awiderangeofdensityvariations,solveXMHDmodelonMHDtimescales,andprovide greatlyimprovedaccuracyoveraFiniteVolumeimplementationofthesamemodel.

©^{2014TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCC} BY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/3.0/).

1. Introduction

In High Energy Density (HED) plasma systems,we usually must deal witha wide dynamic range ofcurrent carrying densities.Whendensityissolowthatthescalelengthsarecomparabletoioninertiallength,thesinglefluidmodelwould breakdown.Fortheseproblemstwo-fluidphysicsisessential,buttheapplicabilityofexistingnumericalmethods [6,16,17, 19,25,34] in thisplasma regime isstill limited. In thispaperwe develop a discontinuous Galerkin(DG)method tosolve an extended Magnetohydrodynamics (XMHD) model based on a relaxation formulation, and will demonstrate that this numericalschemehashighaccuracy,iscomputationallyefficient,andcanhandlethelargedynamicdensityrangeforHED plasmas.

DGschemesarewidelyusedinthehyperbolicalgorithmscommunity,sincethemethodcanhandlecomplexgeometries, hasanarbitraryorderofaccuracy,andisefficientinparallelcalculations[9–12,31].Inaddition,DGhasfurtheradvantages for HED plasmas, because the density range is very wide, one often faces a problem where the initial density profile is nearly a δ-singularity (e.g. a pinched-down wire). Such problems are difficult to approximate numerically, and most previous techniquesare based on the modification ofthe singularities withsmooth kernels in some narrow region (e.g.

[37,40]andthereferencestherein),andhencesmearsuchsingularities.However,DGmethodsdependontheweakformof theequationsandcansolvesuchproblemswithoutmodification,leadingtoavery accurateresult[38].Themainchallenge

*

Correspondingauthor.

E-mailaddresses:[email protected](X. Zhao),[email protected](Y. Yang),[email protected](C.E. Seyler).

http://dx.doi.org/10.1016/j.jcp.2014.08.044

0021-9991/©^{2014TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense} (http://creativecommons.org/licenses/by-nc-nd/3.0/).

with DGimplementationsliesinpreservingthestabilityofthesystem.Neardiscontinuities,strongoscillationsoftenappear thatmightsendthephysicallypositivequantitiesnegative.Whenthisoccurs,thenumericalsimulationmaybreakdown.In oursystem,thequantitiesthatshouldbepreservedpositiveare:density

ρ

,pressure P,energyE^{.Toensurethesequantities} satisfyapositivity-preservingproperty,onecoulduseaTotalVariationDiminishing(TVD)scheme;howeverallTVDschemes willdegenerate tolower orderaccuracyatsmooth extrema[41].In[42], apositivity-preservinghighorderDGscheme is developed forsolving the compressibleEulerequations,and in[39],the authorsdemonstrated the L¹-stability ofsuch a scheme.This scheme hasbeen applied on solving the idealmagnetohydrodynamic (MHD) equations[3,7].In thispaper, we willdemonstratethat thepositivity-preserving DGschemecan establishapositivity-preserving propertyfora system describedbyanXMHDmodeloraresistiveMHDmodel,thuspreservingthestabilityofthesystem.

ForsolvingtheXMHDmodel,weusetherelaxationalgorithmproposedin[28]tosolvethecombinationofgeneralized Ohm’slaw(GOL)andtheMaxwell–Ampérelawfortheelectricfield. Therelaxationmethodisessentially asemi-implicit timedifferencing proventoconvergetothesolutionofthealgebraicequationsthatarethestiffsourcetermssetequalto zero.InthecaseofXMHDthismeansthesolutionconvergestoGOLinwhichtheelectroninertialtermsareneglectedand totheMaxwellequationinwhichthedisplacementcurrenttermisneglected(Ampére’sLaw).Usingthisalgorithm,onecan avoidtheconstraintimposedbytheunder-resolvedstiffsourcetermstostepoverelectronplasmaandcyclotronfrequencies, whichareoftenunder-resolvedinthecharacteristicregime,andtherebyallowonetosolvetheXMHDequationsonMHD timescales.Inthisalgorithm,theHalltermislocallyimplicitavoidingthesubstantialeffortinsolvingalargelinearsystem.

