High order ADER schemes for a unified first order hyperbolic formulation of Newtonian continuum mechanics coupled with electro-dynamics

Contents lists available atScienceDirect

Journal

of

Computational

Physics

www.elsevier.com/locate/jcp

High

order

ADER

schemes

for

a

unified

first

order

hyperbolic

formulation

of

Newtonian

continuum

mechanics

coupled

with electro-dynamics

Michael Dumbser

a

,

∗

,

Ilya Peshkov

b

,

c

,

Evgeniy Romenski

c

,

d

,

Olindo Zanotti

a a_{DepartmentofCivil,EnvironmentalandMechanicalEngineering,UniversityofTrento,ViaMesiano77,38123Trento,Italy} b_{InstitutdeMathématiquesdeToulouse,UniversitéToulouseIII,F-31062Toulouse,France}

c_{SobolevInstituteofMathematics,4Acad.KoptyugAvenue,630090Novosibirsk,Russia} d_{NovosibirskStateUniversity,2PirogovaStr.,630090Novosibirsk,Russia}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received6December2016 Receivedinrevisedform9June2017 Accepted10July2017

Availableonline17July2017

Keywords:

Symmetrichyperbolicthermodynamically compatiblesystems(SHTC)

Unifiedfirstorderhyperbolicmodelof continuumphysics(fluidmechanics,solid mechanics,electro-dynamics)

Finitesignalspeedsofallphysicalprocesses Arbitraryhigh-orderADERDiscontinuous Galerkinschemes

Path-conservativemethodsandstiffsource terms

Nonlinearhyperelasticity

In this paper, we propose a new unified firstorder hyperbolic model of Newtonian continuum mechanics coupled with electro-dynamics. The model is able to describe the behavior of moving elasto-plastic dielectric solids as well as viscous and inviscid fluids in the presence of electro-magnetic fields. It is actually a very peculiar feature of the proposed PDE system that viscous fluids are treated just as a special case of elasto-plasticsolids. Thisisachievedbyintroducingastrainrelaxation mechanisminthe evolutionequations ofthedistortionmatrix A,whichinthecaseofpurelyelasticsolids mapsthe currentconfiguration to the reference configuration. Themodel alsocontains a hyperbolic formulation of heat conduction as well as a dissipative source term in theevolutionequations fortheelectricfieldgivenbyOhm’s law.Viaformalasymptotic analysisweshowthatinthestifflimit, thegoverningfirstorderhyperbolicPDEsystem withrelaxationsourcetermstendsasymptotically tothewell-knownviscousandresistive magnetohydrodynamics(MHD)equations.Furthermore,arigorousderivationofthemodel fromvariational principlesis presented, togetherwith thetransformation ofthe Euler– Lagrange differentialequations associated with theunderlying variationalproblem from Lagrangiancoordinates toEulerian coordinates inafixed laboratoryframe. The present paper henceextends the unified first orderhyperbolic model ofNewtonian continuum mechanicsrecentlyproposedin[110,42]tothemoregeneralcasewherethecontinuum iscoupledwithelectro-magneticfields.ThegoverningPDEsystemissymmetrichyperbolic and satisfiesthefirst and secondprinciple ofthermodynamics,henceit belongs to the so-calledclass of symmetric hyperbolic thermodynamically compatible systems(SHTC), whichhavebeenstudiedforthefirsttimebyGodunovin1961[61]andlaterinaseriesof papersbyGodunovandRomenski[67,69,119].Animportantfeatureoftheproposedmodel isthatthepropagationspeedsofallphysicalprocesses,includingdissipativeprocesses,are finite.The model is discretized using high order accurate ADERdiscontinuous Galerkin (DG)finiteelementschemeswithaposteriori subcellfinitevolumelimiterandusinghigh orderADER-WENOfinitevolumeschemes.Weshownumericaltestproblemsthatexplore

*

Correspondingauthor.

E-mailaddresses:[email protected](M. Dumbser),[email protected](I. Peshkov),[email protected](E. Romenski),[email protected] (O. Zanotti).

http://dx.doi.org/10.1016/j.jcp.2017.07.020

0021-9991/©2017TheAuthor(s).PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

aratherlargeparameterspaceofthemodelrangingfromidealMHD,viscousandresistive MHDoverpureelectro-dynamicstomovingdielectricelasticsolidsinamagneticfield.

©2017TheAuthor(s).PublishedbyElsevierInc.Thisisanopenaccessarticleunderthe CCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

1.1. Electrodynamics of moving media

Inthispaper, wepropose anewunified first order hyperbolic model ofNewtonian continuum mechanicscoupled with electro-dynamics. The model isthe extension ofour previous results [42], hereafter Paper I, on aunified formulation of continuummechanics towardsthecouplingofthetimeevolutionequationsforthematter withtheelectricandmagnetic fields.Theproblemofdeterminingtheforceactingonamediuminanelectromagneticfield,aswellastherelatedproblem ofdeterminingtheenergy-momentumtensorofanelectromagneticfieldinamedium,hasbeendiscussedintheliterature overthe yearssincethe workby Minkowski[93] andAbraham [2]. However,to thebestofourknowledge, auniversally acceptedsolutiontothisproblemhasbeenabsenttodate[83,88,60,51,34].

Inthisrespect,ourworkcanbebroadlyconsideredasacontributiontothemodelingofelectrodynamicsofmoving con-tinuousmedia.Wedonotclaimtogiveanultimatesolutiontotheproblem,butrathertoshowthat,withinourformalism, alltheequationscanbe obtainedinaconsistentwaywithrathergoodmathematicalproperties(symmetrichyperbolicity, firstorderPDEs,wellposedness oftheinitialvalue problem,finitespeedsofperturbationpropagationevenfordissipative processesinthediffusiveregime)andthatthecorrespondingphysicaleffectsarecorrectlydescribed.Byanextensive com-parison withthe numericalandanalyticalsolutions tothewell established modelsastheMaxwell equations,idealMHD equations andviscous resistive MHD(VRMHD) equations,we demonstrate that the proposed nonlinear hyperbolic dissi-pativemodel isable to describedielectrics (

η

→ ∞

), idealconductors (

η

→

0), andresistiveconductors (0

<

η

<

∞

) as particularcases,where

η

istheresistivity.Thus,theapplicabilityrangeoftheproposedmodelislargerthanthoseforideal andresistiveMHDmodels,becausethe electricandmagneticfieldsaregenuinely independentandare governedbytheir owntimeevolutionequationsasintheMaxwellequations.

InPaper Iand[110],weprovidedaunifiedfirst-orderhyperbolicformulationoftheequationsofcontinuummechanics, showing for the first time that the dynamics of fluids and solids can be cast in a single mathematical framework. This becomespossibleduetotheuseofacharacteristicstraindissipationtime

τ

,whichisthecharacteristictimeforcontinuum particle

rearrangements.

Byits definition,thecharacteristictime

τ

,asopposedtothe viscositycoefficient,isapplicableto thedynamicsofbothfluidsandsolids(see thediscussionsin[110] andPaper I)andisacontinuum interpretationofthe seminalideaoftheso-called

particle settled life time (PSI)

ofFrenkel[54],whoappliedittodescribethefluidityofliquids, seealso

[18,16,17]

andreferencestherein.Inaddition,thedefinitionof

τ

assumesthecontinuumparticlestohavea

finite

scale and thustobedeformableasopposedtothe

scaleless mathematical points in

classicalcontinuummechanics.Wenote thatthemodelstudiedin[110] andPaper Iwas usedbyseveralauthors,e.g.[118,91,114,57,10,66,53,115,9,99,109,139]to citejustafew, inthesoliddynamicscontextsinceitsoriginalinventionin1970thbyGodunovandRomenski [68,64]but therecognitionthatthesamemodelisalsoapplicabletothedynamicsofviscousfluidsanditsextensivevalidationinthe fluiddynamicscontextwasmadeonlyrecentlyin[110]andPaper I.

