Second order accurate asynchronous scheme for modeling linear partial differential equations

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Second order accurate asynchronous scheme for

modeling linear partial differential equations

Asma Toumi, Guillaume Dufour, Ronan Perrussel, Thomas Unfer

To cite this version:

Asma Toumi, Guillaume Dufour, Ronan Perrussel, Thomas Unfer. Second order accurate

asyn-chronous scheme for modeling linear partial differential equations. Applied Numerical Mathematics,

Elsevier, 2017, 121, pp.115-133. �10.1016/j.apnum.2017.06.014�. �hal-01881736�

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Second

order

accurate

asynchronous

scheme

for

modeling

linear

partial

differential

equations

✩

Asma Toumi

a

,

∗

,

Guillaume Dufour

a

,

1

,

Ronan Perrussel

b

,

Thomas Unfer

b a_ONERA,₂_avenue_Edouard_Belin,₃₁₄₀₀_Toulouse,_France

b_{LAPLACE-ENSEEIHT}_2,_rue_Charles_Camichel_BP_7122,₃₁₀₇₁_Toulouse_Cedex_7,_France

We propose an asynchronous methodfor the explicitintegration of multi-scale partial differentialequations.Thismethodisrestrictedbyalocal CFL(CourantFriedrichsLewy) condition rather than the traditional global CFL condition. Moreover, contrary to other local time-stepping(LTS) methods,the asynchronousalgorithmpermits the selection of independenttimestepsineachmeshelement.WederivedanasynchronousRunge–Kutta 2(ARK2)schemefromastandardexplicitRunge–Kuttamethodandweproved thatthe ARK2 schemeis secondorderconvergent. Comparing withthe classicalintegration, the asynchronousschemeiseffectiveintermsofcomputationtime.

1. Introduction

Numerical simulationhasbecome a centraltool for themodeling ofmanyphysicalsystems (ﬂuiddynamics,plasmas, electromagnetism,etc.).Multi-scalephenomenamaketheintegrationofthesephysicalsystemsdiﬃcultintermsofaccuracy andcomputation time. Time-steppingintegration techniquesused forsolving suchproblems generallyfall intotwo cate-gories:explicitandimplicitschemes.Intheexplicitschemes,allunknownvariablesarecomputedatthecurrenttimelevel fromquantitiesalreadyavailable.TimestepisthenlimitedbythemostrestrictiveCFLconditionoverthewhole computa-tiondomain.Theimplicitmethodallowtoovercomethistimestepconstraintbutcomeswithanadditionalcomputational costarisingfromsolvinglargelinearsystems.Thustheimplicitapproachbecomeseffectiveintermsofcomputationalcost onlyifthetimestepsusedarequitelargewithrespecttotheCFLcondition.Howeverthisisnotalwayspossible,especially whenstronglycoupledandmulti-scale phenomenasuchasmicrowaveplasmagenerationarestudied.

Toovercomesuchproblems,anumberoflocaltime-steppingapproacheshavebeendeveloped.Thesemethodsare re-strictedby a local CFL condition ratherthan the traditionalglobalCFL condition. In [14], Osher andSanders proposed a local time-steppingscheme forone-dimensional scalar conservation laws.They gavea thorough analysis ofa ﬁrst order spatial discretization witha localforward Euler time-stepping scheme. This scheme allows each element to take either an entire time step or some ﬁxed number ofsmaller steps. Tang andWarneck proposed in [17] a class of high resolu-tion local time step schemesfor hyperbolic conservations by projecting the solutionincrements ateach localtime step. Savcenco et al.[15] constructed a multirate schemefor parabolicproblems. This schemewas obtainedby adaptation of

✩ _This_work_was_funded_by_the_French_National_Research_Agency_under_contract_{No. ANR-11-MONU-0019.}

*

Correspondingauthor.

E-mailaddresses:[email protected](A. Toumi),[email protected](G. Dufour),[email protected](R. Perrussel),

[email protected](T. Unfer). 1 _Fax:₊₃₃₍₀₎₅₆₂₂₅₂₅_93.

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an implicitRosenbrock-type scheme.The ideais tocompute apredictionforall thecells ofthemeshwiththetrapezoid formula. Then the finemeshis updated usinginformationfromthecoarse meshby quadraticinterpolation. Dawsonand Kirby[2],developedupwindmethodsforsolvingconservationlawswhichallowlocaltimerefinementtobecoupledwith localspatialrefinement.Inthatschemealimiterisappliedwhichisadaptedtotheoutcomeofpreviousstages.W. Hunds-dorfer etal.[8],notedthat severalmultirate schemes forconservationlaws haveone ofthefollowing defects: thereare schemes thatare locallyinconsistant,e.g.[14,2],andschemes thatare notmass-conservative,e.g. [17,15].Theydiscussed these two defects for one-dimensional conservationlaws. In the same context, an error analysisis presented in [7], for explicitpartitionedRunge–Kuttamethodsandmultiratemethodsappliedtoconservationlaws.Hundsdorferandal.showed that the differentmultirate methods studiedin[8] leadto orderreduction ofthe schemes.Toguarantee mass conserva-tion, flux-based decompositions are studied. In this case, the accuracy may deteriorate. Another approach developed by BergerandOliger[1]involvesautomaticallytakingsmallertimestepswherethemeshisrefined.Intheirapproachrefined grids arelaid overregions of the coarsemesh. Informationis then passed betweenthe grids by means ofinjectionand interpolation. Flaherty etal.[3]developed a parallel,adaptive discontinuous Galerkinmethodwitha localforward Euler schemewhichreliesoninterpolatingvaluesintime atinterfacesbetweentimestepsofdifferentsizes.Thisscheme, how-ever, doesnot appear to conserveflux along these interfaces.Also, only firstorder intime methodsare discussed. Note that for thesedifferentalgorithms, local time steps are usually selectedto be fractions ofthe globaltime step. Recently a newtime integrationapproachhasbeenappliedtoequationsofnon-linearelastodynamics [9].Itisbasedonadiscrete spacetimeformofHamilton’svariationalprinciple.Thisalgorithmpermitstheselectionofindependenttime stepsineach meshelement.However,thisapproachisonlyapplicabletoHamiltoniansystems.In[11]OmelchenkoandKarimabadi pre-sentedan asynchronousapproachtoa diffusion–advection–reactionequationinone dimension.Theirmethodisbasedon discrete-eventsimulation(DES). Theydefinedtheconcept of“fluxcapacitor”, avariableinwhichtheystoredtheflux be-tween two cells until a later event. Mass conservationis guaranteed whenthe neighboring cell isupdated andthe flux capacitors areemptied.Inordertoimprovetheaccuracy oftheirasynchronous method,Omelchenko andKarimabadi de-velopedin [12]a second orderDESalgorithmandshowedthat,atleastnumerically, thesecond orderisindeedattained on one-dimensionalgasdynamicstestproblems.Morerecently,theyextendedin[13] theirDESalgorithmtomultiple di-mensions inthecaseof“logicallyuniformmeshes”. In[16],V.A.Semiletov andS.A.Karabasovproposed an asynchronous time-stepping algorithm for the Compact Accurately Boundary Adjusting high-REsolutionTechnique (CABARET), which is an Eulermethod fornon-linear aeroacoustic problems. Numericaltests up to 3D show that the asynchronous algorithm maintains thesamesecond-orderconvergencerateastheoriginal single-timesteppingschemeandthat italsodecreases the absolutenumerical error. However, the question of the performance in terms of computational time remains open. In [10]V.A.Semiletov andS.A.Karabasov adaptedtheir asynchronous methodforsolvingfluid dynamicsequations.Their approach isbasedon atransformationof thegoverningequationsinspaceandtime inorderto rewritethe initial prob-lem on a uniform Cartesian grid, with adapted conditions at the grid interfaces. As for the numerical implementation of their scheme, the authors combined their new asynchronous method with their CABARET scheme presented in [16] which results ina quasi-second orderscheme.The correspondingnumericaltestswere limitedto two-dimensional prob-lems.