Furthermore,withXMHDmodel,inlow-densityregion,thecurrentissuppressedby bothHallandelectroninertialterms, one isableto usetheunmodified Spitzerresistivity,which makesthe plasma–vacuumtransitionautomatic andphysical.

ThisalgorithmhasbeenimplementedintoanXMHDcodecalledPERSEUSwithasecond-orderfinitevolume(FV)scheme.

Because ofthe extended stencil,FV isoften toodiffusive to characterize small-scalefluid instabilitiesand toresolve the localdetailsofa shockstructure withoutalargenumberofcells.Ontheother hand,DGschemeismorecompactinthe sensethateverycellistreatedindependently,andthesolutionislocalizedwithinacell.Thisleads toareducednumerical diffusion.Inthispaper,we comparethe performanceof aDGformulationXMHDcode tothat ofan FVformulation, and willdemonstratethroughnumericalteststhatDGhasasignificant advantageoverthesameorderofFVinbothmemory andCPUtime.AdditionallywecompareselectedresultswiththosefoundfromtheMHDmodelcomputedusingthesame algorithm.Thiscomparison demonstratestheviability ofthemethodforsolving XMHDproblemsaswell astopoint out importantdeficienciesinMHD.

Thispaperisorganizedasfollows.InSection2weintroducetheXMHDmodelandtherelaxationalgorithm.InSection3, we construct the DG scheme used in this paper, and demonstrate that a positivity-preserving limiter can preserve the positivity-preservingpropertyinasystemgovernedbyanXMHDmodeloraresistiveMHDmodel,wewillalsogiveabrief introductionon theimplementation ofouralgorithm.In Section 4,we presenttheresultsofnumerical tests,in some of thetests, wedoacomparisonwithFV,toshowthat DGismoreaccurateandmoreefficientthananFV implementation.

Wealsogiveexamplesofthemethodappliedtofundamentalplasmaphysicsproblemsandprovideacomparisonwiththe MHDmodelforthesameproblems.Concludingremarks areprovidedinSection5.

2. XMHDmodel 2.1. Governingequations

InthestudyofHEDplasmas,weareofteninaregimewheretheioninertiallengthoreventheelectroninertiallength areresolvedbythecelldimensions.Whenthisoccurs,standardMHDdoesnotaccuratelydescribethesystem,andamore generalized model is called for. The two-fluid model expressed with a center-of-massformulation, which is the XMHD model,coversmostofthephysicsoccurringinthisregime.Theprimaryexceptionbeingduetospace-chargeeffects,which isneglectedinthequasineutral XMHDmodel.TheXMHDmodelisgivenby:

∂ ρ

∂

t

+ ∇ · ( ρ

u

) =

^{0 (2.1)}

∂

∂

t

( ρ

u

) + ∇ · ( ρ

uu

+

IP

) =

J

×

B (2.2)

∂

En

∂

t

+ ∇ ·

u

(

En

+

^P

)

=

^u

· (

J

×

^B

) + η

J² (2.3)

∂

B

∂

t

+ ∇ ×

^E

=

^{0 (2.4)}

∂

E

∂

t

−

c²

∇ ×

B

= −

¹

0

J (2.5)

∂

J

∂

t

+ ∇ ·

uJ

+

^Ju

−

¹ n_eeJJ

−

^e

m_eIP_e

=

^nee2 m_e

E

+

^u

×

^B

− η

J

−

¹ n_eeJ

×

^B

(2.6)

∂

tS_e

+ ∇ · (

u_eS_e

) = ( γ −

¹

)

n¹_e^−γ

η

J²

.

(2.7)

Where

ρ

, P,u=(ux,uy,uz),En,B=(Bx,By,Bz),E=(Ex,Ey,Ez),J=(Jx,Jy,Jz),and Se denotethedensity,pressure, velocityfield,thesumofthekineticandinternalenergies,magneticfield,electricfield,currentdensity,andelectronentropy density,respectively.The resistivityis

η

.We haveqi= −^Zqe=^{Z e,with Z beingthenetioncharge.Withtheassumption} ofquasineutralityandtoleadingorderinm_e/m_i,we haven_e=^Zni=^{n Z.}

ρ

=^min_i+^men_e,

μ

=^Zme/m_i,u_e=^u−^J/(n_ee), J=^e(Zniui−ⁿeue).WealsohaveP=(

γ

₋1)(En−

ρ

u²/2)andPe=^Senγ₋1

e .