1.2. Symmetric hyperbolic equations, well-posedness, causality

When one deals with nonlinear dynamical phenomena, and thus with nonlinear time-dependent partial differential equations(PDEs),perhaps, thefirstexaminationa newmodelhastopassisto verifyiftheinitialvalue problem(IVP)is well-posed,atleastlocally,i.e.whetherthesolutiontothesystemofPDEswithgiveninitialdataexists,isuniqueandstable (depends continuously on the initial data). We emphasize that the well-posedness ofthe IVP should not be considered as a purely mathematical requirement but as a fundamental physical observation about the time evolution of physical systems,i.e.exactlyasweconsidercausality,conservationandthermodynamicprinciples,GalileanorLorentzinvarianceto be essential features ofmacroscopictime evolution. Inother words,a modeldescribing thetime evolution ofa physical systemandhavinganill-posedIVPshouldberegardedaswrong.Moreover,thewell-posednessoftheIVPisafundamental propertyoftime-dependentPDEsinordertobenumericallysolvable.

Fromthemathematicalviewpoint,thewell-posednessoftheIVPcannotbeguaranteedforageneralnonlinearsystemof PDEs.Thisevencannotbeguaranteedforafirstorderquasi-linearsystem[12,49,96],orformodelswhichwereconsistently derivedfrommicroscopictheoriesas, forexample,theBurnettequationsderived fromthegaskinetictheory[14,128,138]. However,thereisaclassofnonlinearPDEsforwhichtheIVPislocallywell-posedintime,whichistheclassofhyperbolic PDEs.Unfortunately,itishardlypossibletoprovethatagivennon-linearfirstordersystemisglobally(notintime butin the spaceofphysicallyrelevant state parameters)hyperbolic becausethiswould requiretoprove theglobalexistence of thefullbasisofeigenvectorsforamatrixwhoseentriescanbehighlynonlinear.Forexample,themodelconsideredinthis paperhasnonlineartermsuptopower4andtofindanalyticalexpressions fortheeigenvaluesandeigenvectors,andthus

toprovehyperbolicityinthisway,seemsimpossible.SohowcanwebesurethattheIVPfortheproposednonlinearmodel iswell-posed,i.e.thatthemodelishyperbolic?

The modelproposed in thispaperwas developedwithin a very importantsubclass of first ordernonlinearhyperbolic systemswhosenon-dissipativepartcanbewritteninthefollowingquasi-linearform

M(

p

)

∂

p

∂

t

+ H

k

(

p

)

∂

p

∂

xk

=

0

,

(1)

M

T

_{= M >}

_{0 and}

_H

T

k

= H

k,forwhich(local)well-posednessisknowntoholdtrue[80,12,96,121].Thissubclassiscalled

symmetric hyperbolic systems of PDEs and it is a generalization of Friedrichs-symmetrizable linearsystems [55]. One may naturally questionhowrestrictiveitisforamodeltobesymmetrichyperbolic?AsitwasshownbyGodunov[61–63]and later byothers[56,15,122]thereisan intimateconnectionbetweenthesymmetrichyperbolicityandthermodynamics,i.e. a first order systemof PDEs is symmetric hyperbolic ifit admitsan extraconservation lawfor a convex potential which

plays theroleofthetotalenergyforthesystem. Therefore,such asubclassofhyperbolicPDEscanbeassociatedwiththe thermodynamicallycompatiblesystemsoffirstordertime-dependentPDEs.

Theconsiderednonlinearsystemforelectrodynamicsofmovingmediahasbeendevelopedwithinsuchaclassof sym-metric hyperbolic systems with convexenergy. Therefore, one can be certain that, despite the highly nonlinear terms, the proposedmodelisgloballyhyperbolicandtheIVPforitiswell-posed,andhencethemodelcanbesolvednumerically.

What concernsamathematicalguidetoderivesymmetrichyperbolictimeevolutionequations,asin[110]andPaper I, we follow the so-called formalism of SymmetricHyperbolicThermodynamicallyCompatible

systems of conservation laws,

or simplySHTCformalismhere.ThisformalismisdescribedinSection2.

Eventually,werecallthatapartofthewell-posednessoftheIVPthe

hyperbolicity also

naturallyaccountsforanother fun-damentalobservationabouttimeevolutionofphysicalsystems,namelythefinitevelocityforanyperturbationpropagation, i.e.

causality.

1.3. Hyperbolic PDEs with stiff relaxation source terms

Bypassingfromthenon-dissipativedynamicsdescribedbysymmetrichyperbolicPDEsoftype(1)todissipative dynam-ics(viscousmomentum,heatandchargetransfer),wedonotwanttodestroythefundamentalphysicalpropertiesdiscussed above, i.e.thewell-posednessandcausality.Thus, we believethat theonlycompromise isto modeldissipativeprocesses by adding algebraic source terms of relaxation type to the right hand side of (1). Thus, we shall consider the following generalizationof(1)

M(

p

)

∂

p

∂

t

+ H

k

(

p

)

∂

p

∂

xk

= −

1

τ

S

(

p

),

(2)

where

τ

isa dissipationtime scale1 that canbe afunction ofthestate parametersp as well. Thisoptionpreservesboth features (well-posednessandcausality)becausethehyperboliccharacterofthePDEs isdefinedbytheleadingtermsonly, i.e.by thesymmetric matrices

M

and

H

k. AsshowninPaper I, therelaxationsource termscan besuccessfullyused to describeviscousmomentumandheattransferwhich,formanyyears,inclassicalcontinuum mechanicswasbelievedtobe possibleonlyintheframeworkofsecondorderparabolicPDEs.

An

attractive feature

ofusingalgebraicrelaxationsourcetermsisthepossibilityofastraightforwardgeneralizationtoa nonlineartransporttheoryvia

τ

=

τ

(

p

)

(e.g.non-Newtonianfluids,elasto-plasticsolids,non-Fourierheatconduction) with-out anyincreaseinthecomplexity ofthemathematicalformulation,becauseno newanalyticalandnumericaltechniques arerequiredtotreatthemodelwith

τ

(

p

)

.Forexample,thesamefamilyofADERfinitevolumeanddiscontinuousGalerkin methodsthat wasusedinpaperIforthesimulationofNewtonian flowswith

τ

=

const was alsoappliedtothemodeling ofnonlinearelasto-plasticdeformationinsolidsin[139]wherethedissipationtimewasnotconstantbutvariesover

∼

15 ordersofmagnitude.

Atthispoint,westressthattheSHTCformalismis

radically

different from thewell-knownMaxwell–Cattaneoapproachto constructhyperbolicrelaxationmodels

[25]

typicallyusedinextendedirreversiblethermodynamics(EIT)[79].The notice-able differencebetweentheMaxwell–CattaneoandtheSHTCapproachisthat therelaxationaffectsthe differentialterms intheMaxwell–Cattaneoapproachwhileitisintroducedina

purely algebraic manner in

theSHTCformalism.InMaxwell– Cattaneo-typemodelsthefluxJacobian aswellasthecorresponding soundspeedsdependonarelaxationparameter



as



−1_{[142].Inparticular,inordertoapproachNewtonianflowswithaMaxwell–Cattaneo-typemodel,theshearsoundspeed} has to go toinfinity2 _{which ofcourse violates thecausality principle.In the SHTC formalism,as demonstratedin paper}

I via thedispersion analysis, the soundspeedsare always finite,also inthe limit

τ

→

0. Anotherapparent consequence

1 _{Theremaybe,ofcourse,multipledissipationtimescales}

_τ

_1,

_τ

_{2,. . . ,etc.correspondingtodifferentphysicalprocesses(viscousdissipation,dissipation} duetoheattransfer,chemicalreactions,etc.).Inequation(2),wewriteasingletime

τ

justforsimplicityofnotationandonlytoemphasizethemain characterofthesourcetermsusedinthispaper.

