Locally implicit discontinuous Galerkin method for time domain electromagnetics

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domain electromagnetics

Victorita Dolean, Hassan Fahs, Loula Fezoui, Stephane Lanteri

To cite this version:

Victorita Dolean, Hassan Fahs, Loula Fezoui, Stephane Lanteri.

Locally implicit discontinuous

Galerkin method for time domain electromagnetics. [Research Report] RR-6990, INRIA. 2009, pp.37.

�inria-00403741v2�

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Thème NUM

Locally implicit discontinuous Galerkin method

for time domain electromagnetics

Victorita Dolean — Hassan Fahs — Loula Fezoui — Stéphane Lanteri

N° 6990

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Vi toritaDolean

∗

, Hassan Fahs ,Loula Fezoui , Stéphane Lanteri

†

ThèmeNUMSystèmesnumériques ProjetNa hos

Rapportdere her he n°6990Juillet200937pages

Abstra t: In the re ent years, there has been an in reasing interest in dis ontinuous Galerkintimedomain(DGTD)methodsforthesolutionoftheunsteadyMaxwellequations modeling ele tromagneti wavepropagation. One ofthemain features of DGTD methods is theirability to dealwith unstru tured meshes whi h are parti ularly well suited to the dis retizationofthegeometri aldetailsandheterogeneousmediathat hara terizerealisti propagation problems. Su h DGTD methods mostoften relyon expli it time integration s hemes andlead to blo k diagonalmass matri es. However, expli itDGTD methods are also onstrainedbyastability onditionthat anbeveryrestri tiveonhighlyrenedmeshes and when the lo al approximation relies on high order polynomial interpolation. An im-pli it time integration s heme is anatural way to obtain atime domain method whi h is un onditionallystablebut attheexpenseoftheinversionofagloballinearsystematea h timestep. Amoreviableapproa h onsistsin applyinganimpli ittimeintegrations heme lo ally in therened regions of themesh while preserving an expli it time s heme in the omplementarypart,resultinginan hybridexpli it-impli it (or lo allyimpli it) time inte-grationstrategy. Inthis paper,wereport onourre enteortstowardsthedevelopmentof su h ahybrid expli it-impli it DGTD method for solvingthetime domain Maxwell equa-tionsonunstru turedsimpli ialmeshes. Numeri alexperimentsfor2Dand3Dpropagation problemsinhomogeneousandheterogeneousmediaillustratethepossibilitiesofthemethod forsimulationsinvolvinglo allyrened meshes.

Key-words: omputationalele tromagneti s,time domain Maxwell's equations, dis on-tinuousGalerkinmethod,unstru turedtetrahedralmeshes,hybridexpli it-impli its heme.

∗

UniversityofNi e-SophiaAntipolis,J.A.DieudonnéMathemati sLaboratory,CNRSUMR6621, F-06108Ni eCedex,Fran e

†

INRIASophiaAntipolis,Na hos proje t-team, 2004RoutedesLu ioles,BP93,F-06902Sophia An-tipolisCedex,Fran e

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Résumé : Ces dernièresannées, unintérêt roissant aété portéaux méthodes detype Galerkindis ontinuutilisantdesmaillagesnon-stru turés esderniersétantparti ulièrement bien adaptés à ladis rétisation des détails géométriquesqui ara térisentles appli ations réalistes. Lesméthodesde typeGalerkindis ontinu endomainetemporelfontleplus sou-vent appel à des s hémas d'intégration en temps expli ites et sont soumisesà des limites destabilité qui s'avèrent ontraignantesenmaillages lo alementranésoulorsque l'ordre d'interpolationlo alestélevé. Uns hémad'intégrationimpli iteestunevoienaturellepour aboutiràune méthode numérique in onditionnellementstable. Cependant, untels héma onduitàlarésolutiond'unsystèmelinéaireglobalà haquepasdetempseaçantdumême oup undes prin ipaux avantages des formulationsde type Galerkin dis ontinu. Une ap-pro he plus viable onsisteà appliquer un s héma impli ite lo alementdans leszones du maillagesquisontranées,et àpréserverl'utilisationd'un s hémaexpli ite pourles équa-tionsasso iéesauxélémentssituésdanslapartie omplémentairedumaillage. Onobtient ainsiuns hémad'intégrationentempshybrideexpli ite-impli ite(oulo alementimpli ite). Dans e rapport, nousétudionsune méthodeDGTD hybrideexpli ite-impli ite de e type pourla résolution numérique dusystème des équations de Maxwell en domainetemporel sur des maillages non-stru turés en simplexes. Des expérien es numériques en 2D et 3D portant sur des problèmes de propagation en milieux homogènes et hétérogènes permet-tentd'illustrer lespossibilitésd'unetelleméthodepourdessimulationsmettantenjeudes maillageslo alementranés.

Mots- lés : éle tromagnétismenumérique, équationsde Maxwellen domainetemporel, méthodesGalerkindis ontinues,s hémahybrideexpli ite-impli ite.

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Contents

1 Introdu tion 4

2 Continuousproblem 6

3 Dis retization in spa e 7

4 Time dis retization 9

4.1 Expli ittimes heme . . . 10 4.2 Impli ittimes heme . . . 11 4.3 Hybridexpli it-impli it times heme . . . 11

5 Stability studyof thehybrid expli it-impli itDGTD-

P

p

method 13 6 Numeri al assessmentofa ura y in 2D 18 6.1 Propagationofaneigenmodeinaperfe tly ondu ting avity . . . 19 6.2 S atteringofaplanewavebyadiele tri ylinder. . . 19

7 Numeri al and performan eresultsin 3D 22 7.1 S atteringofaplanewavebyanair raft . . . 26 7.2 Exposureof headtissuestoalo alizedsour eradiation . . . 28

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1 Introdu tion

Nowadays, a variety of methods exist for the numeri al treatment of the time domain Maxwell equations, ranging from the well establishedand still prominent nite dieren e time domain (FDTD) methods basedon Yee's s heme [Yee66℄-[TH05℄ to the more re ent niteelementtimedomain(FETD)anddis ontinuousGalerkintimedomain(DGTD) meth-ods[PFC06℄-[HW02℄-[CCR05℄-[FLLP05℄-[MR05℄-[CFP06℄. Theuseofunstru turedmeshes (based on quadrangles ortriangles in twospa e dimensions, and hexahedra ortetrahedra inthree spa edimensions)isanintrinsi feature ofthelattermethods whi h anthus eas-ily dealwith omplex geometriesand heterogeneouspropagation media. Theyalso dene thenaturalroutetotheso- alled

hp

-adaptivesolutionstrategies[DKP

+

07℄. Unfortunately, lo al meshrenement antranslatein averyrestri tivetime stepinorder to preservethe stability of the expli it time integration s hemes whi h are most often adopted in FETD and DGTD methods. There are basi ally two dire tions to ure this e ien y problem. Therstone onsists in usingalo al time steppingstrategy ombinedto anexpli ittime integration s heme, whilethe se ond approa h relies ontheuse of animpli it orahybrid expli it-impli ittimeintegrations heme.