Inthispaperwefocusontheasynchronous methodproposedbyone oftheco-authorT.Unfer.Hismethoduses local stabilitycriteriaandcanbeusedonanarbitrarymesh.In[22],hedevelopedanupwindasynchronousforwardEulerscheme forthetransportequationinonedimension.Westudiedtheextensionofthismethodtoanarbitraryspacedimensionand we proved through a thoroughanalysis that the asynchronous scheme isfirst order convergent. In addition,we noticed thattheasynchronousschemereducesnumericaldiffusionandCPUtimeincomparisontotheclassicalfirstorderscheme. Toimprovetheconvergencerateoftheasynchronousscheme,T.Unferproposed in[21] anextensionoftheasynchronous integration procedure to higher order. This method was explored in some details from a practical viewpoint. We then explored it froma moretheoretical viewpointandweproved that theasynchronous schemeasit was presentedin[21] isatmostoforderone.ThederivationofhigherorderexplicitLTSmethodsisaseriousproblemandonlyafewmethods are available. In [18], A. Taube et al. proposed an ADER-DG scheme withtime-accurate LTS forthe Maxwell equations. Theyfound vianumericalstudythattheir approachmaintainsthe accuracyoftheglobaltime-steppingADER-DGscheme “withaverysmallincreaseincomputationaleffort”.Grote etal.[4]derivedan explicitLTS schemefromstandardexplicit Runge–Kutta(RK)methods.TheyhaveprovedthattheirschemepreservestheaccuracyoftheoriginalRKscheme.However, theirapproachisapplicableonlyinthecaseoftwodistinctregionsinthemesh:a“coarse”regionwiththelargerelements anda“fine”regionwiththesmallerelements.Inthefineregion,thetimestepmustbeafractionofthecoarsetime step. Here,wedevelopanasynchronousRK2schemeinanarbitrarymesh.

This paperisoutlined asfollows.In section 2, wesummarizebasicideas ofthe asynchronousmethodology. Then,we consideran arbitraryspacediscretizationofa givenlinearpartialdifferentialequationandweapplytheARK2method.In section 3,we provethattheasynchronousschemeissecond orderconvergent.Next,numericalexperimentsthatillustrate the convergencerateof theasynchronous schemeand validate thegain incomputation time are presented insection 4. Finally,someconclusionsandﬁnalremarksaregiveninsection5.

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2. Numericalasynchronoussimulation

Forstability reasons,anyexplicitmethod hasto fulﬁlla time-step restriction.Thismayreduce the methodeﬃciency, becauseautomaticgridgenerationtoolsforunstructuredmeshesmayproduceverysmallcellsingeometricallycomplicated partsofthecomputationaldomain.Toovercomethisproblem,weproposeanasynchronousschemewithlocaltimesteps.

We consider

a polygonal domain in

R

d _(d

_≥

_1). _Let

_{T = {}

_K

i

,

i

=

1

,

. . . ,

N

}

be a partition of the domain

in N polyhedral volumes Ki. For all i

∈ J

1

,

N

K

,

N (

i

)

= {

j

,

|

Ki

∩

Kj

|=

0

}

is the set of indices of neighboring volumes of Ki;

|

Ki

∩

Kj

|

denotesthe

(

d

−

1

)

positivemeasureofKi

∩

Kj.Let

N

+

(

i

)

(resp.

N

−

(

i

)

)bethesetoftheindicesoftheneighbors oftheelementKiwhoreceive(resp.send)theoutflow(resp.inflow)fluxesfrom(resp.to) Ki.

Wesupposeanarbitraryspacediscretizationofagivenlinearpartialdifferentialequationwithoutasourceterm,forthe sakeofsimplicity.Thisleadstoasystemofcoupledordinarydifferentialequations

dyi

dt

(

t

)

=

Biy

(

t

),

∀

Ki

∈

T

,

(1)

whereyiandy arerespectivelyvectorsthatcontainthedegreesoffreedom(dofs)intheelementKiandinalltheelements, andBiamatrixcomingfromthespacediscretization.

2.1. Asynchronousmethodology

Theasynchronousmethodologyisbasedontwokeyideas.Firstofall,theasynchronousalgorithmpermitstheselection ofindependenttimesteps ineach meshelement.Then,thesequenceofupdatetimesofthecellsisnolongerdeﬁnedby continuously addinga uniformtimestep asintheclassical case. Weneed tointroduce thesequenceof themosturgent refreshtimetag

(

ti

₎

i≥1 deﬁnedby t1

=

min j

(

tj

)

:=

ti

,

t 2

₌

_min

min j=i

(

tj

),

2

ti

,

. . .

where for all i, the time step

ti of the element Ki is determined from the local stability restriction. Let

(

tri

)

r be the sequenceofupdatetimesofagivenelementKiwhichisdeﬁnedby

tr_i

=

r

ti

,

∀

r

≥

0

.