2 _{Thisdefect,however,wasremovedintheversionofEITbyMüllerandRuggeri[96]duetoamoreelaboratestructureoftheequations,inwhichthe} relaxationisintroducedinapurelyalgebraicmanneranddoesnotaffectthedifferentialterms,asinourformalism.

ofthestructure oftherelaxationtermsoftheMaxwell–Cattaneo-type modelsisthat thegeneralization toamore realis-ticcase with



=

(

p

)

inevitably affectsthe hyperbolicityof amodel [49] because thematricesof thequasi-linear form nowdependon



(

p

)

.Inaddition,thedifferencesbetweentheapproachesbecomealsoapparentifone takesalookatthe physicalmeaningofthestate variablesusedinbothapproaches. Werecallthat intheEITthe

fluxes are

typically usedas theextra state variables(in additiontothe conventionaloneslike mass,momentumandenergy),whichusually leads to thesituationthatthePDEs havenoapparent structure(new

differential terms

mayappearordisappeardependingonthe choiceoftheclosureforthesystem).IntheSHTCformalism,onlydensityfieldsmayserveasstatevariableswhich,infact, duetothefundamental

conservation principle allows

toobtainequationsinarathercompleteformwithanelegant struc-ture, seeSection2.Lastbutnotleast, itiswellknown thatforMaxwell–Cattaneo-typemodels thechoiceofanobjective timederivative (objectiveframerate)forthefluxescannotbeaddressedinauniquemannersincean infinitenumberof objectivetimederivativesispossible[142,49].Thisisnot thecasefortheSHTCformalism becauseitisframeinvariantby construction[65].

1.4. Numerical solution of hyperbolic PDEs with stiff relaxation

AlthoughtheSHTCformalismprovidesaconsistentframeworkforthemodelingoftime-dependentnonlineardissipative phenomenatheuseofhyperbolicrelaxationmodels(2),asitiswellknown,imposes certaindifficultiesforthenumerical solution in the case when the dissipation time scale defined by the relaxation time

τ



T is much smaller than the macroscopictime scale T

∼

1

/

cmax definedby the maximumcharacteristicspeed

c

max ofthenon-dissipative partofthe model(i.e.bythelefthandsideof(2)).Insuchacase,thesourcetermin(2)iscalleda

stiff source

term.Thepresenceof astiffsourcetermmaylettheoriginalsystemtendtowardsanasymptoticallyreducedsystem(see[26,124]) thatmayeven haveadifferentstructurethantheoriginalone,seee.g.

[42]

.

Fornondissipative hyperbolic PDEs (1), onlya numerical flux must be chosen, oran appropriate fluctuation for non-conservative products [24,104]. In this case, the classical properties required are consistency, stability and accuracy. For dissipativehyperbolic PDEs (2) withrelaxationsource termsalsoan appropriate numericalsourceterm mustbe chosen. Here,notonlythethreeclassicalpropertiesarerequired,butsomeadditionalpropertiesareneededfortheglobal numeri-calscheme:Itshouldbe

well-balanced,

i.e.abletopreservecertainrelevantsteadystatesolutionsalsonumerically.Itshould berobustalsooncoarsegridsifthesourcetermisstiff.Acoarsegridisagridwhosesizedoesnottakeintoaccountthe sourceterm,i.e.thecharacteristicspaceandtimestepsarebasedontheassociatedhomogeneoussystem(1)only.Finally, theschemeshouldbe

asymptotically consistent or

inotherwordsasymptoticpreserving(AP) ifthesourcetermisstiff.This meansthattheschemeshouldgivethecorrectasymptoticbehavior evenifthesourcetermis

under resolved.

Inthispaper,we donotaimto providenewdevelopmentsconcerningthenumericaltreatment ofhyperbolicsystems withstiff relaxation.Thereaderisreferredtotheextensiveliteratureonthesubject, seee.g. thefollowingnon-exhaustive listofreferences

[85,107,108,78,21,124,19,22,105,98,87,92,40,75,20]

andreferencestherein.Ournumericalresultsfullyrely on the unified family ofADER finite volume (FV) andADER discontinuous Galerkin (DG) methods developed previously in[40,36,41,46,75] andsuccessfullyapplied tothe modelingof viscousNewtonian flows (i.e. inthestiff relaxationlimit) inPaper Iwithin theSHTC framework,see [42]. TheADERfinite volumemethods arebased ona threestage procedure. First,ahigh-ordernon-oscillatoryWENOreconstruction procedureisappliedtothecellaveragesatthecurrenttimelevel. Second,thetemporalevolutionofthereconstructionpolynomialsiscomputedlocallyinsideeachcellusingthegoverning equations.IntheoriginalENOschemeofHartenetal.

[74]

andintheADERschemesofToroandTitarev

[130,137,131,132]

, thistimeevolutionisachievedviaaTaylorseriesexpansionwherethetimederivativesarecomputedbyrepeated differen-tiationofthegoverningPDEswithrespecttospaceandtime,i.e.byapplyingtheso-calledCauchy–Kowalevskiprocedure. However,thisapproachisnotabletohandlestiffsourceterms,unlessan

implicit Cauchy–Kowalevski

methodisused,see recentdevelopmentsdocumentedin

[136]

.Therefore,anewstrategywasproposedin[40],whichonlyreplacestheCauchy– Kowalevskiprocedurecomparedtothepreviouslymentionedschemes.Forthetime-evolutionpartofthealgorithm,a

local

space–time discontinuous Galerkin(DG) finite element schemewas introduced, which is ableto handlealso stiff source terms.Thisstepistheonlypartofthealgorithmwhichis

locally implicit,

andthusallowstheuseofatimestepforthefinal numericalschemethathastoobeyonlyaclassicalCFL-typestabilityconditionbasedonthemaximumwavespeed

c

max of theadvectiveprocessesandnotbasedontherelaxationtime

τ

.Thethirdandlaststepofthefully-discreteADERfamilyof schemesconsistsofa fairlystandard explicit integration over eachspace–timecontrol volume,usingthelocalspace–time DGsolutionsattheGaussianintegrationpointsfortheintercellfluxesandforthespace–timeintegraloverthesourceterm. ComparedtoADER-FVschemes,inthecaseofADER-DGmethodsthereisnoneedforthehighorderWENOreconstruction operator,sincethediscretesolutionisdirectlyevolvedinthespaceofhigherorderpiecewisepolynomials. However,these schemesrequireproperlimitingatshockwaves,seee.g.

[48,145,144]

forrecentdevelopments.

TheADER-FVandADER-DGframeworkisnowwellestablishedandhasalreadybeenappliedtomanydifferentsystems ofPDEs

[36,39,41,35,46,48,145,144,42]

,includinghyperbolicsystemswithstiffrelaxation.