Lo al time stepping s hemes have been studied by several authors. Fumeaux et al. [FBLV04℄ havedesignedsu h astrategyfor adissipativeFVTDmethodonanon-uniform tetrahedralmesh. Theirlo altimesteppings hemeassumesapartitioningoftheunderlying meshbasedonlo algeometri alpropertieswhileensuringlo alstability. Inea hsubdomain, the applied time step is some power of twoof the smallest time step. In [FBLV04℄, the proposed lo al time stepping strategyis applied in onjun tionwith aLax-Wendro time advan ing s heme but, a ordingto theauthors, the developed ideas ould be adapted to othertimes hemesaswell. Analternativemulti- lasslo altimesteppings hemeisproposed byPiperno in [Pip06℄ in the ontextof anon-dissipativeLeap-FrogbasedDGTDmethod. Given that the Leap-Frog s heme an be reformulatedas athree-step Verlet s heme, the proposed strategyisbasedonare ursiveappli ation oftheVerlet s hemeonamulti- lass arrangementof themesh ellswhere atime step

∆t/2

N −k

is used in lass

k

,

N

beingthe totalnumberof lasses. Thetwo- lassstrategyofthisfamilyisprovedto onserveadis rete ele tromagneti energy andto be stableprovided

∆t

issmallenough, whilethelo al time stepping multi- lass strategy is assessed numeri ally in the ontext of the solution of the 2DMaxwell'sequations byaDGTD methodontriangularmeshes. A two- lassstrategyis also onsideredbyCohenal. [CFP06℄intheframeworkofanon-dissipativeDGTDmethod onhexahedralmeshes. Itmakesuseof interpolationstoapproximateunknown eldswhen updatingthe ellsvalues within agiven lass, howeverinterpolations arere ognized to be tooexpensiveintermsof omputtaionaltimefurther motivatingthedevelopmentof multi- lasslo altimesteppingstrategies. Su has hemeisthendevelopedbyessentiallythesame authors in [MPFC08℄ but this time in the ontext of a dissipative version of the DGTD method onsideredin[CFP06℄. Moreover,thisnewstrategyis onstru tedasanadaptation oftheoneproposedbyPipernoin[Pip06℄andinparti ular,itinvolvesare ursiveappli ation oftheLeap-Frogs hemeonamulti- lassarrangementof themesh ellswhereatimestep

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∆t/3

isused in lass

k

. Theresultingmulti- lass lo al timestepping s heme requires 33%less omputationthantheVerlet-basedre ursivemethod. Aremarkthatappliestoall themulti- lasslo altime steppingstrategiesdis ussedso faris thatastability riterionis di ultto obtainandin some asesaredu tionon thesmallesttimestepis ne essaryfor longtimestability. Morere ently,apromisinglo altimesteppingstrategyofarbitraryhigh orderhasbeenproposedbyDiazandGrote[DG07℄forase ondorders alarwaveequation dis retized in spa eby either a ontinuousoradis ontinuousnite elementmethod. The proposed strategy is based on the se ond order Leap-Frog s heme whi h is extended to arbitrarilyhigh order bya modiedequation approa h. Theresulting lo altime stepping method is provedto onserveadis rete energy and expli itCFL onditionsare exhibited forthese ond orderandfourthorder a uratein timemethods. A ordingtotheauthors, thisstrategyextendseasily totheMaxwellequationsin se ondorder formand forvarious dis retizationinspa emethods.

A fewimpli itvariantsof Yee's FDTDmethod havebeendevelopedamong whi h, the alternatingdire tion impli it nite dieren e time domain (ADI-FDTD)method [Nam00℄ whi h is a non-dissipative impli it FDTD method. The ADI-FDTD method oers un- onditional stability with modest omputational overhead despite its impli it formulation be ause itrelies onafa torizationof theimpli it matrixoperatorleadingto theinversion oftri-diagonallinearsystems. A omprehensiveanalysisof thenumeri aldispersionofthe ADI-FDTDmethod ispresentedin[ZC01℄ whereitisshown that,onanon-uniform arte-sianmesh, thetime step an betakenuniformly thesame asthat dened by the oarsest ellwithoutalteringthe a ura yin the nermesh regions. However,in [GLH02℄, Gar ia atal. exhibit a ura y limitationsofthe ADI-FDTDmethod that havenotbeenrevealed by previous studies on numeri al dispersion, by investigating the trun ation error on the timestepdue tothefa torization. Morepre isely,the ADI-FDTDmethod isexpressed as a

O(∆t

2 )

perturbation of an impli it Crank-Ni olson FDTD formulation and it is shown thatsometermsofthetrun ationerrorgrowwith thesquare ofthetimein rement multi-pliedbythespatial derivativesoftheelds, thusgivingriseto potentiallylargenumeri al errors as the time step is in reased. The development of impli it dis ontinuous Galerkin methodsforthesolutionoftimedependentproblemshasbeenlessimpressivethanfortheir expli it ounterparts. In[CDL09℄, animpli itDGTD method is proposed forthe solution ofthe solutionofthe 2DMaxwell equationson triangularmeshes. This method ombines anarbitraryhighorderdis ontinuousGalerkinmethodforthedis retizationinspa ewitha se ondorderCran k-Ni olsons hemefortimeintegration. Atea htimestep,amultifrontal sparseLUmethod isusedforsolvingthelinearsystemresultingfrom thedis retization in spa e. Whenthesimulationsinvolvelo allyrenedmeshes anddespitethe omputational overheadofthesolutionofalinearsystemat ea h timestep,theresultingimpli itDGTD methodallowsforanoti eableredu tionofthe omputingtimewithregardsto itsexpli it ounterpartbasedonaLeap-Frogs heme,for omparablea ura ylevels. However,inthe 3D ase,agloballyimpli itmethodbasedonasparsedire tsolversuersfromlargememory overheads.

