Convergence of energy stable finite-difference schemes with interfaces

Full text

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Contents lists available atScienceDirect

Physics

www.elsevier.com/locate/jcp

interfaces

a

b

,

c

cDepartmentofMathematicsandAppliedMathematics,UniversityofJohannesburg,P.O.Box524,AucklandPark2006,SouthAfrica

t

Availableonline1December2020

Keywords: Convergencerate Finitedifference Stability Well-posedness Interface

We extendtheconvergenceresultsinSvärdandNordström(2019)[7] forsingle-domain energy-stable high-orderﬁnite differenceschemes,to includedomainssplitintoseveral grid blocks. The analysis also demonstrates that reﬂective boundary conditions enjoy the sameconvergenceproperties.Finally, webrieﬂyindicatethattheseresults (andthe previousonesin[7])alsoholdinmultipledimensions.

1. Background

The convergencerateofﬁnite-differenceschemes thatare closedattheboundarieswithstencils oflower order accu-racy1 thantheinteriorstencilsisalong-standingproblemthathasbeentreatedextensivelyintheliteratureine.g.[3,4,1,5]. Recentlytheproblemwas addressed in[7],whereitwasshownthatenergy-stable schemes,satisfyingsome natural con-straints,automatically“gain”asmanyordersontheboundaryasthehighestspatialderivative.

Thecasewhenthedomainissplitinto(atleast)twocomputationaldomainsthatareconjoinedataninterface,wasnot treatedin[7].Theinterfacecaseisfundamentallydifferentfromtheboundarycase.Inthetheoryforboundaryconditions, the semi-discretisation ofthe initial-boundary value problemis Laplacetransformed. This allowsa decomposition ofthe modesintothose that decayintothedomain fromtheboundary,andshould be suppliedwithboundary conditions,and those thatdonot. Theboundednessofthe boundarydataandtheenergystability ofthescheme,enableda proofofthe convergenceresult.

Thisproof,however,doesnotimmediatelyencompassinterfaces,sincetheytakedatafromtheconjoiningdomain,which mayormaynotbesuﬃcientlybounded.Moreover,thetreatmentataninterfaceneednotabidetotheminimalnumberof boundaryconditionsrequirement,sincedataataninterfacecanbeconsideredexact.Likewise,theprevious theorydidnot consider reﬂectiveboundaries,i.e., those thatfeedout-goingwaves backinto thedomainvia thein-going characteristics, sincesuﬃcientboundednessofthesolutionthatisfedbackhastobeestablished.

*

Correspondingauthor.

1 Convergencerateistheratebywhichthesolutionerrordecreases withgridreﬁnements.Theorderofaccuracyisgivenbythetruncationerrorand canbedifferentatdifferentpoints.Itrepresentshowaccuratelytheequation(notthesolution)isapproximated.

https://doi.org/10.1016/j.jcp.2020.110020

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(1) thatthesemi-discreteschemeisenergystable,hasapolynomialnullspace(possiblyempty),isnullspaceconsistent,and nullspaceinvariant.(Alltheassumptionsusedin[7].)

(2) that thenumber ofinterface conditions,as seenfrom oneside, is consistent withtheminimal number ofboundary conditions.Thatis,datafromacrosstheinterface canbereplaced bysmooth andboundeddata,andresultina well-posedproblem.

Remark.Thesecondrequirementimmediatelyridsoneoftheproblemsdiscussedabove.Itprohibitsover-speciﬁcationattheinterface. Furthermore,itisthemostrobustchoicethatismostoftenusedinpractice.

Theprogrammeforanalysinginterfaceaccuracyistoviewtheinterfaceasanexternalboundaryand,withtheaidofthe requirementsabove,theaprioriestimatesandprevioustheory,infertheoptimalconvergencerates.

Wewilldemonstratethisforafewexamples.Thegeneralisationsareobviousandwillonlybediscussedbrieﬂy. 2. Thehyperboliccase

ut

aux

0

a

0

.

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We splitthedomainintotwo: x

0 and x

>

0 withaninterface atx

=

0 anddiscretise eachwithaSummation-by-Parts (SBP) scheme (see [6]). (We assume appropriate trail-off conditions to the left andright and denote the left and right domain as

L

,R.) The gridis equidistantwith indexingxi

ih, i

0

1

2

...

,such that x0 representsthe interface in bothdomains.

vtL

a P−1Q vL

0 (2)

vtR

a P−1Q vR

P−1a

v0R

v0L

e0 where vL

vL

−1

vL0

T and vR

v0R

vR1

...)

T are the unknowns to the left and right and e0

1

0

0

)

T. The SBP operator has the following properties: P

PT

0, Q

QT

B

diag

1

0

0

1

)

and P−1Q v is a ﬁrst-derivative approximation.(See[7] formoreinformation.)Toadherewiththesecondrequirementabove,thereisonlyapenaltyterm totherightinagreementwiththecharacteristicdirection.