Foreachelement,thesourcetermdependsonlyonthecurrentcellwhereasthefluxesdependalsoonneighbors.Hence thesecondbasicideaconsistsinseparatingthesourcetermandthefluxes.Inotherwords,weneedtosplitthetermBi (1) inthreecontributions: Vi foralocalvolumecontribution, F_i+_,_j foralocalfluxcontributionandF_i−_,_j foraflux contribution fromtheneighbor.Then(1)becomes

dyi

dt

(

t

)

=

Viyi

(

t

)

+

j∈N (i)

F+_i_,_jyi

(

t

)

+

F_i−_,_jyj

(

t

),

∀

Ki

∈

T

.

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Theconcept oftheasynchronous timeintegrationcan besummarizedinthreemainphases.First, atthestart-up time all densities andfluxes are initialized. Then, a second phase iscarried out by continuously applyingthe following steps untiltheglobalsimulationclockisadvancedpastthesimulationfinishtime;seealsoFig. 1:findthemosturgentcelltobe refreshedbyusingtheCFLcondition,computethevaluesofthedensitieswhichareneededtocomputethefluxes,compute thenext refreshtimeofthe currentcellandthenupdate thevaluesofthefluxes. Finally,thethird phaseistobuild the solutionattheoutputtime.

ThenumberofupdatedcellsinthesecondstepinFig. 1dependsonthetreatedsystemandonthespacediscretization method. Forexample, in the caseof an upwind scheme,we need to update only the outﬂowﬂuxes andthen only the currentcell Ki andtheneighborsKj,

∀

j

∈

N

+

(

i

)

are involved;seealso[20].Whereasforacentered scheme,we needat leasttoupdatealltheneighbors.Inallcases,onlyfewcellsareinvolved.

Note that the critical point for speedingup an asynchronous method interms of computation time is the searching algorithmforthemosturgenttimetagtobetreated,somealternativeoptionsarepresentedin[22].

2.2. ARK2algorithm

TopresenttheasynchronousRunge–Kutta2algorithm,westartwiththesemi-discretizedsystemwritteninthefollowing formasexplainedinsection2

dyi

dt

(

t

)

=

Viyi

(

t

)

+

j∈N (i)

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Fig. 1. The four steps of the second phase of the asynchronous algorithm with ﬁve elements.

Wedenoteby ym_i theapproximationofyi

(

m

ti

)

andby

y

t

m i theapproximationof dyi dt

(

m

ti

)

.Pre-computedupdate

timescorrespond tomultiplesof

ti ineach Ki.The timeadvanceisperformedusingtheDiscreteTimeScheduler(DTS) algorithmasdeﬁnedin[22],tcuristhecurrentsimulationtime.

Attcursuchthattcur

/

ti isaninteger p,wehavetoperformthefollowingoperations

1. Computethetentativedofsattcur

˜

yi

(

tcur

)

= ˜

yi

(

tpre,i

)

+ (

tcur

−

tpre,i

)

˜

y

t

i

(

tmid,i

),

where tmid,i

=

tcur

−

_t

cur−tpre,i 2

andtpre,i isthe last time wherethe tentative dofs hasbeen updated. The tentative mid-pointslopeisdeﬁnedasfollows

y

˜

t

i

(

tmid,i

)

=

Vi

y_ip−1

+

ti 2

y

t

p−1 i

+

j∈N+(i)

F_i+_,_j

y_ip−1

+

tmid,i,j

− (

p

−

1

)

ti

y

t

p−1 i

+

j∈N−(i)

F_i−_,_j

ynj j

+

tmid,i,j

−

nj

tj

y

t

nj j

,

where nj

=

tpre,i

tj

, tmid,i,j

=

tnext,i,j

−

tnext,i,j

−

tpre,i,j 2

with tpre,i,j

=

max

(

nj

tj

,

(

p

−

1

)

ti

)

and tnext,i,j

=

min

((

nj

+

1

)

tj

,

p

ti

)

.

2. Foralltheneighborsoftheelementi,i.e. j

∈

N (

i

)

,weupdatethetentativevalues

˜

yj

(

tcur

)

= ˜

yj

(

tpre,j

)

+ (

tcur

−

tpre,j

)

y

˜

t

j

(

tmid,j

),

tpre,jisthelasttimewherethetentativedofshasbeenupdatedandtmid,j isthe“mid-point”timefortheelement j. 3. Affectthedofsandupdatethelastrefreshtimes

yp_i

= ˜

yi

(

tcur

),

tpre,i

=

tcurand tpre,j

=

tcur

,

∀

j

∈

N

(

i

).

4. Afterallelements Ki suchthattcur

/

tiisanintegerhavebeenupdated,updatetheirslopes

y

t

p i

=

Viy_ip

+

j∈N (i) F_i+_,_jy_ip

+

F_i−_,_j

ynj j

+ (

p

ti

−

nj

tj

)

y

t

nj j

,

where nj

=

tcur

tj

.

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Fig. 2. Notation:the temporalevolutionofthesystembetweentwoinstantstri

i and t ri+1

i wheretheelements j andn arethe neighborsofi (left).

DependencyoftheelementKiontheotherelementsofthemesh(right).(Forinterpretationofthereferencestocolorinthisﬁgure,thereaderisreferred

tothewebversionofthisarticle.)

Theﬁnalsimulationtimeshouldcorrespondtoacommonmeetingtimeofallcellsi.e. amultipleofthelowestcommon multiple(LCM)ofallthelocaltimesteps.Ifitisnotthecase,weuseaninterpolationatleastoforder2.

3. Propertiesoftheasynchronousscheme

3.1. ConvergenceoftheARK2scheme

The asynchronousRK2 schemeisderived fromthe standardexplicitRunge–Kutta 2method.We shallnowprove that theARK2schemeissecondorderconvergent.

3.1.1. Notations

1. Let usbegin withsome notations concerning the temporal evolution of the systembetween two successive update timesofanelement Ki whicharemultiplesof

ti;seealsoFig. 2(left)

(a) Lettri+1

i

:= (

ri

+

1

)

ti

=

tcur

≤

T whereT istheﬁnaltimeofthesimulation.Supposethattheelement Kihasbeen

updatedm timesbetweentri

i andt ri+1

i attheinstants:t1pre,i,t2pre,i,...,tmpre,i,0

≤

m

≤

ti minj∈N (i)(tj)

+

1. (b) Let

δ

_ik

=

tk pre,i

−

t k−1

pre,i, for all 2

≤

k

≤

m,

δ

i1

=

t1pre,i

−

t ri i ,

δ

m+1 i

=

t ri+1 i

−

tmpre,i and

m+1 k=1

δ

ki

=

ti. Then, tkmid,i

=

tk_pre_,_i

− δ

k_i

/

2 for1

≤

k

≤

m andtm_mid+1_,_i

=

tri+1

i

− δ

m+1 i

/

2.