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Expli it-impli itmethodsforthesolutionofthesystemofMaxwellequationshavebeen studied byseveralauthors withthesharedgoalof designingnumeri almethodologiesable todealwith hybridstru tured-unstru turedmeshes. Forexample,astable hybrid FDTD-FETD method is onsidered by Rylander and Bondeson in [RB02℄, while Degerfeldt and Rylander[DR06℄proposeaFETDmethodwithstablehybridexpli it-impli ittimestepping working on bri k-tetrahedralmeshes that donot requirean intermediatelayer of pyrami-dalelements. Theimpli itNewmarktime steppings hemeisemployedforthetetrahedral elements whi h allows for lo al mesh renement while avoiding aredu ed time step. For thebri kelements,spatiallumping andexpli ittimesteppingisemployed,resultinginthe appli ationofthestandardnite-dieren etimedomains heme. In[KCGH07℄,theauthors study the appli ation of expli it-impli it Runge-Kutta (so- alled IMEX-RK) methods in onjun tion with high order dis ontinuousGalerkin dis retizations on unstru tured trian-gularmeshes,in theframeworkofunsteady ompressibleowproblems(i.e. thenumeri al solutionofEuler orNavier-Stokesequations). Originallydevelopedto solvethesti oper-atorof onve tion-diusion-rea tion models, IMEX-RK methodsare in this workused for separatingthetimeintegrationof sti andnon-stiportionsof the omputationaldomain withregardstogrid-indu edstiness. Anothernotablefeatureofthisworkistheappli ation ofautomati error-basedtime step ontrollersexploiting thefa t that IMEX-RK methods provideembedded s hemeswhi hallowfortheevaluationofatemporalerror.

Thepresentworkis on ernedwiththedevelopmentofanon-dissipativehybrid expli it-impli it DGTD method for solving the time domain Maxwell equations on unstru tured simpli ial meshes. The hybrid expli it-impli it DGTD method onsidered here has been initially introdu ed by Piperno [Pip06℄. However, to ourknowledge, this hybrid expli it-impli itDGTD method has notbeeninvestigated numeri ally sofar forthe simulationof realisti ele tromagneti wavepropagationproblems. We ondu t su hanumeri al investi-gationherefor2Dand3Dpropagationproblemsinhomogeneousandheterogeneousmedia. We also propose a omplete stability analysis of the method based on energeti onsider-ations, extendinga partial result obtainedin [Pip06℄. The rest of thepaper is organized asfollows: in se tion2,westatetheinitial andboundaryvalueproblem to be solved;the dis retization in spa e by a dis ontinous Galerkin method is dis ussed in se tion 3 while theintegrationintimeis onsideredinse tion4;thestabilityofthehybridexpli it-impli it DGTD method is treated in se tion 5; numeri alresults for 2D and 3D problems are re-spe tivelyreportedinse tions6and7;nally,se tion8 on ludesthis paperanddis usses futureworks.

2 Continuous problem

We onsider the Maxwell equations in three spa e dimensions for heterogeneous linear isotropi media with no sour e. The ele tri eld

E(~x, t) =

~

t

_(E

x

, E

y

, E

z

)

and the mag-neti eld

H(~x, t) =

~

t

_(H

(10)

(

ǫ∂

t

E

~

−

url

H =

~

− ~

J,

µ∂

t

H

~

+

url

E = 0,

~

(1)

wherethe symbol

∂

t

denotesatimederivativeand

J(~x, t)

~

is a urrentsour eterm. These equationsaresetonaboundedpolyhedraldomain

Ω

of

R

3

. Thepermittivity

ǫ(~x)

andthe magneti permeability tensor

µ(~x)

are varying in spa e, time-invariant and both positive fun tions. Our goal is to solve system(1) in adomain

Ω

with boundary

∂Ω = Γ

a

∪ Γ

m

, whereweimposethefollowingboundary onditions:











~n

× ~

E = 0

on

Γ

m

,

~n

_{× ~}

E

₋

r µ

ε

~n

× ( ~

H

× ~n) = ~n × ~

E

in

−

r µ

ε

~n

× ( ~

H

in

× ~n)

on

Γ

a

.

(2)

Here

~n

denotes the unit outward normal to

∂Ω

and

( ~

E

in

, ~

H

in

)

is a given in ident eld. The rst boundary ondition is alled metalli (referring to aperfe tly ondu ting surfa e)whilethese ond onditionis alledabsorbingandtakesheretheformofthe Silver-Müller ondition whi h is a rst order approximation of the exa t absorbing boundary ondition. Thisabsorbing onditionisappliedon

Γ

a

whi hrepresentsanarti ialtrun ation of the omputational domain. Finally, system(1) is supplemented withinitial onditions:

~

E

0 (~x) = ~

E(~x, t)

and

H

~

0 (~x) = ~

H(~x, t)

.

3 Dis retization in spa e

We onsiderapartition

T

h

of

Ω

intoasetoftetrahedra

τ

i

ofsize

h

i

withboundary

∂τ

i

su h that

h = max

τ

_i

∈T

_h

k

isatriangle

a

ik

whi hwewill allinterfa e. Forea hinternalinterfa e

a

ik

,wedenote byS

ik

themeasureof

a

ik

andby

~n

ik

theunitarynormalve tor,orientedfrom

τ

i

to

τ

k

. For theboundaryinterfa es,theindex

h

∪ F

h

B

. Furthermore,weidentify

F

B

h

to

∂Ω

sin e

k∈V

i

S

. Foragivenpartition

T

h

, weseekapproximatesolutionsto(1)inthenitedimensional subspa e:

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V

p

(

T

h

) =

{~v ∈ L

2 (Ω)

3 : v

k|τ

i

∈ P

p

p

insidethe element

τ

i

.