1.5. Computational advantages of using hyperbolic dissipative models

From the computational performance viewpoint, a clear difference between classical parabolic theories such as the Navier–Stokes–Fourier (NSF) equations or the viscous and resistive magnetohydrodynamics (VRMHD) equations and our

hyperbolic approach is thatthe latteruses a much largerset ofstate variables (2 scalarfields,namely themass density and thetotal energydensity,

+

3 componentsof themomentum density

+

9 componentsofthe non-symmetric distor-tion tensor

+

3components ofthethermalimpulse vector+6componentsoftheelectro-magneticfields),thatisatotal numberof23evolutionequations.Thisobviouslyrequiresmorememoryandcomputationaleffort ifcompared withonly 5 equations forthe NSF equations or 8equations forVRMHD. However, we can alsolist some benefits in utilizing first orderhyperbolicmodels.Asdiscussedin[42,89,90],onemaypointoutthefollowingcomputationaladvantageswhenusing hyperbolicmodelsoverconventionalparabolicmodels

•

thekeyadvantageoftheSHTCmodelproposedinthepresentpaperisitsnaturalabilitytodescribemoving

fluids and

solids in anelectromagneticfieldwithinthe

same PDE

systemandthusallowsthesimulationofmagnetizedfluids

and

solidswithinthe

same computer

code,whileimplementationsbasedonclassicaltheoriessuchasNSFandVRMHDonly applytofluidsandare

not able

todescribethemotionanddeformationofelasticsolids;

•

a numerical methodapplied to firstorder hyperbolic PDEs can achieve higherorderof accuracy thanfor high-order PDEs on the samediscrete stencil; inparticular, infirst orderhyperbolic models the sameorder ofaccuracy forthe solutionandthestressesandother fluxesiseasilyachieved,whileinparabolicmodelsthedissipativefluxesarebased onthe derivativesofthestate vector,andthusingeneraloneorderofaccuracyislostifthestenciloftheschemeis notproperlyextended;

•

firstordersystemsarelesssensitivetothequalityofcomputationalgridsandboundarysingularities;

•

no severeparabolic time steprestriction



t

∼ 

x2 arisesforexplicitdiscretizations offirst orderhyperbolic systems andthereforenumericalschemeswithanexplicittimesteppingcanbeused.Thisallowsnotonlyforastraightforward andefficientparallelization,butcan evenleadto

faster simulations

withtheenlargedSHTCsystemcomparedto con-ventionalparabolicmodels,inparticularinthecontext ofexplicitdiscontinuousGalerkinfiniteelementschemes, see theexamplesprovidedin

[42]

;

•

thepossibilityofaneasyandfilter-lessvisualizationofcomplexflowswiththehelpofthedistortiontensor,see[42]. For recentworkon hyperbolic reformulationsofthe steadyviscous andresistiveMHDequationsandtime dependent convection–diffusionequationsbasedonstandardMaxwell–Cattaneorelaxation,seethepapersofNishikawaetal.

[100,101,

89,90,11]andMontecinosandToro

[95,94,135]

.Theseapproaches,however,weremotivatedbycomputationalreasons,while ourhyperbolic theorywasmotivatedbytheconstructionofaconsistentphysicaltheoryfirst.Inparticular,thementioned hyperbolic approachesareattributedto theMaxwell–Cattaneo-typemodelsdiscussedinSection 1.3becausethegradients ofthefields(stress,heatflux)areusedasextrastatevariables,andthussuchapproachesmaysufferfromthesamephysical inconsistencies asdiscussedinSection1.3.Modelsofasimilarnatureare alsousedinnon-equilibriumgasdynamics.See thefollowingsectionwherewediscussotherdifferencesbetweenthissortofmodelsandourapproach.

1.6. Beyond conventional continuum theories

Aftermany discussionsfollowing thepublication ofpaper I, theauthors believethat it isnecessary tostress that the unifiedhyperbolictheoryproposedin[110,42]wasneverthoughttobean

extension of

theconventionalparabolicNavier– Stokes–Fourier (NSF) theory likeit was intended inthe Maxwell–Cattaneo approach.It should be ratherconsidered asa completelyindependentmodelbasedondifferentprinciples.Forexample,theconstitutivelaws(Newton’slawofviscosity andFourier’slawofheatconduction)oftheNSFtheory entirelyrelyonthe

steady state assumption by

completelyignoring the history(even short) ofhow such a steadystate was reached. Such a steadystate assumption isnot used inour hy-perbolictheory,neitherexplicitlynorimplicitly,andthusitshould beregardedasa genuinelytransientmodel.Infact,in order touseourmodel onemay

not even know about

the existenceofthe famous transportlawsofNewton andFourier. Nevertheless,solutions corresponding toNewtonianflows andFourierheatconductionare realizableinourtheory inthe longwave-lengthapproximation(orinthestiffrelaxationlimit)asdiscussedaboveandwasshownin[42],paperI,through aformalasymptoticexpansionuptofirstordertermsinthedifferentrelaxationtimesappearinginthemodel.

Therefore, one may consider the possibility that the parabolic NSF theory with its steady state assumption is justa particular realizationof the more general hyperbolic model [110,42]. This also becomes transparent after recalling that our hyperbolic theory also includes solid mechanics (nonlinear elastic and elasto-plastic solids) asa particular case, see paper I. Clearly,thenon-NewtonianliquidsfallinbetweentheNewtonian-like andsolid-like behaviorandhencecan also be modeled withour approach via thedependence of therelaxation time on the state parameters. Thisis howeverthe subjectofanongoingresearch.

Eventually, there is an apparent similarity betweenthe extended structure (if comparedwith theEuler equations) of ourhyperbolicmodelandthemacroscopicmodelsinnon-equilibriumgasdynamicsderivedwiththehelpofthemethodof moments[138]fromthegaskinetictheoryofBoltzmann.Nevertheless,onthemathematicalside,therearemoredifferences than similarities between these approaches, and a straightforward comparisonof the solutions is not a trivialtask. For example,itisnotacceptableintheSHTCformalismthatthefluxesserveasstatevariables,butonlydensity-likequantities areadmissible,whileinthemethodofmoments,themomentsofthedistributionfunctionareconstructedinasuchaway that themacroscopicfieldshavethemeaningoffluxes,suchasstresstensor,heatflux,etc. Anotherremarkabledifference is the fact that the methodof moments serves asa method toderive macroscopic equationsfor gasflows solely,while

theapplicationofourmodelisnotrestrictedtogasdynamicsonly,butitincludeselasto-plasticsolidsandliquidsaswell, which are not described by thekinetic gastheory ofBoltzmann. Nevertheless,a detailed comparisonof the approaches in the gasdynamics settingmay lead to a better understanding anda further improvementof the theory anddeserves a detailedinvestigationinthe future.Fornow,we only notethat both approachesgive thesamelow-order termsinthe expansionintheso-calledKnudsennumber(relaxationtimeinourcase)correspondingtotheNSFtheory,whilethehigher order terms,as always, depend on the choice of the closure, which in ourcase corresponds to the choice of the scalar functions,thetotalenergypotential(tobeconvextoguaranteethesymmetrichyperbolicity)andtherelaxationtimes.

1.7. Outline of the paper

The restofthe paperisstructured asfollows.In thefirst part,we concentrateon themathematical principlesofthe SHTCformalism.Inparticular,Section2isdedicatedtothedeparturepointoftheSHTCformalism,namelytheLagrangian systemofmasterequations,whileSection3containstheEulerianformulationofthemastersystemafterthechangeofthe LagrangianvariablesintoEulerianones.InSection3.3,wethenproposeatotalenergypotentialwhichservesastheclosure ofthegoverningequations.In Section4,we summarizetheEuleriansystemofgoverningPDEs whichshallbe usedlater forthenumericalsimulationinSection7.InSection5,wedemonstrateviaaformalasymptoticanalysistherelationofthe proposed hyperbolic model tothe idealmagnetohydrodynamics (MHD) andviscous and resistivemagnetohydrodynamics (VRMHD)equations.Here,we alsopartiallyrepeat theresultsofpaperIforcompleteness.Section 6briefly describesthe family of ADER methods employed in this paper. Eventually, the proposed energy potential and the Eulerian systemof governingPDEs are usedinSection 7where wegive extensivenumerical evidenceoftheapplicability ofthe modeltoa widerangeofelectromagneticflows.SomeconcludingremarksandanoutlooktofutureworkisgiveninSection8.

