Contents lists available atScienceDirect

## Journal

## of

## Computational

## Physics

www.elsevier.com/locate/jcp

### Short note

## Convergence

## of

## energy

## stable

## ﬁnite-difference

## schemes

## with

## interfaces

### Magnus Svärd

a*,*

### ∗

### ,

### Jan Nordström

b*,*

c
a_{Dept.}_{of}_{Mathematics,}_{University}_{of}_{Bergen,}_{P.O.}_{Box}_{7803,}_{5020}_{Bergen,}_{Norway}

b_{Computational}_{Mathematics,}_{Department}_{of}_{Mathematics,}_{Linköping}_{University,}_{SE-581}_{83}_{Linköping,}_{Sweden}

c_{Department}_{of}_{Mathematics}_{and}_{Applied}_{Mathematics,}_{University}_{of}_{Johannesburg,}_{P.O.}_{Box}_{524,}_{Auckland}_{Park}_{2006,}_{South}_{Africa}

### a

### r

### t

### i

### c

### l

### e

### i

### n

### f

### o

### a

### b

### s

### t

### r

### a

### c

### t

*Articlehistory:*
Received6May2020

Receivedinrevisedform19November2020 Accepted20November2020

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.

©2020TheAuthor(s).PublishedbyElsevierInc.Thisisanopenaccessarticleunderthe CCBYlicense(http://creativecommons.org/licenses/by/4.0/).

**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.*E-mailaddresses:*Magnus.Svard@uib.no(M. Svärd),jan.nordstrom@liu.no(J. Nordström).

1 _{Convergence}_{rate}_{is}_{the}_{rate}_{by}_{which}_{the}_{solution}_{error}_{decreases with}_{grid}_{reﬁnements.}_{The}_{order}_{of}_{accuracy}_{is}_{given}_{by}_{the}_{truncation}_{error}_{and}
canbedifferentatdifferentpoints.Itrepresentshowaccuratelytheequation(notthesolution)isapproximated.

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

0021-9991/©2020TheAuthor(s).PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

Here,weaddressbothinterfacesandreﬂectiveboundaryconditionsandwediscusspossiblegeneralisationstomultiple dimensions.Speciﬁcally,werequire

(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**

Weconsidertheadvectionequationontherealline,

*ut*

### +

*aux*

### =

0*,*

*a*

*>*

0*.*

(1)
We splitthedomainintotwo: *x*

*<*

0 and *x*

*>*

0 withaninterface at*x*

### =

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 indexing

*xi*

### =

*ih,*

*i*

### =

0*,*

### ±

1*,*

### ±

2*,*

*...*

,such that *x*0 representsthe interface in bothdomains.

**v**_{t}L

### +

*a P*−1

**Q v**L### =

0 (2)**v**_{t}R

### +

*a P*−1

**Q v**R### = −

*P*−1

*a*

*(*

*v*

_{0}

*R*

### −

*v*

_{0}

*L*

*)*

**e**0 where

**v**

*L*

_{= (...}

_{= (...}

_{v}L−1

*,*

*vL*0

*)*

*T*and

**v**

*R*

*= (*

*v*0

*R*

*,*

*vR*1

*,*

*...)*

*T*are the unknowns to the left and right and

**e**0

*= (*

1*,*

0*,*

*...*

0*)*

*T*. The SBP operator has the following properties:

*P*

### =

*PT*

_{>}

_{>}

_{0,}

_{Q}_{+}

_{Q}T_{=}

_{B}_{=}

_{diag}_{(}

_{(}

_{−}

_{1}

_{,}

_{,}

_{0}

_{,}

_{,}

_{...,}

_{...,}

_{0}

_{,}

_{,}

_{1}

_{)}

_{)}

_{and}

*−1*

_{P}

_{Q v is}_{a}

_{ﬁrst-derivative}approximation.(See[7] formoreinformation.)Toadherewiththesecondrequirementabove,thereisonlyapenaltyterm totherightinagreementwiththecharacteristicdirection.