Followingthe dis ontinuous Galerkin approa h, theele tri and magneti elds inside ea h niteelementaresear hedforaslinear ombinations

(~

E

i

, ~

H

i

)

oflinearlyindependent basisve torelds

ϕ

~

ij

, 1

≤ j ≤ d

, where

d

denotes thelo al numberof degreesoffreedom inside

τ

i

. Let

P =

Span

(~

ϕ

ij

, 1

≤ j ≤ d)

. The approximate elds

(~

E

h

, ~

H

h

)

, dened by

(

_{∀i, ~E}

h|τ

i

= ~

E

i

, ~

H

h|τ

i

= ~

H

i

)

are allowed to be ompletely dis ontinuous a ross element boundaries. Forsu h adis ontinuous eld

~

{ ~

U

h

}

ik

. Be auseofthisdis ontinuity,aglobalvariationalformulation annot beobtained. However,dot-multiplying(1)byanygivenve torfun tion

ϕ

~

∈ P

,integrating overea hsingleelement

τ

i

andintegratingbyparts,yields:











Z

τ

i

~

ϕ

_{· ǫ}

i

∂

t

E

~

=

Z

τ

i

url

ϕ

~

· ~

H

−

Z

∂τ

i

~

ϕ

_{· (~}

H

_{× ~n),}

Z

τ

i

~

ϕ

_{· µ}

i

∂

t

H

~

=

−

Z

τ

i

url

ϕ

~

· ~E +

Z

∂τ

i

~

ϕ

_{· (~E × ~n).}

(4)

InEq. (4), wenowrepla ethe exa telds

E

~

∂τ

i

, aspe i treatmentmust beintrodu edsin etheapproximateeldsaredis ontinuousthroughelementfa es,leading to the denition of a numeri al ux. We hoose to use a fully entered numeri al ux, i.e.

∀i, ∀k ∈ V

i

, ~

E

|a

ik

≃ {~E

h

}

ik

, ~

H

|a

ik

≃ { ~

H

h

}

ik

. Themetalli boundary ondition(rst relationof (2)) onaboundaryinterfa e

a

ik

∈ Γ

m

(

k

in the element indexof the titious neighboringelement)isdealtwithweakly,inthesensethattra esof titiouselds

~

E

_k

and

∈ Γ

a

(see [CDL09℄formoredetails). Fromnowon,weassume

Γ

a

=

∅

ex eptinthenumeri alresults se tion where we onsider both internal (i.e. avity) and external propagation problems. Evaluatingthesurfa eintegralsin(4)usingthe enterednumeri alux,andre-integrating bypartsyields:

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









Z

τ

i

~

ϕ

· ǫ

i

∂

t

E

~

i

=

1

2 Z

τ

i

(

url

ϕ

~

· ~

H

i

+

url

H

~

i

· ~ϕ)

−

1 ₂

X

k∈V

i

Z

a

ik

~

ϕ

· (~

H

_k

_{× ~n}

_ik

_),

Z

τ

i

~

ϕ

· µ

i

∂

t

H

~

i

=

−

1

2 Z

τ

i

(

url

ϕ

~

· ~E

i

+

url

E

~

i

· ~ϕ)

+

1

2 X

k∈V

i

E

i

and

H

i

respe tivelythe olumnve tors

(E

il

)

1≤l≤d

and

(H

il

)

1≤l≤d

. Then,(5) isequivalentto:











M

ǫ

i

dE

i

dt

=

K

i

H

i

−

X

k∈V

i

S

ik

H

k

,

M

i

µ

dH

i

dt

=

−K

i

E

i

+

X

k∈V

i

S

ik

E

k

,

(6)

wherethesymmetri positivedenitemassmatri es

M

σ

i

(

σ

standsfor

ǫ

or

µ

),thesymmetri stinessmatrix

K

i

andthesymmetri interfa ematrix

S

ik

(allofsize

d

× d

)aregivenby:

(M

σ

i

)

jl

= σ

i

Z

τ

i

t

_ϕ

_~

ij

· ~ϕ

il

,

(K

i

)

jl

=

1

2 Z

τ

i

t

_ϕ

_~

ij

·

url

ϕ

~

il

+

t

_ϕ

_~

il

·

url

ϕ

~

ij

,

(S

ik

)

jl

=

1

2 Z

a

ik

t

_ϕ

_~

ij

· (~ϕ

kl

× ~n

ik

).

4 Time dis retization

The hoi eofthetimedis retizationmethodisa ru ialstepfortheglobale ien yofthe numeri almethod. Thetemporalintegrationmethods aredividedintotwomajorfamilies: impli itandexpli its hemes. Impli its hemesrequirethesolutionoflargematrixsystems resultinginahigh omputationaleortpertimeiterationandtheviabilityofsu has heme stronglydependsonthee ien yoftheusedlinearsystemsolver.Theadvantageofimpli it s hemesistheirrobustness on erningthe hoi eofthetimestepusedthan anbe hosen

(13)

arbitrarilylargeinthe aseofanun onditionallystables heme. Thus,asimulationrequires onlyasmallnumberoftimeiterations,buteverytimestepisburdenedbyahighnumeri al eort. Expli it s hemes in ontrastare easyto implement, produ egreatera ura y with less omputational eortthan impli itmethods, but are restri tedby astability riterion enfor ing a lose linkage of the time step to the spatial dis retization parameter. This restri tion may result in a large number of iterations per analysis, ea h iteration with a low omputational eort. Then, apossiblealternativeis to ombine thestrengthsof both s hemesbyapplyinganimpli ittimeintegrations hemelo allyintherenedregionsofthe mesh whilepreserving anexpli it times hemein the omplementary part,resultingin an hybridexpli it-impli it (orlo allyimpli it) timeintegrationstrategy.

Thesetoflo alsystemofordinarydierentialequationsforea h

τ

i

(6) anbeformally transformedin aglobalsystem. Tothis end, wesuppose that allele tri (resp. magneti ) unknownsaregatheredina olumnve tor

E

(resp.

H

)ofsize

d

g

= N

T

h

d

where

N

T

h

stands forthenumberofelementsin

T

h

. Thensystem(6) anberewrittenas:











M

ǫ

dE

dt

=

KH

− AH − BH,

M

µ

dH

dt

=

−KE + AE − BE,

(7)

wherewehavethefollowingdenitions andproperties:

M

ǫ

_{, M}

µ

and

K

are

d

g

× d

g

blo k diagonal matri es with diagonal blo ks equal to

M

ǫ

i

, M

µ

i

and

K

i

respe tively. Therefore

M

ǫ

and

M

µ

are symmetri positive denite matri es,and

K

isasymmetri matrix.