We have chosen to model the electron equation of state using the electron entropy density given by S_e =^Pe/nγ−¹ e because we are unable to prove that the electron pressure will remain positive when the electron energy equation is used.Theissuearisingintheproof ofelectronpressurepositivityisduetothenon-positive natureoftheelectronenergy equationsourcetermswhichdirectlyinvolve theelectricfield;however,theelectronentropydensitysourcetermispositive andwecaneasilyproveelectronpressurepositivity.Theproofofpositivityofthetotalpressureisalsocomplicatedbythe non-positivenatureofthetotalenergysourceterm,butinthatcasetheproofgoesthroughifcertaintime-steprestrictions areapplied.Thedisadvantageoftheuseoftheelectronentropydensityovertheelectronenergydensityisthattheentropy densityonlyallowsforadiabaticchangesinthethermodynamicvariablesandhencewecannot captureweaksolutions in theformofelectronshocks.SuchcircumstancescouldariseforinstanceintheformofLangmuirshocks.Asweareprimarily concerned withlow-frequencyphenomena, well belowtheelectronplasma frequency,we donot considertheuse ofthe electronentropydensitytobeagreatrestriction.

Thereisalsoaconstraintimpliedin(2.4).Takingthedivergenceof(2.4)showsthatif∇ ·^{Bisinitially0,itisalways0.}

Thisgivesusaconstraint∇ ·^B=^0.Forlocalpreservationof∇ ·^B=^{0,weusethelocally}divergence-freeDGmethod[23], whicheliminatestheneedforanexpensivedivergencecleaningschemeortheuseofglobaldivergence-freeelementsora globaldivergencefreereconstruction.

By switchingoffthe electroninertia term, electronpressure andtheHall termin Eq.(2.6),we get theresistiveMHD model,whichischaracterizedby(2.1)–(2.5),and

E

+

u

×

B

= η

J

.

(2.8)

Ifnotspecified,forsimplification,MHDmodelisusedtorefertoresistiveMHDmodelthroughoutthispaper.

FortheclassofproblemsofinteresttousXMHDhasseveralimportantadvantagesoverthetwo-fluidmodel.Theseare:

1.Adirect comparisonto MHDcanbemadeby simplyswitchingoffsome termsinGeneralizedOhm’sLaw (2.6); 2.The relaxationmethod (tobe discussedin Section2.2) canbe usedto stepoverthe plasmafrequencyandelectron cyclotron frequencywhentheydonothavetoberesolved.Thisgreatlyreducesthecomputationalcostof XMHDmodeling.Wehave not beensuccessfulthusfarinimplementing asimilarschemeforthefull two-fluid model.Thereasonforthisis dueto finitechargeseparationwhichisatopicbeyondthescopeofthispaper.Thustherewouldappeartobenoadvantagetothe two-fluidmodelforquasineutralproblemsandmanyadvantagesfortheXMHDmodel.

2.2. Relaxationmodel

Eqs. (2.5) and(2.6) are non-dimensionalized to exhibit the dimensionlessparameters that characterize therelaxation.

Thisgivesthefollowing[28]:

∂

E

∂

t

=

^c2

v²

(∇ ×

B

−

J

)

(2.9)

∂

J

∂

t

+ ∇ ·

uJ

+

Ju

− λ

_i

L0nJJ

−

^miL0 me

λ

i

IPe

=

^L20n

λ

²_e

E

+

u

×

B

− λ

_i

L0nJ

×

B

− η

J

,

(2.10)

where v=^L0/t0 isthecharacteristicspeedandL0,t0 arerepresentativelengthandtimerespectively.Theelectronandion inertiallengthsareλ²_j=^mj/n0e²

μ

0.Therelaxationparametersarec²/v² andL²₀/λ²_e.