IntherestofthepaperweusetheEinsteinsummationconventionoverrepeatedindices.

2. SHTCformalismandthemastersystem

ThehyperbolicdissipativetheorydiscussedinthispaperreliesontheSHTCformalism.Thedevelopmentofthe formal-ismstartedin1961afteritwasobserved byGodunov[61,62] thatsome systemsofconservationlawsadmitting an

extra

conservation law also admitaninterestingparametrization

∂

Mpi

∂

t

+

∂

Npji

∂

yj

=

0 (3)

whichallowstorewritethegoverningequationsina

symmetric form

Mpipk

∂

pk

∂

t

+

N j pipk

∂

pk

∂

yj

=

0

,

(4)

where t is the time, yj are the spatial coordinates, pk is the vector of state variables, M

(

pi

)

and Nj

(

pi

)

are the scalar potentials ofthe state variables. Here andin the rest ofthe paper, a potential with the state variables in the subscript should be understood asthe partialderivatives ofthe potential withrespect to thesestate variables. Thus, forexample,

Mpk, N j

pk, Mpipk andN j

pipk in(3)–(4)shouldbe understoodasthefirstandsecondpartial derivativesofthepotentials M and

N

j_{withrespecttothestatevariables}

_p

i,e.g.

M

pk

= ∂

M

/∂

pk,

M

pipk

= ∂

2_M

_/(∂

_p

i

∂

pk

)

,etc. Inthisparametrization,theextraconservationlawhasalwaysthefollowingform

∂(

piMpi

−

M

)

∂

t

+

∂(

piNpji

−

N j

₎

∂

yj

=

0 (5)

and, infact,it isjusta straightforwardconsequenceofthe governingequations(3) andcan be obtainedasa

linear

com-bination of theseequations. Indeed,(5) can be obtained asa sumof the equations(3) multiplied by the corresponding factors pi.

Ifthe potential M

(

pi

)

is a strictly convexfunctionof thestate variables then thesymmetric matrix Mpipk ispositive definiteand(4)becomesa

symmetric hyperbolic system

ofequations[55].

Usually, the generating potential M has the meaning of the generalized pressure while its Legendre transformation

piMpi

−

M has themeaning ofthe totalenergy andthus,(5) isthe totalenergy conservationlaw.

3 _{Hence, the}

observa-tionofGodunovestablishesthe veryimportantconnectionbetweenthewell-posedness oftheequationsofmathematical physicsandthermodynamics.

It wasunderstoodlaterontheexampleoftheidealMHDequations[63] thattheoriginalobservationofGodunov[61]

relates onlyto conservationlawswritten inthe Lagrangian frame, which indeedadmitsa fully

conservative formulation,

4

while thetime evolutionequationsintheEulerianframe haveamorecomplicatedstructure,exceptforthecompressible Eulerequationsofidealfluids.

ThestructureoftheEulerianequationsanditsrelationtothefullyconservativestructureoftheequationsinLagrangian formwasrevealedinaseriesofpapersbyGodunovandRomenski[69,70,65,71,119,120,72].Inparticular,in[65],basedon thegrouprepresentationtheory[59],arathergeneralformoffirstorderPDEswiththefollowingpropertieswasproposed:

•

PDEsareinvariantunderrotations

•

PDEsarecompatiblewithanextraconservationlaw

•

PDEsaregeneratedbyonlyonepotentiallike

M

•

PDEsaresymmetrichyperbolic

•

PDEsareconservativeandgeneratedbyinvariantdifferentialoperatorsonly,suchasdiv,gradandcurl.

OnemaynaturallyquestionhowthisclassofPDEs,whichshallbereferredtoas the

master

system, relatestothemodelsthat describecontinuum mechanicsandwhetheritistoorestrictivetodealwithdissipativeprocessessuchasviscous momen-tumtransfer,heattransfer,resistiveMHD,etc.,typicallydescribedbysecondorderparabolicequations.First,itisimportant to emphasize that invariance underorthogonal transformations andthe existence ofan extra conservationlaw, which is typically thetotal energyconservation,are compulsoryrequirementsforcontinuum mechanics models.Second,asshown recently[110,42],thereisnophysicalreasonimposingthatthedissipativetransportprocessessuchasviscousmomentum transferorheatconductionshouldbeexclusivelymodeledbythesecondorderparabolicdiffusiontheory,buttheycanalso beverysuccessfullymodeledbyamoregeneralframeworkbasedonfirstorderhyperbolicequationswithrelaxationsource terms. Third,afteranalysisofa ratherlargenumberofparticular examplesofcontinuum models [69–71,119,120],itwas shown thatmanymodels fall intotheclass ofSHTCsystems. Amongthem are thecompressibleEulerequationsof ideal fluids,theidealMHDequations,theequationsofnonlinearelasto-plasticity,theelectrodynamicsofmovingmedia,amodel describing superfluidhelium,theequationsgoverningcompressiblemulti-phaseflows,elasticsuperconductors,andfinally also theunifiedfirst orderhyperbolic formulationforfluidandsolid mechanics introducedin [110,42].Inthispaper,we showthatalsotheviscousandresistiveMHDequationscanbecastintotheformofafirstorderSHTCsystem.

The startingpointoftheSHTCformalismisasub-systemoftheLagrangian conservationlawsgivenineqs. (1) of[65], which willbe refereedtoasthe

master system from

nowon.The final governingPDEswritten intheEulerianframe will thenbetheresultofthefollowingsystemofLagrangianmasterequations:

dMvi dt

−

∂

Pi j

∂

yj

=

0

,

(6a) dMPi j dt

−

∂

vi

∂

yj

=

0

,

(6b) dMdi dt

−

ε

i jk

∂

bk

∂

yj

=

0

,

(6c) dMbi dt

+

ε

i jk

∂

dk

∂

yj

=

0

.

(6d)

Here,

v

i isthevelocityofthematter, Pi j isthestresstensor,while

d

i and

b

i aresomevectorsdescribingtheelectricand magneticfields,respectively.

Incontrasttotheclassicalparabolictheoryofdissipativeprocesses,thegoverningequationsinourapproachareall

first

order hyperbolic PDEsandthedissipativeprocesseswill

not be

modeledby

differential terms,

but

exclusively via algebraic

re-laxation source terms, whichwillbespecifiedlaterintheEuleriancase.Thishasthe

important consequence that

thestructure ofthedifferentialtermsandthetypeofthePDEisthe

same in

both,thedissipativeaswellasinthenon-dissipativecase. We recall,that ifthedissipationisexcludedintheclassicalsecondorderparabolicdiffusiontheory,thisthenchangesnot onlythestructureofthePDEs,butalsotheir

type.

Becauseofthisfact,withintheSHTCformalismwecanstudythestructureofthegoverningequationsbyrestrictingour considerationstothenon-dissipativecaseonly.Wealsonotethatifthedissipationsourcetermsareswitchedoff,thenthe modeldescribesanelasticmedium,see

[110,42]

.

4 _{Inthispaper,underfullyconservativeformoftheequationsweunderstandnotonlythedivergenceformoftheequations,i.e.generatedbythe} divergencedifferentialoperator,butratherthattherearenospacederivativesmultipliedbyunknownfunctions,whilealgebraicproductionsourceterms canbepresent.