A

isalsoa

d

g

(S

ik

)

jl

= (S

ki

)

lj

and then

S

ki

=

t

S

ik

;thus

A

isasymmetri matrix.

B

is a

d

g

× d

g

blo k diagonal matrix, whose non-zeroblo ks are equal to

S

ik

when

a

ik

∈ F

_h

B

. Inthat ase,

(S

ik

)

jl

=

−(S

ik

)

lj

;thus

B

isaskew-symmetri matrix. Consequently,ifweset

S

= K

− A − B

,thesystem(7)rewritesas:











M

ǫ

dE

dt

=

SH

,

M

µ

dH

dt

=

−

t

_SE

_.

(8)

4.1 Expli it time s heme

Thesemi-dis retesystem(8) anbetimeintegratedusingase ond-orderLeap-Frogs heme as:

(14)











M

ǫ

E

n+1

− E

n

∆t

=

SH

n+

1

2 ,

M

µ

H

n+

3

2 − H

n+

1

2 ∆t

!

=

₋

t

_SE

n+1

_.

(9)

Theresultingfullyexpli itDGTD-

P

p

methodisanalyzedin[FLLP05℄whereitisshown thatthemethodisnon-dissipative, onservesadis reteformoftheele tromagneti energy andisstableundertheCFL-like ondition:

∆t

_≤

2 α

,

with

α =

k (M

−µ

₎

1

2 t

S

(M

−ǫ

)

1

2 k,

(10)

wherethematrix

(M

σ

₎

−

1

2

istheinversesquarerootof

M

σ

.

4.2 Impli it time s heme

Alternatively, the semi-dis rete system (8) an be time integrated using a se ond-order Crank-Ni olsons hemeas:











M

ǫ

E

n+1

− E

n

∆t

=

S

H

n

_{+ H}

n+1

2 ,

M

µ

H

n+1

− H

n

∆t

=

−

t

_S

E

n

_{+ E}

n+1

2 .

(11)

Su hafullyimpli itDGTD-

P

p

methodis onsideredin [CDL09℄ forthesolutionofthe 2DMaxwellequations. Inparti ular, theresultingmethodisun onditionallystable.

4.3 Hybrid expli it-impli it time s heme

Asmentionedabove,expli itandimpli ittimes hemebasedmethodshavetheirownadv an-tagesand drawba ks. Whentheunderlyingmesh islo allyrened amoreviableapproa h onsists in applying an impli it time s heme lo ally in the rened regions of the mesh, whilepreservinganexpli ittimes hemeinthe omplementarypart,resultinginanhybrid expli it-impli it (orlo ally impli it) timeintegration strategy. We onsider hereamethod ofthiskindthatwasre entlyproposedbyPipernoin[Pip06℄. Theset ofelements

τ

i

ofthe meshisnowassumedtobepartitionedintotwosubsets: onemadeofthesmallestelements andtheotheronegatheringtheremainingelements. Inthefollowing,thesetwosubsetsare respe tivelyreferred as

S

i

and

S

e

. The distin tion between the twosubsets an bedone a ordingtoageometri althreshold,or/andaphysi al riterionaswell. Notethatthereis noneedofaparti ularassumptiononthe onne tivityofthetwosubsets. Intheproposed hybridtimes heme,thesmallelementsarehandledusingaCrank-Ni olsons hemewhileall otherelementsaretimeadvan ed usingavariantofthe lassi alLeap-Frogs hemeknown astheVerletmethod. IntheVerletmethod,theelds

H

and

E

aredenedatthesametime

(15)

stationandtimeintegrationpro eedsinthreesub-steps: (1)

H

istimeadvan ed from

t

n

to

t

n+

1

2

withtime step

∆t/2

,(2)

E

istime advan edfrom

t

n

to

t

n+1

withtime step

∆t

and, (3)

H

is time advan ed from

t

n+

1

2

to

t

n+1

withtime step

∆t/2

. Then, starting from the valuesoftheeldsattime

t

n

_{= n∆t}

,theproposed hybridexpli it-impli ittimeintegration s heme onsistsinthree sub-steps:

(1) the omponentsof

H

and

E

asso iatedtotheset

S

e

aretimeadvan edfrom

t

n

to

t

n+

1

2

withtimestep

∆t/2

usingapseudo-forwardEulers heme,

(2) the omponentsof

usingthereversedpseudo-forwardEulers heme.

Inorder tofurther des ribethis s heme, weintrodu eadditionaldenitions. First,the problemunknownsarereorderedas:

E

₌

E

_e

E

_i

and

H

=

H

_e

H

_i

,

where sub-ve tors with an e subs ript (respe tively, an i subs ript) are asso iated to the lements of the set

S

e

(respe tively, the set

S

i

). We dedu efrom this partitioning of the unknownve torsthefollowingde ompositions ofthesystemmatri es:

M

ε

₌

_M

ε

e

O

M

i

ε

, M

µ

=

_M

µ

e

O

M

µ

i

,

K

₌

_K

e

O

K

_i

, B =

_B

e

O

B

_i

.

where

M

ε

e/i

and

M

µ

e/i

aresymmetri positivedenitematri es,

K

e/i

aresymmetri matri es and

B

e/i

areskew-symmetri matri es. Thematrix

A

whi hinvolvestheinterfa ematri es

S

ik

isde omposedas:

A

₌

_A

ee

A

ei

A

_ie

A

_ii

,

where

A

ee

and

A

(16)











M

ε

e

dE

e

dt

=

S

e

H

e

− A

ei

H

i

,

M

µ

e

dH

e

dt

=

−

t

_S

e

E

e

+ A

ei

E

i

,











M

ε

i

dE

i

dt

=

S

i

H

i

− A

ie

H

e

,

M

µ

i

dH

i

dt

=

−

t

_S

i

E

i

+ A

ie

E

e

.

(12)

Then,theproposedhybridexpli it-impli it algorithm onsistsinthefollowingsteps:











M

µ

_e

H

n+

1

2 e

− H

n

e

∆t/2

!

=

₋

t

_S

e

E

n

e

+ A

ei

E

n

i

,

M

ε

_e

E

n+

1

2 e

− E

n

e

∆t/2

!