(c) Supposethatforall j

∈

N (

i

)

wehavethefollowingassumptions: i. betweentri

i andt ri+1

i ,the element j has beenupdated z ri

j timesatinstantswhich are multiplesof

tj then 0

≤

zri

j

≤

m.Forexample,inFig. 2(left),z ri

n

=

1 andzrji

=

5 wheren and j aretwoneighborsofi.

ii. rj

tj

≤

ri

ti andattheinstanttkpre,i,rkj

tj denotethelast updatetimeoftheelement j whichismultipleof

tj justbefore tkpre,i.Notethat theremaybe situationswhere r k+1

j

tj

=

rkj

tj.In Fig. 2forinstance,forthe neighborn,r4 j

tj

=

r3j

tj

=

r2j

tj

=

r1j

tj

=

rn

tn. iii. tk mid,i,j

=

tknext,i,j

−

tk next,i,j

−

tkpre,i,j 2

=

t k next,i,j

+

tkpre,i,j 2 ,with: A. tk_pre_,_i_,_j

=

max

(

rk_j

tj

,

ri

ti

)

; B. tk_next_,_i_,_j

=

min

((

rk_j

+

1

)

tj

,

(

ri

+

1

)

ti

)

.

2. Theupdate ofan element Ki involvesitsneighbors.Theneighborsof Ki dependontheir neighbors,theneighborsof the neighborsof Ki involves their neighbors and soon. Then, we introduce thefollowing notations; see also Fig. 2 (right).

(a) Theelements Kj, j

∈

N (

i

)

,aretheneighborsofKi.

(b) Theelements K_{i,1},

{

i

,

1

} ∈

N (

j

)

,aretheneighborsofKj andthentheneighborsoftheneighborsofKi. (c) Theelements K_{i,2},

{

i

,

2

} ∈

N ({

i

,

1

})

,aretheneighborsoftheneighborsoftheneighborsofKi.

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Lethri

{i,ts} bethenumberofdependenciespresentedbytheelement Ki betweent

ri

i andts,wherets isthetimeofthe last synchronizationofthesolutionwhichcanparticularlybe theinitialtime t0_._In_other _words,_the_elements _K

{i,hri_i,ts} usetheirvaluescomputedatthetimeofthelastsynchronizationofthesolution.Notethatthestudyofthedependency will benecessaryforthetheoreticalstudy oftheARK2schemewhichneeds thecontrolofthetemporal evolutionof the system.However, forthe asynchronousalgorithm,theupdateofa givenelementinvolvesonlytheneighborsand then noneedoftheneighborsoftheneighbors.Thisdependencyisimplicitanditishiddenbehindtheintermediate stepsofthealgorithm.

3. Letusintroducesomeelementsfortheidentiﬁcation:

(

Bx

)

i

=

Vixi

+

j∈N+(i) F_i+_,_jxi

+

j∈N−(i) F_i−_,_jxj

(

B Pix

)

i

=

Vixi

+

j∈N+(i) F_i+_,_jxi

where Piisadiagonalprojectionmatrixonthedofoftheelementi.

Proposition1.Theevolutionofanygivencelli betweenthetwoinstantstri

i

=

ri

tiandtiri+1

=

t ri

i

+

tiisgivenbythefollowing

expression yri+1 i

=

y ri i

+

ti

Vi

yri i

+

ti 2

y

t

ri i

+

m

+1 k=1

δ

k_i

j∈N+(i) F_i+_,_j

yri i

+

tk_mid_,_i_,_j

−

ri

ti

_y

t

ri i

+

m

+1 k=1

δ

_ik

j∈N−(i) F_i−_,_j

yr k j j

+

tk_mid_,_i_,_j

−

rk_j

tj

_y

t

rk j j

.

Proof. Bydeﬁnition,wehave

yri+1 i

= ˜

y tm pre,i i

+ δ

m+1 i

˜

y

t

i

(

tm_mid+1_,_i

)

= ˜

yt m−1 pre,i i

+ δ

m i

˜

y

t

i

(

tm_mid_,_i

)

+ δ

m_i +1

y

˜

t

i

(

tm_mid+1_,_i

)

..

.

=

yri i

+

m

+1 k=1

δ

k_i

˜

y

t

i

(

tk_mid_,_i

)

whereforall1

≤

k

≤

m

+

1

y

˜

t

i

(

tk_mid_,_i

)

=

Vi

yri i

+

ti 2

y

t

ri i

+

j∈N+₍i) F_i+_,_j

yri i

+

tk_mid_,_i_,_j

−

ri

ti

_y

t

ri i

+

j∈N−(i) F_i−_,_j

yr k j j

+

tk_mid_,_i_,_j

−

rk_j

tj

_y

t

rk j j

Toconclude,wejustusethat

_km₌+₁1

δ

k_i

=

ti.

2

Proposition2.Foranygivenelementi,andbetweentwoinstantstri

i andt ri+1

i ,wehavethefollowingresult

m

+1 k=1

δ

k_i

×

tk_mid_,_i_,_j

=

(

2ri

+

1

)(

ti

)

2

,

∀

j

∈

N

(

i

).

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Inparticular,betweenthetwoinstantsts,anytimeofthesynchronizationofthesolution,andts

+

tiwehave

∀

j

∈

N (

i

)

m

+1 k=1

δ

k_i

×

tk_mid_,_i_,_j

=

(

ti

)

2

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Proof. Weﬁrstprovethat,foragivenelement j

∈

N (

i

)

,betweentwosuccessivetimesofupdateoftheelement Kjwhich are multiples of

tj and which are betweent_iri and tr_ii+1, t_midk _,_i_,_j is constant. In other words, if ri

ti

≤ (

rj

+

1

)

tj

<

(

rj

+

l

)

tj

≤ (

ri

+

l

+

1

)

ti,l integer,thenforallk suchthat

(

rj

+

l

)

tj

<

tk_pre_,_i

≤ (

rj

+

l

+

1

)

tjwehave

tk_pre_,_i_,_j

=

max

(

rk_j

tj

,

ri

ti

)

=

max

((

rj

+

l

)

tj

,

ri

ti

)

= (

rj

+

l

)

tj and tk_next_,_i_,_j

=

min

((

rk_j

+

1

)

tj

, (

ri

+

1

)

ti

)

=

min

((

rj

+

l

+

1

)

tj

, (

ri

+

1

)

ti

)

= (

rj

+

l

+

1

)

tj

.