InHigh-Energy-Density(HED)plasmasthephenomenaofinterestoftenoccurontimescalesmuchslowerthanthechar- acteristicelectronplasmaandelectroncyclotronfrequencies,whichmeans,v^candλe^L0,forcingthat(2.9)and(2.10) relaxtotheequilibrium:

∇ ×

B

=

J (2.11)

E

+

^u

×

^B

− λ

i

L₀nJ

×

^B

+ λ

i

L₀n

∇

^Pe

= η

J

.

(2.12)

This means that,when the electroninertial scaleis under-resolved, thesolution relaxesto theinertia-less GOLand J willbe constrainedby J= ∇ ×^{B.Thismodelis}implementedintoanalgorithm inwhichwe useanimplicit–explicittime advance,allowingtimestepscomparabletoMHD.ThuswecansolveproblemsinwhichtheHalleffectisimportantwithout significantcomputationalcostoverthatofstandardMHDwiththecaveatthattheCFLconditionforthespeedoflightmust berespected.Ourexperiencehasshownformostproblemswecanuseamuchreducedspeedoflightwithoutaffectingthe resultssignificantly.Typicallythisreductionfactorisintherangeof15–30forHEDproblems.Formoreinformationonthe relaxationmethod,pleasereferto[28].

The formulation ofthe XMHDmodelgiven by (2.1)–(2.7)is not standard [27,30].The differencesare relevant forthe implementationofthenumericalalgorithm.Notethatthemomentumandenergyequationshavesourcetermsandastrue

sourcetermstheydependonlyontheindependentvariableswithnoderivatives.InmostformulationsofXMHDthecurrent isadependentvariabledeterminedthroughAmpére’slawandtheelectricfieldisdeterminedbythegeneralizedOhm’slaw.

Inthepresentformulationinwhichdisplacementcurrentisretained,theelectricfieldandcurrentdensityareindependent variables havingseparate evolution equations.Hence thetermson theRHSof (2.2), (2.3),(2.6) and(2.7) are truesource terms.Thereareissuesthatarepotentialcauseforconcernbynothavingafullyconservativeforminwhichalltermsare indivergenceform.Theseare:1.theJ×^{Bforcetermmayaffectshockcapturingabilityofthenumerical}discretizationand 2.thenon-positiveformoftheenergysourcetermcouldaffectthepositivityofthepressure.Weaddressthefirstconcern byproviding teststhat showtheDGschemedoeswell incapturingshocks anddiscontinuities andthe secondconcernis addressedbyprovingthatthepositivity-preservinglimiterpreservesthestabilityofthesystem.

3. DiscontinuousGalerkinformulation

Inthissection,weconstructtheDGmethodsthatwillbeusedfortheXMHDmodel.

3.1. Localdivergence-freediscontinuousGalerkinformulation

Inthispaper,we onlyconsiderproblemswhich canbe consideredinvariant in z direction. Therefore,wehave∂z=^0, and∇ =(∂x,∂y,0).Giventhis,wecansummarizeEqs.(2.1)–(2.7)withthefollowingform:

U_t

+

^f

(

U

)

_x

+

^g

(

U

)

_y

=

^s

(

U

)

(3.1)

where

U

= ( ρ ,

m_x

,

m_y

,

m_z

,

En

,

B_x

,

B_y

,

B_z

,

E_x

,

E_y

,

E_z

,

J_x

,

J_y

,

J_z

,

S_e

)

^T

,

f

(

U

) =

mx

,

mxux

+

^P

,

myux

,

mzux

, (

En

+

^P

)

ux

,

0

, −

^Ez

,

Ey

,

0

,

c²Bz

, −

^c2By

,

uxJx

+

Jxux

−

¹

ZneJxJx

−

^e me

Pe

,

uxJy

+

Jxuy

−

¹

ZneJxJy

,

uxJz

+

Jxuz

−

¹

ZneJxJz

,

Seuex

T

,

g

(

U

) =

m_y

,

m_xu_y

,

m_yu_y

+

^P

,

m_zu_y

, (

En

+

^P

)