**Remark.***Generally,onecanpenalisebothequationsatx*0*and,withaproperscaling,stabilitycanbeproven.Althoughstableand*

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

Weapplytheenergymethodinasomewhatunconventionalway.Multiplytheﬁrstequationin(2) by**v***L _{P to}*

_{obtain}

**v**

*L*2

*t*

### +

*a*

*(*

*v*

*L*0

*)*

2### =

0*.*

The left problemis well-posed asa stand aloneproblemand we obtainthe bound *vL*_{0}

### ∈

*L*2

*(*

0*,*

*T*

*)*

. Next,we multiply the
secondequationin(2) by*(*

**v**

*R*

_{)}

_{)}

*T*

_{P ,}**v**

*R*2

*t*

### −

*a*

*(*

*v*

*R*0

*)*

2### = −

*2av*0

*R*

*(*

*v*0

*R*

### −

*vL*0

*).*

Byrecastingtheboundaryterms,weobtain
**v**

*R*2

*t*

### +

*a*

*((*

*v*

*R*0

### −

*v*0

*L*

*)*

2### =

*a*

*(*

*v*0

*L*

*)*

2*.*

This problemis bounded, andwell-posed,since *vL*_{0}

### ∈

*L*2

*(*

0*,*

*T*

*)*

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

### +

*Aux*

### =

0 discretisedby**v**_{t}L

*+ (*

*A*

### ⊗

*P*−1

*Q*

*)*

**v**

*L*

*= +(*

*A*−

### ⊗

*P*−1

*)((*

*vL*

_{0}

### −

*vR*

_{0}

*)*

### ⊗

**e**

_{0}

*L*

*)*

(3)
**v**_{t}R

*+ (*

*A*

### ⊗

*P*−1

*Q*

*)*

**v**

*R*

*= −(*

*A*+

### ⊗

*P*−1

*)((*

*vR*

_{0}

### −

*v*

_{0}

*L*

*)*

### ⊗

**e**

_{0}

*R*

*)*

where**e***L,R* _{are}_{both}_{one}_{at}_{x}

_{=}

_{0 and}

_{zero}

_{elsewhere.}

_{(For}

_{more}

_{information}

_{on}

_{Kronecker}

_{products,}

_{see}

_{[}

_{6}

_{].)}

_{The}

_{division}

*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 independentlyboundedin

*L*2

_{(}

_{(}

_{0}

_{,}

_{,}

_{T}_{)}

_{)}

_{such}

_{that}

_{optimal}

_{rates}

_{are}

_{obtained.}

*2.1.* *Reﬂectiveboundaryconditions*

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

*<*

0, to *ξ >*

0. Thatis, by *x*

*= −ξ*

and
*∂x*

*= −∂*

*ξ*,suchthat,

*u _{t}L*

### −

*au*

_{ξ}L### =

0*,*

*ξ >*

0*.*

(4)
Relabelling

*ξ*

as*x,uL*

### =

*u*1,

*uR*

### =

*u*2,weobtainthesystem,

*u*1

*u*2

*t*

### +

### −

*a*0 0

*a*

*u*1

*u*2

*x*

### =

0*,*

*x*

*>*

0
withtheboundarycondition, *u*1

*(*

0*,*

*t*

*)*

### =

*u*2

*(*

0*,*

*t*

*)*

.(Inparticularwe note thattheproblemiswell-posed if*u*2

*(*

0*,*

*t*

*)*

### =

*g*

*(*

*t*

*)*

### ∈

*L*2

*(*

0*,*

*T*

*)*

.) The previously derived apriori estimate guarantees that *u*1

*(*

0*,*

*t*

*)*

### ∈

*L*2

*(*

0*,*

*T*

*)*

andwe canview theproblemasa
well-posedproblemwitha*reﬂective external*boundarythatconsequentlyenjoyoptimalconvergencerates.