Asaresult, tk_mid_,_i_,_j

=

(

2

(

rj

+

l

)

+

1

)

tj 2

.

(5)

Now,weproveEquation(4).Supposethatforall j

∈

N (

i

)

,zri

j isthenumberoftimesofupdateoftheelement Kjattimes thataremultiplesof

tj betweenri

tiand

(

ri

+

1

)

ti.Therearethreecases:

1. If zri

j

=

0 thenrj

tj

≤

ri

ti

≤ (

ri

+

1

)

ti

≤ (

rj

+

1

)

tj andforall1

≤

k

≤

m

+

1,

t_midk _,_i_,_j

=

(

ri

+

1

)

ti

+

ri

ti 2

=

(

2ri

+

1

)

ti 2

,

therefore, m

+1 k=1

δ

k_i

×

tk_mid_,_i_,_j

=

(

2ri

+

1

)(

ti

)

2 2

.

2. If zri j

=

1 thenwehave m

+1 k=1

δ

k_i

×

tk_mid_,_i_,_j

=

((

rj

+

1

)

tj

+

ri

ti

)

2

× ((

rj

+

1

)

tj

−

ri

ti

)

+

((

ri

+

1

)

ti

+ (

rj

+

1

)

tj

)

2

×

(

ri

+

1

)

ti

− (

rj

+

1

)

tj

=

1 2

(

ri

+

1

)

2

ti2

−

r2i

ti2

=

(

2ri

+

1

)(

ti

)

2 2 3. If zri

j

≥

2 thenrj

tj

≤

ri

ti

< (

rj

+

1

)

tj

< ...

< (

rj

+

zrji

)

tj

≤ (

ri

+

1

)

ti

< (

rj

+

zrji

+

1

)

tj.Using(5)we havethe followingresult m

+1 k=1

δ

k_i

×

tk_mid_,_i_,_j

=

((

rj

+

1

)

tj

+

ri

ti

)

2

× ((

rj

+

1

)

tj

−

ri

ti

)

+

zri_j−1

s=1

(

2

(

rj

+

s

)

+

1

)

tj 2

×

tj

+

((

ri

+

1

)

ti

+ (

rj

+

z ri j

)

tj

)

2

× ((

ri

+

1

)

ti

− (

rj

+

z ri j

)

tj

)

=

((

rj

+

1

)

tj

)

2 2

−

(

ri

ti

)

2 2

+

((

ri

+

1

)

ti

)

2 2

−

((

rj

+

zrji

)

tj

)

2 2

+

zri_j−1

l=1

(

2

(

rj

+

l

)

+

1

)

×

(

tj

)

2 2

.

Notethat

z ri j−1

l=1

(

2

(

rj

+

l

)

+

1

)

isasumofanarithmeticsequencewithacommondifferenceequaltotwo.Therefore, zri_j−1

l=1

(

2

(

rj

+

l

)

+

1

)

=

zri_j−1

l=0

(

2

(

rj

+

l

)

+

1

)

−(

2rj

+

1

)

=

zri j

(

4rj

+

2zr_ji

)

2

− (

2rj

+

1

)

=

zri j

2

+

2rj

(

zr_ji

−

1

)

−

1

(9)

andthen m

+1 k=1

δ

_ik

×

tk_mid_,_i_,_j

=

(

2ri

+

1

)(

ti

)

2 2

.

Theproofisthencomplete.

2

Proposition3.Foranyelementi andateachtimetri

i

=

ri

ti,rianinteger,wehavethefollowingexpressionoftheslopei

y

t

ri i

=

B Piyri

i

+

j∈N−(i) F_i−_,_j

yrj j

+ δ

i,j

⎛

⎝

B Pjyrj

j

+

{i,1}∈N−(j) F_{−_i_,₁_},_jyr{i,1} {i,1}

⎞

⎠

+

hri_i_,_ts−1

z=1

⎛

⎝

z n=1

δ

_{i,n−1},{i,n}

{i,n}∈N−₍_{i,n−1}) F_{−_i_,_n₋₁_},{_i_,_n_}

⎞

⎠

×

⎛

⎝

B P_{i,z}yr{i,z}

{i,z}

+

{i,z+1}∈N−({i,z}) F_{−_i_,_z_},{_i_,_z₊₁_}y_{r{_ii_,,_zz+₊1₁}_}

⎞

⎠ ,

whereforall j

∈

N

−

(

i

)

,

δ

i,j

=

ri

ti

−

rj

tj

>

0 andrj

tjisthelasttime,multipleof

tj,ofupdatingtheelement j justbefore tr_ii.

Forall1

≤

z

<

hri

i,ts

−

1,r{i,z}

t{i,z}

<

t

ri

i and

δ

{i,z−1},{i,z}

=

r_{i,z−1}

t_{i,z−1}

−

r_{i,z}

t_{i,z}

>

0 wheretsisthetimeofthelast

syn-chronizationofthesolutionandhri

{i,ts}isthenumberofdependenciespresentedbytheelementKibetweent

ri

i andts.Wealsosupposed

thatK_{i,0}

≡

Kj,j

∈

N (

i

)

aretheneighborsofKi. Proof. Theexpressionoftheslopei attri

i isgivenbythefollowingformula

y

t

ri i

=

Viyr_ii

+

j∈N+(i) F_i+_,_jyri i

+

j∈N−(i) F−_i_,_j

yrj j

+ δ

i,j

y

t

rj j

=

B Piyri

i

+

j∈N−(i) F_i−_,_j

yr_jj

+ δ

i,j

y

t

rj j

,

theelements j aretheneighborsoftheelementi andforall j

∈

N

−

(

i

)

,

δ

i,j

=

ri

ti

−

rj

tj

>

0 whererj

tj

=

t rj

j

<

t ri

i is thelasttime,multipleof

tj,wheretheelement j hasbeenupdated.UsingthenotationsillustratedinFig. 2wehavethe followingresult

y

t

rj j

=

B Pjyrj

j

+

{i,1}∈N−(j) F−_j_,_{_i_,₁_}

yr_{_i{i_,,₁1}_}

+ δ

j,{i,1}

y

t

r_{i,1} {i,1}

,

where for all j

∈

N (

i

)

and for all

{

i

,

1

}

∈

N (

j

)

, r_{i,1}

t{i,1}

=

t r_{i,1}

{i,1}

<

t

rj

j is the last time, multiple of

t{i,1}, where the element K_{i,1}hasbeenupdated.Then

δ

j,{i,1}

=

rj

tj

−

r{i,1}

t_{i,1}

>

0 and

y

t

r{i,1} {i,1}

=

B P_{i,1}yr{i,1}

{i,1}

+

{i,2}∈N−({i,1}) F_{−_i_,₁_},{_i_,₂_}

yr{i,2} {i,2}

+ δ

{i,1},{i,2}

y

t

r{i,2} {i,2}

.