u_y

,

E_z

,

0

,−

^Ex

,−

^c2Bz

,

0

,

c²B_x

,

u_yJ_x

+

^Jyu_x

−

¹

ZneJ_yJ_x

,

u_yJ_y

+

^Jyu_y

−

¹

ZneJ_yJ_y

−

^e

m_eP_e

,

u_yJ_z

+

^Jyu_z

−

¹

ZneJ_yJ_z

,

S_eu_ey

T

,

s

(

U

) =

0

,

J_yB_z

−

^JzB_y

,

J_zB_x

−

^JxB_z

,

J_xB_y

−

^JyB_x

,

ux

(

JyBz

−

^JzBy

) +

^uy

(

JzBx

−

^JxBz

) +

^uz

(

JxBy

−

^JyBx

) + η

J²_x

+

^J2y

+

^J2z

,

0

,

0

,

0

, −

^Jx

0

, −

^Jy

0

, −

^Jz

0

,

^Zne

2

m_e

E_x

+

^uyB_z

−

¹

ZneJ_yB_z

− η

J_x

,

Zne² m_e

E_y

+

^uzB_x

−

¹

ZneJ_zB_x

− η

J_y

,

^Zne

2

m_e

E_z

+

^uxB_y

−

¹

ZneJ_xB_y

− η

J_z

,

( γ −

¹

)

n¹_e^−γ

η

J²_x

+

^J2y

+

^J2z

^T

(3.2)

with

u_x

=

^mx

ρ ^,

^uy

⁼

m_y

ρ ^,

^uz

⁼

m_z

ρ ^,

uxe

=

^ux

−

^Jx

/(

Zen

),

uye

=

^uy

−

^Jy

/(

Zen

),

uze

=

^uz

−

^Jz

/(

Zen

).

Weconsiderthecomputationaldomain[⁰,Lx]× [⁰,Ly]^,anduserectangularmeshdefinedas 0

=

x1

2

< · · · <

x_N

x+¹₂

=

Lx

,

0

=

y1

2

< · · · <

y_N

y+¹₂

=

Ly

,

withI_{i j}= [^x_i−¹

2,x_i+1

2]× [^y_j−¹

2,y_j+1

2]representingthecells.Forsimplicity,weapplyuniformmeshesonly,anddenote x and y tobethemeshsizesinxandy directions,respectively.

Following[23],wedefinethefiniteelementspaceas V_h^k

=

v

:

^v

|

Ii j

∈

^Pk

(

I_{i j}

), ∂

v₆

∂

x

+ ∂

v₇

∂

y

=

⁰

,

whereP^k(Ii j)=(P(Ii j))¹⁵,withP(Ii j)beingthespaceofpolynomialsofdegreeatmostkincellIi j.v6 andv7arethesixth andseventhcomponentsofthebasisfunctionsusedtoexpand B_x andB_y.Notethat,byusingsuchafiniteelementspace,

∇ ·^B=^{0 islocallypreserved}automatically.Althoughalocaldivergencefreeconstraintdoesnotguaranteeglobaldivergence freesolutions,ithasbeenshownthatalocalconditionisveryaccurate[8],although[1]and[2]reportsomeadvantagesin usingaglobaldivergencefreereconstructionoveralocalconstraint. Weprovideanumericaltest oftheglobaldivergence ofthemagneticfieldinSection4.6.2,whichsupportstheconclusionof[8].

ThentheDGschemeisthefollowing:findQ∈^V_h^{k,suchthatforanyv}∈^V_h^k

(

Q_t

,

v

)

i j

=

f

(

Q

),

v_x

i,j

+

y_j+1

2 yj−¹₂

f_i−1 2,jv⁺

i−¹₂,jdy

−

y_j+1

2 yj−¹₂

f_i+1 2,jv⁻

i+¹₂,jdy

+

g

(

Q

),

vy

i,j

+

x

i+¹₂

xi−¹₂

g_i,j−1 2v⁺

i,j−¹₂dx

−

x

i+¹₂

xi−¹₂

g_i,j+1 2v⁻

i,j+¹₂dx

+

s

(

Q

),

v

i,j

,

(3.3)

where(u,v)_{i j}=

Ii j

15

=¹uvdxdy andv⁺

i−¹₂,j(y),v⁻

i+¹₂,j(y),v⁺

i,j−¹₂(x),v⁻

i,j+¹₂(x)arethetracesofvonleft,right,lower,up- peredgeofthecellIi j,respectively.Moredetailscanbefoundin[42].Withthedefinitionofthetraces,theone-dimensional numerical fluxes are:f_i−1