Remark.Generally,onecanpenalisebothequationsatx0and,withaproperscaling,stabilitycanbeproven.Althoughstableand

convergent,sucha“two-way”couplingrenderstheleftproblemill-posedviewedasastand-aloneproblem.Hence,Requirement(2)is notsatisﬁed.

Weapplytheenergymethodinasomewhatunconventionalway.Multiplytheﬁrstequationin(2) byvLP toobtain

vL

2

t

a

v L 0

2

0

.

The left problemis well-posed asa stand aloneproblemand we obtainthe bound vL0

L2

0

T

)

. Next,we multiply the secondequationin(2) by

vR

TP ,

vR

2

t

a

v R 0

2

2av0R

v0R

vL0

).

Byrecastingtheboundaryterms,weobtain

vR

2

t

a

v R 0

v0L

2

a

v0L

2

.

This problemis bounded, andwell-posed,since vL0

L2

0

T

)

. Hence,the theory of[7] appliesto both theleft andright problemsindividuallyandweconcludethat(2) willconvergewithoptimalrates.

Thisprocedurecanbegeneralisedtosymmetricsystems,ut

Aux

0 discretisedby

vtL

A

P−1Q

vL

A

P−1

vL0

vR0

e0L

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vtR

A

P−1Q

vR

A+

P−1

vR0

v0L

e0R

)

whereeL,R arebothoneatx

=

0 and zeroelsewhere.(FormoreinformationonKroneckerproducts, see[6].)Thedivision

A

A+

+

A− refers to thesigns ofthe eigenvalues such thatonly in-going characteristicsare penalisedon each side of the interface. As in the scalar case, the outgoing characteristics on each side serve as data for the other side and are independentlyboundedinL2

0

T

)

suchthatoptimalratesareobtained.

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2.1. Reﬂectiveboundaryconditions

Another wayof looking at (1) is obtained by transforming the left domain, x

0, to

0. Thatis, by x

and

ξ,suchthat,

utL

auξL

0

0

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Relabelling

asx,uL

u1,uR

=

u2,weobtainthesystem,

u1 u2

t

a 0 0 a

u1 u2

x

0

x

>

0

withtheboundarycondition, u1

0

t

u2

0

t

)

.(Inparticularwe note thattheproblemiswell-posed ifu2

0

t

g

t

L2

0

T

)

.) The previously derived apriori estimate guarantees that u1

0

t

L2

0

T

)

andwe canview theproblemasa well-posedproblemwithareﬂective externalboundarythatconsequentlyenjoyoptimalconvergencerates.

3. Theparaboliccase

aux



uxx,withaninterfaceatx

=

0 andtrail-off conditionstotheleftandright.TheSBP second-derivativeoperatorisgivenas:D2

P−1

A

B S

)

where A is symmet-ric positive semi-deﬁnite and S holds aﬁrst-derivative stencilsat theboundary points. Following [2], theapproximation becomes,

P vL

a Q vL

A

B S

vL

1e0

v0L

v0R

2

0

SvL

0

SvR

0

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P vR

a Q vR

A

B S

vR

3e0

vL0

v00

4

0

SvR

0

SvL

0

).

In[2] thestablechoicesofthe

σ

’sarederived.Furthermore,these

σ

’shavethefollowingdependenciesofh:

1,3

h−1

and

2,4

1

.

LuL

h

LuLx

LuR

h

LuxR

f

uR

uRx

).

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Asuﬃcientconditionforthisstand-aloneproblemtobewell-posedisuR

L2

0

T

anduxR

L2

0

T

)

.(Ormoreprecisely,

huRx

L2

0

T

)

.)Furthermore,wesimultaneouslydemandthemirroredconditionsontherightside.

Remark.NotethattheinterfaceconditionisanapproximationofuL

0

t

uR

0

t

)

andnotaRobin-typeconditionsince

1,3are

h−1

and

2,4are

1

)

.Furthermore,parabolicityrequiresaboundaryconditiononallboundariesimplyingthataninterface

conditiononbothsidesoftheinterfacedoesnotimplyover-speciﬁcationofthestand-aloneproblems,inaccordancewithRequirement (2).

Next, we investigatethe stability of the stand-aloneproblems. Thatis, we want thenumerical approximationsof (6) to be bounded in L2

0

T

)

.By perusing the stability proof in[2], itis clearthat

SvL

0 and

SvR

)

0 are indeed bounded in L2

0

T

)

. (See p. 358-9. The eigenvalues associated with

SvL

0

SvR

0 and

SvL

0

SvR

)

0 are non-zero and non-vanishing.) However, a bound on vR

0 or vL0,which isneeded forstand-alonewell-posedness of thetwo problems,is not obtainedfromtheinterface stability.(Thecorresponding eigenvalueiszeroonp.359.)Torecoverthenecessaryestimates on vR0 orvL0,weproceedviatheenergyestimateof(5) whichgivesthebounds

sup t∈{0,T}

v L

T

0

vL

AvLdt

(7) sup t∈{0,T}

v R

T

0

vR

AvRdt

0

T

L

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D

h−1

1 1 0

0

1 1 0

1 1 0 0

,

such that Dv is an approximation of the derivative in the domain and

Dv

2h

=

hvTDTDv bounds the L2-norm of the derivativeofv (interpretedasapiecewiselinearfunction).