2.1. Variational nature of the field equations in a moving elastic medium

ItiswellknownthatmanyequationsofmathematicalphysicscanbederivedastheEuler–Lagrangeequationsobtained bytheminimizationofaLagrangian.Asanexample,onecanconsiderthenonlinearelasticityequationsinLagrangian co-ordinates[73].TheclassicalMaxwellequationsofelectrodynamicscanalsobederivedbytheminimizationofaLagrangian withtheuseofthegauge theory[58].Itturns outthat thecouplingofthesetwophysicalobjects inasingle Lagrangian gives usa straightforward way to derive the equations for the electromagnetic field in a moving medium. We start by introducingtwovectorpotentialsandascalarpotential:

x

= [

xi

(

t

,

y

)

],

a

= [

ai

(

t

,

y

)

],

ϕ

(

t

,

y

),

(7) sothat

ˆ

vi

=

∂

xi

∂

t

, ˆ

Fi j

=

∂

xi

∂

yj

,

(8)

ˆ

ei

= −

∂

ai

∂

t

−

∂

ϕ

∂

yi

, ˆ

hi

=

ε

i jk

∂

ak

∂

yj

,

(9)

Here,

t is

time, y

= [

yi

]

andx

= [

xi

]

aretheLagrangianandEulerianspatialcoordinatesrespectively,whilea and

ϕ

arethe conventionalelectromagneticpotentials.

Then,wedefinetheactionintegral

L

=



d ydt

,

(10)

where

= (ˆ

vi

,

F

ˆ

i j

,

e

ˆ

i

,

h

ˆ

i

)

istheLagrangian.

Firstvariationof

L

givesustheEuler–Lagrangeequations

∂

v_ˆi

∂

t

+

∂

_Fˆ i j

∂

yj

=

0

,

(11)

∂

e_ˆi

∂

t

+

ε

i jk

∂

_hˆ k

∂

yj

=

0

,

(12)

∂

_ˆej

∂

yj

=

0

.

(13)

Tothissystem,thefollowingcompatibilityconstraintsshouldbeadded(theyaretrivialconsequencesofthedefinitions

(8)

and

(9)

)

∂ ˆ

Fi j

∂

t

−

∂

v

ˆ

i

∂

yj

=

0

,

∂ ˆ

Fi j

∂

yk

−

∂ ˆ

Fik

∂

yj

=

0

,

(14)

∂ ˆ

hi

∂

t

+

ε

i jk

∂

e

ˆ

k

∂

yj

=

0

,

∂ ˆ

hj

∂

yj

=

0

.

(15)

Inordertorewriteequations

(11)

–(15)intheformofsystem

(6)

,letusintroducethepotential

U as

apartialLegendre transformationoftheLagrangian

dU

=

d

(

v

ˆ

i

ˆvi

+ ˆ

ei

ˆei

− ) = ˆ

vid

vˆi

+ ˆ

eid

eˆi

−

Fˆi jdF

ˆ

i j

−

hˆid

ˆ

hi

=

ˆ

vid

v_ˆi

+ ˆ

eid

eˆi

+

Fˆi jd

(

− ˆ

Fi j

)

+

hˆid

(

−ˆ

hi

).

(16)

Hence,denoting

m

i

=

vˆi,

e

i

=

ˆei, Fi j

= − ˆ

Fi j,

h

i

= −ˆ

hi,wegetthethermodynamicidentity dU

=

Umidmi

+

UFi jdFi j

+

Ueidei

+

Uhidhi

.

Eventually,intermsofthevariables

q

= (

mi

,

Fi j

,

ei

,

hi

)

(17)

dmi dt

−

∂

UFi j

∂

yj

=

0

,

(18a) dFi j dt

−

∂

Umi

∂

yj

=

0

,

(18b) dei dt

−

ε

i jk

∂

Uhk

∂

yj

=

0

,

(18c) dhi dt

+

ε

i jk

∂

Uek

∂

yj

=

0

,

(18d)

whichshouldbesupplementedbystationaryconstraints(13),

(14)

2 and

(15)

2whichnowreadas

∂

Fi j

∂

yk

−

∂

Fik

∂

yj

=

0

,

∂

ei

∂

yi

=

0

,

∂

hi

∂

yi

=

0

.

(19)

System (18) is, in fact, identical to (6). In order to see this, one needs to introduce fluxes as new (conjugate) state variables p

= (

Umi

,

UFi j

,

Uei

,

Uhi

),

(20) whichwedenoteas vi

=

Umi

,

Pi j

=

UFi j

,

di

=

Uei

,

bi

=

Uhi

,

(21)

andanewpotential

M

(

p

)

asaLegendretransformof

U

(

q

)

,i.e.

M

=

miUmi

+

Fi jUFi j

+

eiUei

+

hiUhi

−

U

,

(22)

orbriefly

M

(

p

)

=

q

·

p

−

U

(

q

).

(23)

Afterthat,system

(18)

transformsexactlyto

(6)

,whiletheconstraints

(19)

readas

∂

MPi j

∂

yk

−

∂

MPik

∂

yj

=

0

,

∂

Mdi

∂

yi

=

0

,

∂

Mbi

∂

yi

=

0

.

(24)

Onemayclearlynoteasimilaritybetweentheequations(6c)–(6d)(or

(18c)

–(18d))andtheMaxwellequations.However, becausenoassumptionsabouttheLagrangian

,andthus,aboutthepotentials

U

(

q

)

andM

(

p

)

havebeenmadeyet,these equationsshouldbeconsideredasa

nonlinear generalization of

theMaxwellequations,e.g.see[28,126,1].

We notethat equations(6a)–(6b)and

(6c)

–(6d)(or

(18a)

–(18b)and

(18c)

–(18d)) arenotindependentasitmayseem. They are coupled via the dependenceof the potential M

(

p

)

(or U

(

q

)

) on all the state variables (17). Thiscoupling will emergeinamoretransparentwaywhenweshallconsidertheseequationsintheEulerianframeinSection3.

2.2. Properties of the master system 2.2.1. Energy conservation

Acentralroleinthesystemformulationisplayedbythethermodynamicpotential

U

=

U

(

mi

,

Fi j

,

ei

,

hi

),

(25)

oritsdual

M

=

M

(

vi

,

Pi j

,

di

,

bi

)

(26)

asoneofthemgeneratesthefluxesin

(18)

,whiletheothergeneratesthedensityfieldsin(6).Thepotential

U typically

has themeaningofthetotalenergydensityofthesystem,whileM has themeaningofapressure.

Inaddition,solutionsofthesystem(18)satisfyanextraconservationlaw

dU dt

−

∂

∂

yj



UmiUFi j

+

ε

i jkUeiUhk



=

0 (27)

whichshouldbeinterpretedasthetotalenergyconservation.Intermsofthedualpotential

M and

dualstatevariables(21)

itreadsas d dt



viMvi

+

Pi jMPi j

+

diMdi

+

biMbi

−

M



−

∂

∂

yj



viPi j

+

ε

i jkdibk



=

0

.

(28)

The energyconservationlaw(27)isnot independentbuta consequenceofall theequations (18). Indeed,ifwemultiply eachequationin(18)byacorrespondingfactorandsumuptheresult,weobtainequation (27)identically:

Umi

·

(18a)

+

UFi j

·

(18b)

+

Uei

·

(18c)

+

Uhi

·

(18d)

≡

(27)

.

(29)

Thesameistruefor

(28)

and

(6)

.