Likewise,

y

t

r_{i,2} {i,2}

=

B P_{i,2}yr{i,2}

{i,2}

+

{i,3}∈N−({i,2}) F_{−_i_,₂_},{_i_,₃_}

yr{i,3} {i,3}

+ δ

{i,2},{i,3}

y

t

r_{i,3} {i,3}

.

In the same way, we deﬁne for all 3

≤

z

<

hri

i,ts

−

1, the slopes

y

t

r{i,z} {i,z} and

δ

_{i,z−1},{i,z}

=

r_{i,z−1}

t_{i,z−1}

−

r_{i,z}

t_{i,z}

>

0.Inparticular,forz

=

hri

(10)

Fig. 3. The temporal evolution of the system between two instants of synchronization tq_{and t}q+1₌_tq₊_t

max; example of ﬁve elements.

y

t

r {i,hri i,ts−1} {i,hri_i_,_ts−1}

=

B P_{ i,hri_i_,_ts−1}y r {i,hri i,ts−1}

{i,hri_i_,_ts−1}

+

{i,hri_i_,_ts}∈N−({i,hri_i_,_ts−1}) F− {i,hri_i_,_ts−1},{i,hri_i_,_ts}

yts {i,hri_i_,_s}

+ δ

_{_i_,_hri i,ts−1},{i,h ri i,ts}

y

t

ts {i,hri_i_,_ts}

,

where,t r {i,hri_{i,ts }} {i,hri_i,ts}

=

ts.Consequently,

y

t

ri i

=

B Piyri

i

+

j∈N−(i) F−_i_,_j

yrj j

+ δ

i,j

B Pjyrj

j

+

{i,1}∈N−(j) F−_j_,_{_i_,₁_}

yr{i,1} {i,1}

+ δ

j,{i,1}

B P_{i,1}yr{i,1}

{i,1}

+

{i,2}∈N−({i,1}) F_{−_i_,₁_},{_i_,₂_}

yr{i,2} {i,2}

+ δ

{i,1},{i,2}

...

+ ... + δ

_{_i_,_hri i−2},{i,hrii−1}

B P_{ i,hri_i−1}y r {i,hri_i−1}

{i,hri_i−1}

+

{i,hri_i}∈N−({i,hri_i−1}) F− {i,hri_i −1},{i,hri_i}

y0 {i,hri_i}

+ δ

{i,hri_i −1},{i,hri_i}

(

B y 0

₎

{i,hri_i}

andthen

y

t

ri i

=

B Piyri

i

+

j∈N−(i) F−_i_,_j

⎛

⎝

yr_jj

+ δ

i,j

⎛

⎝

B Pjyrj

j

+

{i,1}∈N−(j) F_{−_i_,₁_},_jyr{i,1} {i,1}

⎞

⎠

⎞

⎠

+

h

ri_i−1 z=1

⎛

⎝

z n=1

δ

_{i,n−1},{i,n}

{i,n}∈N−({i,n−1}) F_{−_i_,_n₋₁_},{_i_,_n_}

⎞

⎠

×

⎛

⎝

B P_{i,z}yr{i,z}

{i,z}

+

{i,z+1}∈N−₍_{i,z}) F_{−_i_,_z_},{_i_,_z₊₁_}y_{r{_ii_,,_zz₊+1₁}_}

⎞

⎠ ,

wesupposedthat K_{i,0}

≡

Kj, j

∈

N (

i

)

aretheneighborsofKi.

2

Tostudytheconvergencerateoftheasynchronous scheme,we ﬁrstneedtosupposethatthesolutionissynchronized at the instants ts which are multiples of a ﬁxed time step

tmax. In the case of meshes where all the elements have a common meeting time, the solution is automatically synchronized at the instants which are multiples of the lowest common multiple (LCM) of all the local time steps and then

tmax is equal to the LCM. If it is not the case, so that the local time steps are completely independent, we have to suppose that the solution is regularly synchronizedat the instantsmultiplesof

tmax where,inthiscase,

tmax iscomputedinfunctionofthetimesteps

ti.Forinstance,wecan choose

tmax

=

maxi

(

ti

)

. Note that the hypothesis of regular synchronizationof the solution hasno inﬂuence on the asynchronousaspect;seeFig. 3.Infact,thecellsadvanceindependentlybetweentq _and_tq+1_._It_is_important_to _note_that

(11)

theregularsynchronizationofthesolutionwasimposedonlyfordemonstrationpurposes.Numerically,thesynchronization is only madeattheﬁnal time of thesimulation independentlyof themesh; seethe ARK2algorithm. Let

(

tq

₎

q≥0 be the sequence deﬁned by tq+1

=

tq

+

tmax. The studyof the convergence of the ARK2 scheme will be done between two instantsofsynchronization:tq andtq+1.Let

(

tnq

)

1≤n≤Nq bethe sequenceofthemosturgentrefreshtime tags betweent

q andtq+1,where Nq

=

isi andforalli, si

=

tmax

ti

.

(

tk_i_,_q

)

k is thesequenceofthe temporalevolutionof theelement i betweentq _and_tq+1 _deﬁned_by_tk

i,q

=

tq

+

k

ti(Fig. 3). Theorem1.Foralltheelementsi andatanyinstanttri

i,q

=

tq

+

ri

tisuchthattq

<

tri,iq

≤

tq+1,thefollowingexpressionholdstrue

yt ri i,q i

=

yt q i

+ (

ri

ti

)(

B yt q

)

i

+

(

ri

ti

)

2 2

(

B 2_ytq

₎

i (6)

+

2ri

n=3

(

ti

)

n

α

n,ri

(

B Pi

)

n−2

₍

_B2_ytq

)

i

+

Rri,iasyn

,

(7) where:

α

n,0

=

0

∀

n

≥

3,

α

3,ri

=

α

3,ri−1

+

(ri−1)2+(ri−1) 2 ,

α

4,ri

=

α

3,ri−1

+

α

4,ri−1

+

(ri−1)2 2 ,

α

n,ri

=

α

n,ri−1

+

α

n−1,ri−1

+

αn_−2,ri−1 2

∀

5

≤

i

≤

2

(

ri

−

1

)

,

α

2ri−1,ri

=

α

2(ri−1),ri−1

+

α_{2(ri −1)−1,ri −1} 2 ,

α

2ri,ri

=

α_{2(ri −1),ri −1} 2 ,and

Rri i,asyn

∞

≤

Mri_i

n=3 Cn,ri

(

ti

)

n

_Bn

∞

ytq

_∞ (8) Mri

i

=

max

(

2ri

,

hiri

)

andforall3

≤

n

≤

M ri

i,Cn,riisaboundedconstant.