2,j=f(Q⁻

i−¹₂,j(y),Q⁺

i−¹₂,j(y)) and^g_i,j−¹

2 =^g(Q⁻

i,j−¹₂(x),Q⁺

i,j−¹₂(x)). The local Lax–Friedrichs (LLF) fluxisusedtoevaluate_f

i−¹₂,j,g_i,j−1 2:

_f

i−¹₂,j

=

¹ 2

f

Q⁻

i−¹₂,j

(

y

) +

^f

Q⁺

i−¹₂,j

(

y

) +

¹

2

| λ |

_i−¹

2,j

Q⁻

i−¹₂,j

(

y

) −

^Q+_i−1 2,j

(

y

)

g_i,j−1

2

=

¹

2

g

Q⁻

i,j−¹₂

(

x

) +

g

Q⁺

i,j−¹₂

(

x

) +

¹

2

|λ|

_i,j−¹

2

Q⁻

i,j−¹₂

(

x

) −

Q⁺

i,j−¹₂

(

x

)

.

(3.4)

We donot presentlyhaveasolutionforafullRiemannsolverorevenanapproximateRiemannsolver.Thereforewehave used anad-hocformofan LLFfluxbasedonanestimate ofthemaximumeigenvaluesofthefluxJacobian computedre- spectivelyfromtheEuler((2.1)–(2.3)),Maxwell((2.4)–(2.5)),andGOL((2.6)–(2.7))sub-blocks.Thesemaximumeigenvalues, sometimescalledfreezingspeeds[35],arethelocalvaluesof|^u|+^cs,c,|^ue|+^cesfortheEuler,Maxwell,andGOLsub-blocks respectively, wherec²_es=^m_mⁱ_e^c2s istheelectronsoundspeedandc²_s=

γ

P

ρ

⁻¹ istheacousticspeed.Wehavefound the LLF fluxbasedonthesevaluestobeentirelysatisfactoryandwenotethatthesamevaluesareusedinthefinitevolumecode andthatwithDG,farlessdiffusionisexhibitedthanFV,whichwillbeshowninSection4.

|λ|i,j arethefreezingspeedsevaluatedoncell{ⁱ,j}^{,andinEq.(3.4)weset}

| λ |

_i−¹

2,j

=

^max

| λ |

i−¹,j

, | λ |

i,j

| λ |

_i,j−¹

2

=

^max

| λ |

i,j−¹

, | λ |

i,j

.

3.2. Positivity-preservinglimiter

Physically,thedensity

ρ

andpressureP arepositive.Therefore,theexactsolutionisinaconvexset[7]

G

=

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎩

w

=

⎛

⎜ ⎜

⎜ ⎜

⎜ ⎝ ρ

mx

m_y m_z En

S_e

⎞

⎟ ⎟

⎟ ⎟

⎟ ⎠

ρ >

0 andP

= ( γ ₋

1

)

En

−

¹ 2

m²_x

+

^m2y

+

^m2z

/ ρ

>

0 andS_e

>

0

⎫ ⎪

⎪ ⎪

⎪ ⎪

⎬

⎪ ⎪

⎪ ⎪

⎪ ⎭

.

(3.5)

We wanttoconstructthenumericalsolutionwhichisalsointheset(3.5).WeuseforwardEulerfortimeintegrationand maintainthepositivityofdensity,pressureandelectronentropydensityattimeleveln+^{1,providedthattheyarepositive} attime leveln.It isclearthat,En>0 is automaticallyestablishedifwe have

ρ

>0 and P>0.Inthissection,we prove thatthepositivity-preservingpropertycanbeguaranteedwithaspeciallimiterinourXMHDmodel.