2.2.2. Possible interpretation of the state variables

Usually,thederivation ofamodelbegins withthe choiceofstate variables.Inthecontext ofclassicalhydrodynamics, theanswerisuniversal.Thestatevariablesaretheclassicalhydrodynamicfieldssuchasmass,momentum,entropy,ortotal energy.Inanycasewhichisbeyondtheinviscidhydrodynamicssettings,thechoiceofextrastatevariablesisnotuniversal. IntheSHTCformalism,wehoweverfollowadifferentstrategy,whichconsistsoftwostages.Inthefirststage,thegoverning equationsareformulatedbeforeanychoiceofextrastate variableshasbeenmade. ThestructureofthegoverningPDEsis a consequenceofthefivefundamental requirementsformulated earlierinthissection.The physicalmeaningofthe state variablesbecomes clearatthesecond stage,whenwe trytocompare asolutiontothemodelwithspecific experimental observations.Atthisstage,wesimultaneouslyclarifythemeaningofthestatevariablesandlookforanappropriateenergy potential whichcanbe seenalsoasthechoiceoftheconstitutiverelations intheclassicalcontinuum mechanics.Forthe proposedmodel,thisstrategyisrealizedinSection3.3.

Thus,inthissection, wegive onlyapproximateinterpretationsofthestate variableswhile theirprecise meaningswill be givenin Section3.3.Asstatedabove, thespacevariables y

= [

yi

]

canbe treatedastheLagrangian coordinateswhich areconnectedtotheEuleriancoordinatesx

(

t

)

= [

xi

(

t

)]

measuredrelativetoalaboratoryframe bytheequality yi

=

xi

(

0

)

. Itisalsoimpliedthat vi

=

dxi_dt in(6)isthevelocityofthematterrelativetothelaboratoryframe,while

m

i

=

Mvi in(18) hasameaningofageneralizedmomentumdensitywhichmayincludecontributionsfromotherphysicalprocessesandin generaldependsonthespecificationofthepotential

M,

orU . As itwillbe showninSection 3.3,

m

i couplesthematerial momentumand

electromagnetic momentum (Poynting

vector).ThetensorialvariableFi j

=

_∂∂_yxi_j isthedeformationgradient,

e

i and

h

i aretheelectricandmagneticfields,respectively.However,theexactmeaningofthefields

e

i and

h

iwillbeclarified laterinSection3whenweshalldistinguishamongdifferentreferenceframes.

2.2.3. Symmetric hyperbolicity

System

(6)

canberewritteninasymmetricquasilinearfrom

M

∂

p

∂

t

+ N

j

∂

p

∂

yj

=

0

,

(30)

withthesymmetricmatrix

M(

p

)

=

Mpp

= [∂

2M

/∂

pi

∂

pj

]

and

constant symmetric

matrices

N

jconsistingonlyof1,

−

1 and zeros.Moreover,system

(6)

issymmetrichyperbolicifM

(

p

)

isconvex.Inotherwords,theCauchyproblemfor

(6)

(aswell asfor

(18)

)isautomaticallywellposedlocallyintimeforsmoothinitialdata

[29]

.Werecallthattheconvexityof M

(

p

)

is equivalenttotheconvexityof

U

(

q

)

duetothepropertiesoftheLegendretransformation.

2.2.4. q andp-type

state variables

Weemphasizetheverydistinctnatureofthevariablesq andp. Thevariablesq appearinthetimederivativeandhave themeaningofdensities(volumeaveragedquantities).Wethusshallrefertocomponentsofq as

density fields.

Ontheother hand,thevariablesp appearasfluxesinthemastersystem(18)(or(6)),andthuswillbereferredtoas

flux fields (surface

definedquantities),seealsothediscussionin[109].IntheSHTCformalism, thepotential M

(

p

)

(thegeneralizedpressure) andthefluxfieldsp areconjugatequantitiestothepotential

U

(

q

)

(totalenergydensity)andthedensityfieldsq,i.e.they areconnectedbythefollowingidentities

pi

=

Uqi

,

qi

=

Mpi (31)

and

M

=

qiUqi

−

U

,

U

=

piMpi

−

M

.

(32)

Thus,itfollowsfrom(31)thatifwewantthenonlinearchangeofvariables(31)tobeaone-to-onemap,oneshouldrequire thatthepotentials

U

(

q

)

and

M

(

p

)

beconvexfunctionsbecause

Mpp

=

∂

q

∂

p

=



∂

p

∂

q



₋1

=

Uqq

.

(33) 2.2.5. Stationary constraints

Solutionstosystem

(18)

satisfysomestationaryconservationlawsthatarecompatiblewithsystem

(18)

andconditioned bythestructureofthefluxterms:

∂

Fi j

∂

yk

−

∂

Fik

∂

yj

=

0

,

∂

ek

∂

yk

=

0

,

∂

hk

∂

yk

=

0

.

(34)

Thesestationarylawsholdforevery

t

>

0 iftheyarevalidat

t

=

0,andthusshouldbeconsideredastheconstraintsonthe initialdata.Indeed,applyingthedivergenceoperator,forinstance,toequations

(18d)

weobtain

∂

∂

t



∂

hk

∂

yk



=

0

,

(35)

whichyieldsthethirdequationin(34)ifitwasfulfilledattheinitialtime.Theotherlawscanbeobtainedinasimilarway. As weshalldiscusslaterontheexampleoftheEulerianequations,thesituationisratherdifferentintheEuleriansetting, andthe stationaryconstraintslike (34)arenot separate butan intrinsicpartofthestructure ofthegoverning equations writtenintheEulerianframe.

2.2.6. Complimentary structure

Wealsonotea

complimentary structure

ofequations(18)and

(6)

,i.e.thePDEsaresplitintopairs.Ineachpair,avariable appearing inthetimederivative,say ui in(18a),thenappearsinthefluxofthecomplimentaryequationas

U

ui in

(18b)

. Thus,

u

i and Fi j are complimentaryvariables,aswell asei and

h

i.This means,that aphysicalprocess shouldbe always presentedatleastby twostatevariablesandhenceby twoPDEsintheSHTCformalism.Onemaynote acloserelationof such acomplimentarystructure oftheSHTCformalismandtheoddandevenparityofthestatevariableswithrespectto thetime-reversaltransformationinthecontextoftheGENERIC(generalequation ofnonequilibriumreversible–irreversible coupling)formalismdiscussedin[106].

3. MastersystemintheEulerianframe

In this section, we formulate a system of governing equations describing motion of a heat conducting deformable medium (fluid or solid) in the electromagnetic field in the Eulerian frame. This system is obtained as a direct conse-quence ofthemastersystem(18)bymeansoftheLagrange-to-Eulerchangeofvariables: y

→

x. Thistransformationisa nontrivialtask, andthedetailsaregiveninAppendix BfortheelectromagneticfieldequationsandinAppendix Cforthe momentum conservationlawwhilethedetails aboutthe derivationoftheother equationscanbe foundinthe Appendix in [109] orin[65].We give theEulerianformulations usingboth densityfieldsq and fluxfields p. Aswe shallsee, the EulerianequationsdonothavesuchasimplestructureastheLagrangianequations.Nevertheless,westressthatnoneofthe differentialtermswasprescribed“by

hand”,

butallofthemareadirectconsequenceofthe y

→

x variabletransformation solely.

3.1.