Lemma3.1.Foranygridfunctionv,vTAv

c

Dv

2

hforsomec

>

0.

Proof. By assumptionofnullspaceconsistency, A has onezeroeigenvalue associatedwithits rowssummingtozero.(By symmetryitscolumnsthereforealsosumtozero.)Furthermore,itscomponentsare

h−1

)

andweassumethat A isN

×

N. ACholeskyfactorisationisgivenasUTU

=

A.Since A issemi-deﬁnite,itwillresultinonezeroonthediagonalofthe upper-triangularmatrixU .Furthermore,thebandedstructureofA ispreservedinU .Since A1

0,wemusthaveU 1

=

0. Hence,theelementinthelowerrightcornerhastobe0.

SincetherowsofU sumtozero,wecanwriteU

=

R D,whereR isan N

×

N matrixwiththelastrowbeingzero.(The lastcolumnisalsotakentobezerosinceitdoesnotcontributeanywayduetothebandstructure.)Furthermore,noother rowof R sumto zero,since (most) rowsof A

=

DTRTR D aresecond derivative approximationsandthe two derivatives areaccountedforby DT andD.Hence, RTR issymmetricpositivesemi-deﬁnitewithonezeroeigenvalueassociatedwith the last rowandcolumn. Moreover, we denotethe upper-left

N

1

N

1

)

submatrix of RTR as M, which isthen symmetricpositivedeﬁniteandhaselementsof

h

)

.Furthermore,since R isbanded, M

chI forsome c

0 whereI is

the

N

1

)

2-identitymatrix.Consequently,

Dv

TRTR

Dv

c

Dv



2

h,sincethezerointhelast rowof RTR coincidewith thezeroinDv andM deﬁnesanorm.



By(7) andLemma3.1,wehaveDvL,R

L2

0

T

L2

,R

))

which,togetherwiththeL2boundsin(7) onvL,R andSobolev embedding,impliesvLi,R

L2

0

T

L

,R

SvL,R

0

L2

0

T

)

,therightproblemboundsthe“data” f in(6) fortheleftproblem(andviceversafor therightproblem).

Remark.Above, we used thebound on

SvL,R

)

0 obtained from the stabilityproof. However, since onlyh

SvL,R

0

L2

0

T

)

wasneeded, we couldalsohaveobtainedthatdirectlyfrom theL2

0

T

L2

,R

))

bounds on DvL,R,whichgives

hDvL,R

L2

0

T

L

,R

.Seeingthat

SvL,R

)

0arelinearcombinationsof

DvL,R

,theL2

0

T

boundson

h

SvL

0and

h

SvR

)

0 follow.

Weendthissectionwithafewremarks.ThesamereasoningappliestoparabolicsystemssincetheytooyieldL2

0

T

)

boundsonboththevariablesandtheirgradients,whichistheonlyrequirementneededtoapplythetheoryin[7]. Further-more,by mirroringone domainandrewritinga parabolicequation(scalar orsystem)such thattheinterface turnsintoa reﬂectiveboundary,theconvergencerateofthelatterfollowsimmediately.

4. Multi-dimensions

Apossible problemwiththeprevious procedureis that L∞ isnot embedded in H1 inmultipledimensions. However, in multipledimensionsthe boundednessrequirements are suitablymilder.Consider the 2D advection-diffusion equation,

ut

aux

uxx

+

uy y),withperiodicityinthey-direction(andN points).

Using von Neumann analysisin the y-direction reduces the problemto k

1

...

N 1-D problems,one foreach mode,

ˆ

vLk,R inthey-direction.They-derivativeisreducedtoalow-ordertermandcanbeignored.The H1 embeddingimpliesan

L2

L

boundonv

k,forallk

1

...

N.EachmodeconvergeswithoptimalrateandbyParseval’srelation,sodoesvL,R. Thisreasoningshowsthataboundon 0T∂

L,R

u

L,R

)

2dsdt issuﬃcientintheperiodic(intheextradimensions)case, andgivenbythe H1 embedding.Theoryforthenon-periodiccaseislacking,butitappearsreasonablethatthesamebound wouldbesuﬃcient.

CRediTauthorshipcontributionstatement

MagnusSvärd:Writingoriginaldraft.JanNordström:Reviewingandediting. Declarationofcompetinginterest

The authors declare that they haveno known competingﬁnancial interests or personal relationships that could have appearedtoinﬂuencetheworkreportedinthispaper.

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