Weconsiderthefirstfiveequationsandthelastoneinthesystem(3.3).Following[42],attimeleveln,weuseavector of polynomials of degree k, wⁿ_{i j}=(

ρ

ⁿ_{i j},m_xⁿ_{i j},m_yⁿ_{i j},m_zⁿ_{i j},Enn

i j,S_eⁿ_{i j}) to approximate the exact solution, and define the cell averagewⁿ_{i j}=(

ρ

ⁿ_{i j},m_xⁿ_{i j},m_yⁿ_{i j},m_zⁿ_{i j},En

n

i j,S_e).Inthissection,wealwaysuseuⁿ_{i j}=(u_xⁿ_{i j},u_yⁿ_{i j},u_zⁿ_{i j})^T for ^m

n i j

ρⁿ_{i j} =(^mx

n i j

ρⁿ_{i j} ,^my

n i j

ρ_{i j}ⁿ ,^mz

n i j

ρⁿ_{i j} )^T

asthenumericalvelocityincellIi j attimeleveln.Forsimplicity,ifweconsideragenericnumericalsolutiononthewhole computationaldomain,thenthesubscripti j willbeomitted.Bytakingthetestfunctionv=^{1 in(3.3)wehavethescheme} satisfiedbythecellaverages

wⁿ_{i j}⁺¹

=

¹ 2H₁

+

¹

2H₂

,

(3.6)

where

H₁

=

^wn_{i j}

+

^{2 t} x y

y

j+¹₂

yj−¹₂

ˆ

f

w⁻

i−¹₂,j

(

y

),

w⁺

i−¹₂,j

(

y

)

− ˆ

^f

w⁻

i+¹₂,j

(

y

),

w⁺

i+¹₂,j

(

y

)

dy

+

^{2 t} x y

x

i+¹₂

xi−¹₂

ˆ

g

w⁻

i,j−¹₂

(

x

),

w⁺

i,j−¹₂

(

x

)

− ˆ

^g

w⁻

i,j+¹₂

(

x

),

w⁺

i,j+¹₂

(

x

)

dx (3.7)

H2

=

^wni j

+

^{2 t} x y

I_{i j}

s

(

x

,

y

)

dxdy

,

(3.8)

withˆf(·,·)^andgˆ(·,·)^beingtheone-dimensionalnumericalfluxesand s

=

0

,

J

×

^B

,

u

· (

J

×

^B

) + η

J²

, ( γ −

¹

)

n¹_e^−γ

η

J²

T

.

Foraccuracy,weuseL-pointGaussquadratureswithL≥^k+^{1 to}approximatetheintegralin(3.7).Wereferthereadersto [11]formoredetailsofthisrequirement.TheGaussquadraturepointson[x_i−1

2,x_i+1

2]and[y_j−1 2,y_j+1

2]aredenotedby p^x_i

=

x^β_i

: β =

¹

, · · · ,

L

and p^y_j

=

y^β_j

: β =

¹

, · · ·,

^L

,

respectively.Also,wedenote

ω

β asthecorrespondingweights ontheinterval [−¹,1]^{.ThentheGaussquadratureformula} ontheinterval[^x_i−¹

2,x_i+1

2]^{canbewrittenas}

x

i+¹₂

xi−¹₂

f

(

x

)

dx

≈

L β=1

ω

_βf

x^β_i

hi

/

2

,

wherehi=^x_i+¹

2−^x_i−¹

2.Moreover,weuse

ˆ

p^x_i

=

ˆ

x^α_i

: α ₌

0

, · · · ,

M

and p

ˆ

^y_j

= ˆ

y^α_j

: α ₌

0

, · · · ,

M

with2M−¹≥^k,astheGauss–Lobatto points on[^x_i−¹

2,x_i+1

2] ^and[^y_j−¹

2,y_j+1

2]^, respectively.Also, we denote

ω

ˆα asthe correspondingweightsontheinterval[−¹,1]^.ThentheGauss–Lobattoquadratureformulaontheinterval[^x_i−¹

2,x_i+1 2]^can bewrittenas

x

i+¹₂

xi−¹₂

f

(

x

)

dx

≈

M α=0

ˆ ω

_αf

ˆ

x^α_i

h_i

/

2

.