=

S

_e

H

n+

1

2 e

− A

ei

H

n

i

,











M

ε

i

E

n+1

i

− E

n

i

∆t

=

S

_i

H

n+1

i

+ H

n

i

2 − A

ie

H

n+

1

2 e

,

M

µ

i

H

n+1

i

− H

n

i

∆t

=

₋

t

_S

i

E

n+1

i

+ E

n

i

2 + A

ie

E

n+

1

2 e











M

ε

e

E

n+1

_e

_{− E}

n+

1

2 e

∆t/2

!

=

S

_e

H

n+

1

2 e

− A

ei

H

n+1

i

,

M

µ

_e

H

n+1

e

− H

n+

1

2 e

∆t/2

!

=

₋

t

_S

e

E

n+1

e

+ A

ei

E

n+1

i

.

(13)

5 Stability study of the hybrid expli it-impli it

DGTD-P

_p

method

In[Pip06℄,theauthorshowsthatthehybridexpli it-impli its heme(13)fortimeintegration of thesemi-dis rete system (8) asso iated to the DGTD-

P

p

method exa tly onservesthe followingquadrati formofthenumeri alunknowns

E

n

e

,

E

n

i

,

H

n

e

and

H

n

i

:

(17)

E

n

=

_E

e

n

+

E

i

n

+

E

h

n

with











E

n

e

=

t

E

n

e

M

ε

e

E

n

e

+

t

H

n+

1

2 e

M

µ

e

H

n−

1

2 e

,

E

n

i

=

t

E

n

i

M

ε

i

E

n

i

+

t

H

n

i

M

µ

i

H

n

i

,

E

n

h

=

−

∆t

2

4 t

_H

n

i

t

A

ei

(M

ε

e

)

−1

A

ei

H

n

i

,

(14)

asfar as

Γ

a

=

∅

. However,the ondition under whi h

E

n

is a positive denite quadrati formandthusrepresentsadis reteform oftheele tromagneti energyisnotgiven. Inthe followingwestatea onditionontheglobaltimestep

∆t

∆t

_≤

2 α

e

+ max(β

ei

, γ

ei

)

with











α

e

=

k (M

ε

e

)

−

1

2 S

e

(M

µ

e

)

−

1

2 k,

β

ei

=

k (M

ε

e

)

−

1

2 A

ei

(M

µ

i

)

−

1

2 k,

γ

ei

=

k (M

µ

e

)

−

1

2 A

ei

(M

ε

i

)

−

1

2 k,

(15)

where

k . k

denotesamatrixnormandthematrix

(M

σ

e/i

)

−

1

2

istheinverseofthesquareroot ofthe matrix

M

σ

e/i

(

σ

standsfor

ε

or

µ

).

Proof. Werewritethelast andtherstrelationsofthehybridexpli it-impli it algorithm (13)respe tivelyas:











M

µ

_e

H

n−

1

2 e

= M

µ

e

H

n

e

−

∆t

2 (

−

t

_S

e

E

n

e

+ A

ei

E

n

i

) ,

M

µ

_e

H

n+

1

2 e

= M

µ

e

H

n

e

+

∆t

2 (

−

t

_S

e

E

n

e

+ A

ei

E

n

i

) .

(16)

Multiplyingtherstrelationof(16)by

t

_H

n+

1

2 e

andusingtheexpressionof

t

_H

n+

1

2 e

from these ondrelationof(16)leadsto:

t

_H

n+

1

2 e

M

µ

e

H

n−

1

2 e

=

t

H

n

e

M

µ

e

H

n

e

−

∆t

2

4 t

₍

−

t

_S

e

E

n

e

+ A

ei

E

n

i

) (M

µ

e

)

−1

(

−

t

S

e

E

n

e

+ A

ei

E

n

i

) .

(18)

E

n

e

=

k (M

ε

e

)

1

2 E

n

e

k

2 +

t

_H

n+

1

2 e

M

µ

e

H

n−

1

2 e

=

k (M

ε

e

)

1

2 E

n

e

k

2 +

k (M

µ

e

)

1

2 H

n

e

k

2 −

∆t

2

4 k (M

µ

e

)

−

1

2 (

−

t

S

e

E

n

e

+ A

ei

E

n

i

)

k

2 .

Dentingby

Π =

k (M

µ

e

)

−

1

2 (

−

t

S

e

E

n

e

+ A

ei

E

n

i

)

k

2

,wehavethat:

Π

_{≤ k (M}

µ

e

)

−

1

2 t

S

e

E

n

e

k

2 +

k (M

µ

e

)

−

1

2 A

ei

E

n

i

k

2 +

2 _{k (M}

µ

e

)

−

1

2 t

S

e

E

n

e

kk (M

µ

e

)

−

1

2 A

ei

E

n

i

k

≤ α

2 e

k (M

ε

e

)

1

2 E

n

e

k

2 +γ

ei

2 k (M

ε

i

)

1

2 E

n

i

k

2 +

2α

e

γ

ei

k (M

ε

e

)

1

2 E

n

e

kk (M

ε

i

)

1

2 E

n

i

k

≤ (α

2 e

+ α

e

γ

ei

)

k (M

ε

e

)

1

2 E

n

e

k

2 +(γ

2 ei

+ α

e

γ

ei

)

k (M

ε

i

)

1

2 E

n

i

k

2 ,

(17)

where the last relationhas beenobtained using theinequality

2ab

≤ a

2 _{+ b}

2

. Wededu e from(17)that:

E

n

e

≥

1 ₋

∆t

2

4 (α

2 e

+ α

e

γ

ei

)

k (M

ε

e

)

1

2 E

n

e

k

2 ₊

k (M

µ

e

)

1

2 H

n

e

k

2 −

∆t

2

4 (γ

2 ei

+ α

e

γ

ei

)

k (M

ε

i

)

1

2 E

n

i

k

2 _.

(18)

Fortheimpli itpartofthedis reteele tromagneti energy(14)wehave:

E

i

n

=

k (M

ε

i

)

1

2 E

n

i

k

2 +

_{k (M}

µ

_i

)

1

2 H

n

i

k

2 .