Proof. TheproofofTheorem 1isverytechnicalandmanyformulasarelong,itisthenplacedintheappendix.

2

Theorem2.TheasynchronousRunge–Kutta2schemeisasecondorderaccuratescheme. Proof. Let yi

(

tq+1

)

bethesemi-discrete exactsolutionoftheODE(3)attq+1, yt

q+1

i,syn isthesolutionoftheclassicalglobal timestepRunge–Kutta2methodattq+1and yt_iq+1 theasynchronoussolution.Then,

yi

(

tq+1

)

−

yt q+1 i

∞

≤

yi

(

tq+1

)

−

yt q+1 i,syn

∞

+

yt q+1 i,syn

−

yt q+1 i

∞ (9)

TheclassicalglobaltimestepRK2schemeissecondorderconvergentthen,

yi

(

tq+1

)

−

yt

q+1

i,syn

∞

≤

C0

(

tmax

)

3

,

where,for

tmaxsuﬃcientlysmall,C0isaconstantindependentof

tmax.Tocompletetheproofwethusneedtotreatthe secondtermontherightof(9).First,usingTheorem 1,wegetforalli

yt si q,i i

=

y tq i

+ (

si

ti

)(

B yt q

)

i

+

(

si

ti

)

2 2

(

B 2_ytq

)

i

+

2si

n=3

(

ti

)

n

α

n,si

(

B Pi

)

n−2

₍

_B2_ytq

)

i

+

R t_qsi_,_i i,asyn

whereforalli,tsi

q,i

=

tq

+

si

ti,si

=

E

tmax

ti

.Asaﬁrststep,supposethat tmax

ti isaninteger.Asaresult, yt_iq+1

≡

yt si q,i i

=

yt q i

+ (

tmax

)(

B yt q

)

i

+

(

tmax

)

2 2

(

B 2_ytq

)

i

+

2si

n=3

(

ti

)

n

α

n,si

(

B Pi

)

n−2

₍

_B2_ytq

)

i

+

Rt q+1 i,asyn Consequently,

yt_iq_,_syn+1

−

y_itq+1

_∞

=

Et_iq+1

_∞

,

where Et_iq+1

=

2si

n=3

(

ti

)

n

α

n,siB n−2

₍

_B2_ytq

)

i

−

2si

n=3

(

ti

)

n

α

n,si

(

B Pi

)

n−2

₍

_B2_ytq

)

i

−

Rt q+1 i,asyn

=

2si

n=3

(

ti

)

n

α

n,si

j∈N−₍i) Fi,j

n−2

(

B2ytq

)

j

−

Rt q+1 i,asyn

,

B is linear

.

(12)

Therefore,

Et_iq+1

_∞

≤

C1

(

tmax

)

3

,

(10)

wherefor

tmax suﬃcientlysmall,C1 isaconstantindependentof

tmax.Finally,notethatforthecasewhere tmax_t_i isnot an integerwe justneed tosynchronizethesolutionattq+1 using asecond orderinterpolationasitwas explainedwhen theARK2algorithmhasbeenpresented.

2

3.2. ComparisonwiththeLTS-RK2schemepresentedin[4]

In[4],Grote etal.derived an explicitLTS-RK schemefromstandardexplicitRunge–Kutta(RK) methodsinthe caseof twodistinctregionsinthemesh:a“coarse”regionwiththelargerelementsanda“fine”regionwiththesmallerelements. Inthefine region,thetime stepmustbe afractionofthecoarse time step.TheyusetheclassicalRK methodofagiven orderinsidethefineregionandquadratureformulainsidethecoarseregion.Theintermediatevaluesneededatthecoarse andfine mesh interface are obtainedthrough a judicious combinationof interpolation andTaylor expansion. Theyhave proved that their schemepreserves the accuracy of the original RK scheme. We started from their LTS-RK scheme and we adapted the algorithm presented in [4] to the asynchronous aspect in order to apply the three main phases ofthe asynchronousintegration.Inparticular,eachcellistreatedindependently.Inthispaper,weonlyfocusonordertwo.Inthe caseofthemeshimposedin[4],weprovedthattheARK2degeneratesintotheLTS-RK2scheme.Moreprecisely,usingthe asynchronousalgorithm, wegetthe sameresultateachglobaltime step

t, where

t denote thetime stepdictatedby theCFL-conditioninthecoarserpartofthemesh.Whereas,ateachtimestepofsizen

t

/

p where

t

/

p isthelocaltime stepinsidethereﬁnedregionofthemeshwithp andn areintegersand0

<

n

<

p,theresultisdifferent.Infact,wehave thefollowingresult

Proposition4.Inthecaseoftwodistinctregionsinthemeshwherethetimestepoftheﬁneregionisafractionofthetimestepofthe

coarseregion,theARK2schemedegeneratesintotheLTS-RK2schemepresentedin[4].

Proof. Resumingthenotationusedin[4]:

=

c

∪

f where

c isthecoarseregionwiththelargetimestep

tc and

f istheﬁneregionwiththesmalltimestep

tf

=

_ptc,p isaninteger.Introducesomeelementsfortheidentiﬁcation

(

Bx

)

i

=

Vixi

+

j∈N+(i) F+_i_,_jxi

+

j∈N−(i) F_i−_,_jxj

(

B P x

)

i

=

Vixi

+

j∈N+(i) F+_i_,_jxi

+

j∈N− f(i) F_i−_,_jxj

(

B

(

I

−

P

)

x

)

i

=

j∈Nc−(i) F_i−_,_jxj

where

N

c

(

i

)

and

N

f

(

i

)

aretheneighborsoftheelement i whichare respectivelyinthecoarsemesh andtheﬁne mesh. Inotherwords, j

∈

N

c

(

i

)

then j

∈

N (

i

)

∩

c and j

∈

N

f

(

i

)

then j

∈

N (

i

)

∩

f. P isadiagonalmatrix,itsdiagonalentries areequaltozeroorone.Itidentifytheunknownsassociatedwiththelocallyreﬁnedregion.