(

E,

q

)

-formulation

The mainsystemofgoverningequationsstudiedinthispaperisformulated intermsofq-typestate variables(density fields,seeSection2.2.4)

q

= (

ρ

,

m

,

A

,

e

,

h

,

w

,

σ

),

(36)

andthe totalenergydensity

E(

q

)

=

w−1U , where U is the Lagrangiantotal energydensityintroducedinSection 2,

ρ

is the mass density,

σ

=

ρ

s is theentropy density,s is the specificentropy, m

= [

mi

]

isa generalizedmomentum density whichcouplestheordinarymattermomentumdensity,

ρ

v, withtheelectromagneticmomentumdensity,i.e.thePoynting vector. The exactexpression form willbe givenlater. Matrix A

= [

Aik

]

isthedistortionfield5 (see Paper I), e

= [

ei

]

and

h

= [

hi

]

are thevectorfieldswhichrelate totheelectro-magneticfieldsandwillbespecifiedlater, w

=

ρ

J isthethermal impulsedensity(seePaper I),whichcanbeinterpretedasanaveragemomentumdensityoftheheatcarriers.Thevelocity of themedium,v, isnot aprimary state variableandshouldbe computedfromthe generalizedmomentumm,butalso, according totheSHTC formalism, thevelocity andthegeneralizedmomentum relate toeach other as vi

=

E

mi (see (21) andthediscussionbelow).IntheEuleriancoordinates

x

k,thesystemofgoverningequationsreadsas

∂

ρ

∂

t

+

∂(

ρ

vk

)

∂

xk

=

0

,

(37a)

∂

mi

∂

t

+

∂

∂

xk



mivk

+ δ

ik



ρ

E

ρ

+

σ

E

σ

+

ml

E

ml

+

el

E

el

+

hl

E

hl

−

E



+

Ali

E

Alk

−

ek

E

ei

−

hk

E

hi



=

0

,

(37b)

∂

Aik

∂

t

+

∂(

Ailvl

)

∂

xk

+

vj



∂

Aik

∂

xj

−

∂

Ai j

∂

xk



= −

E

Aik

ρ

θ

1

(

τ1

)

,

(37c)

5 _{Rigorouslyspeaking,}

_{A is}

_{notatensorfieldofrank2,sinceittransformslikeatensorofrank1withrespecttoachangeofcoordinates.Thus,weshall} avoidtocallitthedistortiontensor,butinsteadcallitsimplythedistortionfield.

∂

ei

∂

t

+

∂



eivk

−

viek

−

ε

ikl

E

hl



∂

xk

+

vi

∂

ek

∂

xk

= −

E

ei

η

,

(37d)

∂

hi

∂

t

+

∂



hivk

−

vihk

+

ε

ikl

E

el



∂

xk

+

vi

∂

hk

∂

xk

=

0

,

(37e)

∂

wi

∂

t

+

∂ (

wivk

+

E

σ

δ

ik

)

∂

xk

= −

ρ

E

wi

θ

2

(

τ2

)

,

(37f)

∂

σ

∂

t

+

∂



σ

vk

+

E

wk



∂

xk

=

1

E

σ



1

ρ

θ

1

E

Aik

E

Aik

+

ρ

θ

2

E

wi

E

wi

+

1

η

E

ei

E

ei



≥

0

.

(37g)

Theenergyconservationlaw

∂E

∂

t

+

∂

∂

xk

vk

E

+

vi



ρ

E

ρ

+

σ

E

σ

+

ml

E

ml

+

el

E

el

+

hl

E

hl

−

E



δ

ik

+

Ali

E

Alk

−

ek

E

ei

−

hk

E

hi



+

ε

i jk

E

ei

E

hj

+

E

σ

E

wk



=

0 (38)

isaconsequenceofequations(37),i.e.itcanbeobtainedbymeansofthesummationrule(29).Weemphasizethatinthe numericalcomputationsshownlaterinSection4,wesolvetheenergyequation

(38)

insteadoftheentropyequation

(37g)

, butfromthe point ofview of themodel formulation, the entropyshould be considered amongthe vector ofunknowns becauseitisthecomplementaryvariabletothethermalimpulsew

=

ρ

J,seetheremarkinSection2.2.6andPaper I.

Allequationsinsystem(37)exceptthecontinuityequation6 (37a)andtheheatconductionequation7 (37f)–(37g) orig-inate from the Lagrangian equations with the structure (18). The momentum equation and the distortion equation are derivedfromthepair(18a)–(18b),theelectromagneticfieldequations(37d)–(37e)arederivedfromthepair

(18c)

–(18d).

Theenergyconservationlaw

(38)

istheconsequenceofequations(37),sinceitcanbeobtainedasalinearcombination of all equations(37) withcoefficientsintroduced in the followingsection. As in the Lagrangian frame,these coefficients (multipliers)arethethermodynamicallyconjugatestatevariablesandhavethemeaningoffluxes.

Asdiscussedin[110,42],thedistortionfield A describesdeformabilityandorientationofthecontinuumparticleswhich weassume tohaveafinite(non-zero)lengthscale.Macroscopicflow isnaturallyconsidered astheprocess ofcontinuum particlesrearrangementsintheSHTCmodel.Becauseoftherearrangementsofparticles,thefield A isnotintegrableinthe sense thatitdoesnotrelateEulerianandLagrangiancoordinatesofthecontinuum.Asa result,thefield A islocalandit relatestothedeformationgradient F introducedinSection2.1onlyvia

det

(

F

)

=

1

/

det

(

A

).

(39)

However,ifweconsideraparticularcaseofsystem(37)whenthedissipationtermintherighthandsideof(37c)isabsent, whichcorrespondstoanelasticsolid(e.g.seethelastnumericalexamplein Paper I),thenwehavethat A

=

F−1_.

Forsimplicity,weusethesamenotations

m

i,

e

i and

h

i forthegeneralizedmomentum,electricandmagneticfieldsin both theLagrangian andtheEulerianframework. However, thesefieldsare different,see Appendix B.Forexample,ifwe denotebymL,eL andhL theLagrangianfields,i.e.exactlythosefieldswhichareusedinequations(18a),

(18c)

and

(18d)

, thentheyarerelatedtom,e andh appearingintheEulerianequations(37b),

(37d)

and

(37e)

as

mL

=

w m

,

eL

=

w F−1e

,

hL

=

w F−1h

,

(40) where w

=

det

(

F

)

=

1

/

det

(

A

)

,and F is thedeformationgradientintroduced inSection2.1.Subsequently,theLagrangian totalenergydensity

U relates

totheEuleriantotalenergydensity

E

as

w−1U

(

mL

,

F

,

eL

,

hL

)

=

w−1U

(

w m

,

F

,

w F−1e

,

w F−1h

)

=

E(

ρ

,

m

,

F

,

e

,

h

).

(41)

Here, for brevity, we omit other state variables. For example, ifwe denote by



= [

i j

]

all the terms in the momen-tum flux (37b) except the advective term mivk, it can be shown (e.g. see [65] or Appendix in [109]) that, after the Lagrange-to-Euler transformation,the Lagrangian momentum flux UFi j,see (18a),transforms to the Eulerianmomentum flux



ik

=

ρ

FkjUFi j whichinturn,becauseofthechangeofthestatevariableslike(40),expandsas



ik

= −δ

ikP

−

Ali

E

Alk

+

ek

E

ei

+

hk

E

hi

,

(42)

wherethescalar

6 _{Thecontinuityequationis,infact,aconsequenceofthedistortionequation(37c),e.g.see[72,109],butitisconvenienttoconsiderdensityasan} independentstatevariablewiththecompatibilityconstraint

ρ

=ρ0det(A).

7 _{Adifferentformofthehyperbolicheatconductionispossible,seesystem (38)in[119],whichisfullycompatiblewiththeSHTCformalisminthe} sensethatitsLagrangianequationsbelongtothemastersystem[65].However,bothformsareconsistentintheFourierapproximationandbecausewedo notconsidernon-FourierheatconductionwefollowthehyperbolicheatconductionformulationfromPaper Iinthisstudy.Thedetailedcomparisonofthe heatconduction(37f)–(37g)and[119]isthesubjectofanongoingresearchandwillbepresentedsomewhereelse.