(19)

Gathering(18)and (19)yields:

E

n

e

+

E

i

n

≥

1 −

∆t

2

4 (α

2 e

+ α

e

γ

ei

)

k (M

ε

e

)

1

2 E

n

e

k

2 +

1 −

∆t

2

4 (γ

2 ei

+ α

e

γ

ei

)

k (M

ε

i

)

1

2 E

n

i

k

2 +

k (M

µ

e

)

1

2 H

n

e

k

2 +

k (M

µ

i

)

1

2 H

n

i

k

2 .

(20)

(19)







α

2 e

+ α

e

γ

ei

=

(α

e

+ γ

ei

)

2 − α

e

γ

ei

− γ

ei

2 γ

2 ei

+ α

e

γ

ei

=

(α

e

+ γ

ei

)

2 − α

e

γ

ei

− α

2 e

,

wemodify(20)as:

E

n

e

+

E

i

n

≥

1 −

∆t

2

4 (α

e

+ γ

ei

)

2 k (M

ε

e

)

1

2 E

n

e

k

2 +

1 −

∆t

2

4 (α

e

+ γ

ei

)

2 k (M

ε

i

)

1

2 E

n

i

k

2 +

∆t

2

4 γ

ei

(α

e

+ γ

ei

)

k (M

ε

e

)

1

2 E

n

e

k

2 ₊

∆t

2

4 α

e

(α

e

+ γ

ei

)

k (M

ε

i

)

1

2 E

n

i

k

2 ₊

k (M

µ

e

)

1

2 H

n

e

k

2 +

k (M

µ

i

)

1

2 H

n

i

k

2 _.

(21)

Finally,thehybridpartofthedis reteele tromagneti energy(14)yields:

E

n

h

=

−

∆t

2

4 t

_H

n

i

(M

µ

i

)

1

2 (M

µ

i

)

−

1

2 t

A

ei

(M

ε

e

)

−

1

2 (M

ε

e

)

−

1

2 A

ei

(M

µ

i

)

−

1

2 (M

µ

i

)

1

2 H

n

i

≥ −

∆t

2

4 k (M

ε

e

)

−

1

2 A

ei

(M

µ

i

)

−

1

2 k

2 k (M

µ

i

)

1

2 H

n

i

k

2 .

(22)

(20)

E

n

₌

E

n

e

+

E

i

n

+

E

h

n

≥

1 ₋

∆t

2

4 (α

e

+ γ

ei

)

2 k (M

ε

e

)

1

2 E

n

e

k

2 ₊

1 ₋

∆t

2

4 (α

e

+ γ

ei

)

2 k (M

ε

i

)

1

2 E

n

i

k

2 ₊

1 ₋

∆t

2

4 β

2 ei

k (M

µ

i

)

1

2 H

n

i

k

2 +

∆t

2

4 γ

ei

(α

e

+ γ

ei

)

k (M

ε

e

)

1

2 E

n

e

k

2 +

∆t

2

4 α

e

(α

e

+ γ

ei

)

k (M

ε

i

)

1

2 E

n

i

k

2 +

k (M

µ

e

)

1

2 H

n

e

k

2 +

k (M

µ

i

)

1

2 H

n

i

k

2 .

(23)

Finally,usingtherelation:

1 ₋

∆t

2

4 β

2 ei

= 1

−

∆t

2

4 (α

e

+ β

ei

)

2 + (α

2 e

+ 2α

e

β

ei

),

allowsto obtainthatunder the onditions:

∆t

_≤

2 α

e

+ γ

ei

and

∆t

≤

2 α

e

+ β

ei

,

thatis:

∆t

≤

_{max (α}

2 e

+ γ

ei

, α

e

+ β

ei

)

=

2 α

e

+ max (γ

ei

, β

ei

)

P

p

methodisdedu edfroma riterionwhi hisessentiallytheoneobtainedforthefullyexpli it methodhererestri tedtothesubsetofexpli itelements

S

e

,augmentedbytwoterms involv-ingelementsoftheimpli itsubset

S

ε

and

µ

are pie ewise onstantthenwehavethat:

M

ε

e,i

= D

ε

M

e,i

and

M

µ

e,i

= D

µ

M

e,i

,

where

D

ε

and

D

µ

arediagonalmatri eswhoseentries aretheelementwisevalues

ε

i

and

µ

i

respe tively,and:

(21)

α

e

=

k D

c

kk (M

e

)

−

1

2 S

e

(M

e

)

−

1

2 k , β

ei

= γ

ei

=

k D

c

kk (M

e

)

−

1

2 A

ei

(M

i

)

−

1

2 k,

where

D

c

denotesthediagonalmatrixwhoseentriesaretheelementwisevaluesofthe prop-agationspeed

c

i

= 1/√ǫ

i

µ

i

. Wehavethat

k D

c

k

isequalto

c

max

= max (c

i

)

i.e. tothe spe tralradiiof

D

c

. Inthat ase,thestability onditionwrites:

c

max

∆t

_≤

2 α

e

+ β

ei

≤

2 α

e

,

whi h showsthatthestabilityoftheexpli itpartofthealgorithm isguaranteed. In[FLLP05℄,thefollowinglo al ondition:

∀j, ∀k ∈ V

j

,

c

j

∆t

2α

j

+ β

jk

max

r µ

j

µ

k

,

r ε

j

ε

k

<

4V

j

∈ V

i

)

aredimensionless onstantssu h that:

∀~

X

_{∈ P,}











k

url

X

~

k

τ

i

≤

α

i

P

i

V

i

k~

X

_k

_τ

i

,

k~

X

_k

2 a

ik

≤

β

ik

S

ik

V

i

k~

X

_k

2 τ

i

,

(25)

where and

k~

X

k

a

ik

su hthat

τ

i

∈ S

e

and

τ

k

∈ S

i

orvi eversa).

6 Numeri al assessment of a ura y in 2D

In this se tion, the a ura y of the hybridexpli it-impli it DGTD-

P

p

method is assessed numeri ally by applying to the solution of the homogeneous sour e free 2D TM Maxwell equations:

∂H

x

∂t

+

∂E

z

∂y

= 0 ,

∂H

y

∂t

−

∂E

z

∂x

= 0 ,

∂E

z

∂t

−

∂H

y

∂x

+

∂H

x

∂y

= 0,

(26) andby onsideringpropagationproblemsforwhi hananalyti alsolutionisknown.