Letλ1= _x^{t and}λ2= _y^t,thenH1becomes

H1

=

^wni j

+

²

λ

1

L β=1

ω

_β

^ˆ

f

w⁻

i−¹₂,β

,

w⁺

i−¹₂,β

− ˆ

^f

w⁻

i+¹₂,β

,

w⁺

i+¹₂,β

+

2

λ

2

L β=¹

ω

β

g

ˆ

w⁻

β,j−¹₂

,

w⁺ β,j−¹₂

− ˆ

g

w⁻

β,j+¹₂

,

w⁺ β,j+¹₂

,

(3.9)

wherew⁻

i−¹2,β=^w−_i−1

2,j(y^β_j)isapointvalueintheGaussquadrature.Likewisefortheotherpointvalues.Thenwehavethe followinglemma

Lemma3.1.ForthediscontinuousGalerkinmethodwiththeLLFflux,ifwⁿ∈^G,thenH1∈^{G undertheCFLcondition} t

xmax

ux

+

C0

_∞

,

uy

_∞

,

uz

_∞

+

^t

ymax

ux

_∞

,

uy

+

C0

_∞

,

uz

_∞

≤ ω ˆ

0

2

,

(3.10)

and t

xmax

(

ue

)

x

∞

+

^t

ymax

(

ue

)

y

∞

≤ ω ˆ

0

2

,

(3.11)

whereC0= ^γ_ρ^P,v∞isthestandardL^∞-normofv onthewholecomputationaldomain,and(ue)xand(ue)yarethex andy componentsofue.

Proof. Theprooffollowsfrom[42]withsomeminorchanges,soweomitit. 2

Now we considerthe sourceterms.Weusea Gauss quadraturewithL points to computethe integralof thesource.

Then H₂becomes

H2

=

L α=1

L β=1

ω

_α

ω

βh_αβ

,

where

h_αβ

=

^wn_αβ

+

^{2 ts}αβ

andwⁿαβ denotes thepointvalueofwⁿ_{i j} at(xα

i,y^β_j).Likewiseforsαβ. Lemma3.2.Supposewⁿ∈^{G,thenwehaveh}αβ∈^{G underthecondition}

t

≤

^min

i,j,α,βA_s

x^α_i

,

y^β_j

,

(3.12)

where

A_s

= η

J²

ρ + !

η

²J⁴

ρ

²

+

^2P

(

J

×

^B

)

²

ρ /( γ −

¹

)

2

(

J

×

^B

)

²

.

Proof. Supposehαβ=(

ρ

˘,

ρ

u˘,E˘n,˘Se)^T,withu˘=^m_ρ^˘_˘^.Forsimplicity,inthislemma,ifweconsiderthepointvalueat(xα i,y^β_j) attime leveln,thecorresponding indexwillbe omitted.Then itiseasy toshowthat

ρ

˘₌

ρ

ⁿ>0 and S˘e>0.Thesecond inequalityisduetothepositivityofthesourcetermin(2.7).SoweonlyneedtoshowP˘=(

γ

₋1)(E˘n−¹₂

ρ

˘u˘²)>0.Clearly, wehave:

˘

m

=

^m

+

^{2 t}

(

J

×

^B

)

E

˘

n

=

En

+

^{2 t}

u

· (

J

×

^B

) + η

J²

whichfurtheryields P

˘

γ −

¹

=

En

+

^{2 t}

u

· (

J

×

^B

) + η

J²

− [

^m

+

^{2 t}

(

J

×

^B

)]

² 2

ρ

=

^P

γ −

¹

+

2 t

η

J²

−

2 t²

(

J

×

B

)

²

ρ ^.

Therefore, P˘ >0 provided t

≤ η

J²

ρ + !

η

²J⁴

ρ

²

+

^2P

(

J

×

^B

)

²

ρ /( γ −

¹

)

2

(

J

×

^B

)

²

.

Thisconditioncanbemadephysicallytransparentasfollows:

t

< !

¹ 2

( γ −

1

)

v_th

vAv_A or t

< η μ

0

1 v²_A

issufficientfor P˘>0,wherev_th isthethermalspeed,v_A istheAlfvénspeed,v_A≡^√∇×_μ0nm^B_i^. 2

Noticing thefactthatwⁿ⁺¹istheconvexcombinationofH1andhαβ wehavethefollowingresult.