(

I

−

P

)

identifytheunknowns associated withthe coarse region. Theprevious resultsare proved for anygiven ofmesh, inparticular fora mesh with two regions. In fact,using the previous notationswe have

tmax

=

tc.This isa natural choicebecause the solutionis regularlysynchronizedattheinstantsmultiplesof

tc.Thenthesequence

(

tq

)

qisdeﬁnedbytq+1

=

tq

+

tc,foralli

∈

f,

(

tk_i_,_q

)

0≤k≤p is deﬁnedby tki,q

=

k

tf andfor all j

∈

c,

(

tk_j_,_q

)

0≤k≤1 is deﬁnedby t0j,q

=

tq andt 1

j,q

=

tq+1.Let ususethe previousresultsofProposition 1andProposition 2foraﬁneelement,theproofforthecoarseelementscanbedoneinthe sameway.Foralli

∈

f wehave

yt ri+1 i,q i

=

y tri_i_,_q i

+

tf

B P ytrii,q

i

+

(

tf

)

2 2

⎛

⎝

B P

y

t

tri_i_,_q

⎞

⎠

i

+

tf

B

(

I

−

P

)

ytq

i

+

(

2ri

+

1

)(

tf

)

2 2

B

(

I

−

P

)

y

t

tq

i

,

with

y

t

tri_i_,_q i

=

B P ytrii,q

i

+

B

(

I

−

P

)

ytq

+

ri

tf

y

t

tq

i

(13)

ComparingwiththeexpressionoftheslopegiveninProposition 3,thedependencieswiththeneighborsaresimpliﬁed.In fact,inthisparticularcase,thereareonlytwotypesofelementstheﬁnecells whicharegroupedinthematrix P andthe coarse cellswhichare groupedinthe matrix

(

I

−

P

)

.Alltheﬁneelements hasbeenupdatedinthesametimesandit is alsothecaseforthecoarseelements.Consequently,theexpressionof yi (6),i a ﬁneelement,inTheorem 1iswrittenin thefollowingform

yt ri i,q i

=

yt q i

+ (

ri

tf

)(

B yt q

)

i

+

(

ri

tf

)

2 2

(

B 2_ytq

₎

i

+

2ri

m=3

(

tf

)

m

α

m,ri

(

B Pi

)

m−2

₍

_B2_ytq

₎

i

,

andthenweﬁndthesameexpressiongivenin[4].

2

3.3. OriginalityoftheasynchronousformulationinARK2

The originality ofour approachwithrespect to theLTS schemedeveloped in[4],andmore generallywithrespect to anyotherstandardlocaltime-steppingapproach,lieswithinitsabilitytoadapttoanyarbitrarymeshanditsresiliencefor beingsecondorderconvergentschemeindependentlyofthechosenmesh.Indeed,forthestandardLTSschemes,localtime stepsareusuallyselectedtobefractionsoftheglobaltimestepwhereastheasynchronousschemepermitstheselectionof independenttimestepsineachmeshelement.

However, asmentioned inthe introduction,the ideaof usinga differenttime-step inevery element alreadyexists in the literature.In[11],OmelchenkoandKarimabadiintroduced such anasynchronous schemefordrift-diffusion transport. The major difference between their asynchronous approach and the one presented herelies within the way the fluxes are updated. In[11],theconcept of“flux capacitor”, avariablein whichtheflux betweentwo cellsis stored,is used.In ourapproach,desynchronizedfluxescanbeusedwhichallows fluxesrecomputationsinordertoensurethesecond-order convergence.

In [18], A. Taube etal. proposed an ADER-DG scheme with independent time steps forthe Maxwell equations with arbitrary high order of accuracy in space and time, although no proof of the convergence rate is given. Their method consistsinreplacingthetimederivativeswithspacederivatives.Thecomputationoftheﬂuxbetweenthegridcellsisthen basedonthesolutionofgeneralizedRiemannproblems.Ourasynchronousmethodisdifferent,sinceitisderivedfromthe standardexplicitRunge–Kutta2methodandthusthenumericalsolutionneedstogothroughintermediatestagesinorder tobeadvancedintime.

4. Numericaltests

Weconsiderthetwo-dimensionalMaxwellequationsintransverseelectricmode:

⎧

⎪

⎨

⎪

⎩

∂ ∂tEx

=

1∂∂yHz in

× (

0

,

T

)

∂ ∂tEy

= −

1 ∂∂xHz in

× (

0

,

T

)

∂ ∂tHz

=

1 μ∂∂yEx

−

1 μ∂∂xEy in

× (

0

,

T

)

(11) where E

=

Ex

,

Ey

,

0

T

andH

= (

0

,

0

,

Hz

)

T arerespectivelytheelectricandmagneticﬁelds.

and

μ

are,respectively,the electricpermittivity andmagneticpermeability.

isthecomputationaldomainandT is theﬁnal timeofthesimulation. Forthespatialdiscretization,weapplythesecondorder P1-discontinuousGalerkinschemewithupwindﬂuxes.

4.1. NumericalconvergenceoftheARK2scheme

We shall now present numerical experiments which validate the order of convergence of the ARK2 method derived in section 3. The one-dimensional tests assessing the second order convergence ofthe ARK2 scheme havealready been published in [20]. Inthispaper, weconsider thetwo-dimensional Maxwell equations(11). Theinitial condition:

(

Ex

)

0

=

(

Ey

)

0

= (

Hz

)

0

=

cos

(

2

π

x

)

,

is a rectangular domain of size

[

0

,

1

]

× [

0

,

1

]

and the boundary conditions are supposed periodic.

First, we divide

[

0

,

1

]

in X -direction, respectivelyin Y -direction,into treeequalparts. Theleft andrightintervals are discretized with an equidistantmesh of size

x,respectively

y.The middle interval isdiscretized withan equidistant mesh of size p

x

/

q, respectively p

y

/

q where p andq are integers; seealso Fig. 4 left. We start with a simplecase, mesh 1, wherethelocaltime steps arefractionsof theglobaltimesteps with p

=

4 andq

=

1.Then foramoregeneral case, mesh 2, we supposethat p

=

2, q

=

3 and then p

/

q

<

1.Finally, mesh 3is totest the ARK2algorithmin thecase wherethetimestepsareindependent,wesupposethatthepositionsxiand yi followapolynomiallowandthus

xi

=

4

(

1

−

α

)

i

−

1 Nx

−

1

−

1 2

3

+

α

i

−

1 Nx

−

1

−

1 2

+

1 2

,

(12)

Second order accurate asynchronous scheme for modeling linear partial differential equations

HAL Id: hal-01881736

https://hal.archives-ouvertes.fr/hal-01881736

Submitted on 28 Nov 2019

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