(22)

6.1 Propagation of an eigenmode in a perfe tly ondu ting avity Thersttest problem is thepropagation ofaneigenmode in aunitarysquare avitywith perfe tly ondu tingwalls. Althoughsimpleatrstglan e,thistestproblemisinterestingin thepresent ontextbe ausethereis nophysi alme hanismofdissipationof thenumeri al error and thus it is parti ularly hallenging for assessing the a tual a ura y of a given method. Thehybridexpli it-impli it DGTD-

P

p

method isusedin onjun tionwitha non-uniformtriangularmesh (seeFig.1)whi h onsistsof1400verti esand2742triangles. In this ase, the minimum and maximum values of the time step are respe tively equal to

(∆t)

m

= 1.44

pi ose onds and

(∆t)

M

= 235.33

pi ose onds (the ratio between the two values

δ = (∆t)

M

/(∆t)

m

is equalto162).

Forthisproblem,thegeometri riterion

c

g

usedforthedenitionofthesubsets

S

e

and

S

i

is the areaof a triangle. Thedistribution of the element areafor the triangular mesh athand isshownonFig.2. Fromthisdistribution,severalvaluesof

c

g

havebeensele ted for the separationthreshold whi h are summarized in Tab. 1. In this table,

|S

i

|

and

|S

i

inthetable). Forthepurelyexpli ittimes heme, the time step used in the simulations is CFL

e

× (∆t)

m

where CFL

e

is the CFL number dening thestabilitylimitofthe fullyexpli itDGTD-

P

p

methodand isrespe tivelyequal to0.2and0.3for

p

method is not a trivial task. It requires a detailed analysis of the numeri al dispersionerror of thehybridexpli it-impli it DGTD-

P

p

method and thedenition of an errorestimatorallowingthedevelopmentofanauto-adaptivenumeri almethodology. These taskshavenotbeen onsideredsofarandarepostponedforafuture study.

6.2 S attering of a plane wave by a diele tri ylinder

Wepresenthereresultsforthes atteringofaplanewavewithfrequen yF=300MHzbya diele tri ylinder. Thistest problem hasbeen onsidered in severalworks su h as[CD03℄ wheretheexpressionoftheanalyti alsolutionisdetailed. Theinternal ylinderhasaradius

r

0 = 0.6

mandbounds amaterialwithrelativepermittivity

ǫ

2 = 2.25

. The omputational domain

Ω

is bounded by asquare of side length

a = 3.2

entered at

(0, 0)

. A rst order Silver-Müllerabsorbing onditionisappliedontheboundaryofthesquare. Anon-uniform meshisusedwhi h onsistsof4108verti esand8054triangles. Theminimumandmaximum

(23)

X

Y

0

0.5

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure1: Non-uniformtriangularmesh ofaunitarysquare avity.

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0

500 1000

1500

2000

2500

3000

Geometric criterion

Triangle index

(24)

0.001

0.01

0.1

0 1e-08

2e-08

3e-08

4e-08

5e-08

6e-08

7e-08

8e-08

9e-08

L2 error

Time (in s)

Explicit

Hybrid - c=1.0e-05

Hybrid - c=5.0e-05

Hybrid - c=1.0e-04

Hybrid - c=5.0e-04

Hybrid - c=1.0e-03

1e-05

0.0001

0.001

0.01

0.1

1

0 1e-08

2e-08

3e-08

4e-08

5e-08

6e-08

7e-08

8e-08

9e-08

L2 error

Time (in s)

Explicit

Hybrid - c=1.0e-05

Hybrid - c=5.0e-05

Hybrid - c=1.0e-04

Hybrid - c=5.0e-04

Hybrid - c=1.0e-03

Figure 3: Eigenmode in a unitarysquare avity: time evolution of the L2 error. Hybrid expli it-impli itDGTD-

P

1

method (top)andDGTD-

P

2

method (bottom).

(25)

Method

c

g

threshold

|S

i

|

|S

e

|

(∆t)

r

(pi ose onds) CFL

i

DGTD-

P

1

Table 1: Eigenmode in aunitary square avity. Simulation ongurations for the hybrid expli it-impli itDGTD-

P

p

method.

valuesoftherenormalizedtimesteparerespe tivelyequalto

(∆t)

m

= 2.09

pi ose ondsand

(∆t)

M

= 309.63

pi ose onds (the ratiobetweenthe twovalues

δ = (∆t)

M

/(∆t)

m

is equal to 148). As forthe previousproblem, the geometri riterion

c

g

used forthe denition of thesubsets

S

e

and

S

i

istheareaofatriangle. Thedistributionoftheelementareaforthe triangularmeshathandisshownonFig.4. Theminimumandmaximumvaluesofthearea ofatrianglearerespe tivelyequalto

0.25 × 10

−6

m

2

and

0.65 × 10

−2

m

2

. Wehavesele ted

10 −4

m

2

as athreshold valueforthe geometri riterion

c

g

whi h yields

|S

e

| = 5, 745

and

|S

i

| = 2, 309

. Thesimulationtimehasbeensetto10periodsofthein identwave.

Contourlinesof

E

z

fortheanalyti alsolutionandthesolutionresultingfromthehybrid expli it-impli it DGTD-

P

2

method areshownonFig.5. On Fig.6we omparethe

x

7 Numeri al and performan e results in 3D

(26)

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0 1000

2000

3000

4000

5000

6000

7000

8000

9000

Geometric criterion

Triangle index

Figure4: S atteringofaplanewavebyadiele tri ylinder: distribution ofthegeometri riterion.

X

Y

-1

0

1 -1.5

-1

-0.5

0

0.5

1

1.5 EZ

1.85

1.7

1.55

1.4

1.25

1.1

0.95

0.8

0.65

0.5

0.35

0.2

0.05 -0.1

-0.25

-0.4

-0.55

-0.7

-0.85

-1

-1.15

-1.3

-1.45

X

Y

-1

0

1 -1.5

-1

-0.5

0

0.5

1

1.5 EZ

1.95

1.8

1.65

1.5

1.35

1.2

1.05

0.9

0.75

0.6

0.45

0.3

0.15 -0.15

-0.3

-0.45

-0.6

-0.75

-0.9

-1.05

-1.2

-1.35

-1.5

Figure 5: S atteringofaplane wavebyadiele tri ylinder. Contourlinesof

E

z

after 10 periods. Analyti alsolution(left)andhybridexpli it-impli itDGTD-

P

2

method(right).