**3.** **Theparaboliccase**

Weconsiderthesamesetupfortheadvection-diffusionequation,*ut*

### +

*aux*

### =

*uxx*,withaninterfaceat

*x*

### =

0 andtrail-off conditionstotheleftandright.TheSBP second-derivativeoperatorisgivenas:*D*2

### =

*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 v**L

### +

**a Q v**L### =

*(*

### −

*A*

### +

*B S*

*)*

**v**

*L*

### +

*σ*

1**e**0

*(*

*v*0

*L*

### −

*v*0

*R*

*)*

### +

*σ*

2*e*

0*((*

**Sv**L*)*

0*− (*

**Sv**R*)*

0*)*

(5)
**P v**R

### +

**a Q v**R### =

*(*

### −

*A*

### +

*B S*

*)*

**v**

*R*

### +

*σ*

3**e**0

*(*

*vL*0

### −

*v*00

*)*

### +

*σ*

4*e*

0*((*

**Sv**R*)*

0*− (*

**Sv**L*)*

0*).*

In[2] thestablechoicesofthe

* σ*

’sarederived.Furthermore,these* σ*

’shavethefollowingdependenciesof*h:*

* σ*

1*,*3

### ∼

*O(*

*h*−1

*)*

and* σ*

2*,*4

### ∼

*O(*

1*)*

.
Considertheleftproblem.Itisanapproximationoftheadvection-diffusionequationwithboundarycondition

*α*

*LuL*

### +

*h*

*β*

*LuL*

_{x}### =

*α*

*LuR*

### +

*h*

*β*

*Lu*

_{x}R### =

*f*

*(*

*uR*

*,*

*uR*

_{x}*).*

(6)
Asuﬃcientconditionforthisstand-aloneproblemtobewell-posedis*uR*

### ∈

*L*2

*(*

0*,*

*T*

*)*

and*u*

_{x}R### ∈

*L*2

*(*

0*,*

*T*

*)*

.(Ormoreprecisely,
*huR _{x}*

### ∈

*L*2

*(*

0*,*

*T*

*)*

.)Furthermore,wesimultaneouslydemandthemirroredconditionsontherightside.
**Remark.***NotethattheinterfaceconditionisanapproximationofuL*

*(*

0*,*

*t*

*)*

### =

*uR*

*(*

0*,*

*t*

*)*

*andnotaRobin-typeconditionsince*

* σ*

1*,*3

*are*

*O(*

*h*−1

_{)}

_{)}

_{and}_{ σ}

_{ σ}

2*,*4*are*

*O(*

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 *L*2

*(*

0*,*

*T*

*)*

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

**Sv**L*)*

0 and*(*

**Sv**R*)*

0 are indeed bounded
in *L*2

_{(}

_{(}

_{0}

_{,}

_{,}

_{T}_{)}

_{)}

_{.}

_{(See}

_{p.}

_{358-9.}

_{The}

_{eigenvalues}

_{associated}

_{with}

_{(}

_{(}

**L**_{Sv}_{)}

_{)}

0

*+ (*

**Sv**R*)*

0 and*(*

**Sv**L*)*

0*− (*

**Sv**R*)*

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

0 or *vL*0,which isneeded forstand-alonewell-posedness of thetwo problems,is not
obtainedfromtheinterface stability.(Thecorresponding eigenvalueiszeroonp.359.)Torecoverthenecessaryestimates
on *vR*_{0} or*vL*_{0},weproceedviatheenergyestimateof(5) whichgivesthebounds

sup
*t*∈{0*,T*}

**v**

*L*

_{ ≤}

_{C}

_{C}

_{,}

_{,}

*T*0

*(*

**v**

*L*

*)*

**Av**Ldt### ≤

*C*

*,*

(7)
sup
*t*∈{0

*,T*}

**v**

*R*

_{ ≤}

_{C}

_{C}

_{,}

_{,}

*T*0

*(*

**v**

*R*

*)*

**Av**Rdt### ≤

*C*

*.*

Sincetheestimatesobtainedwiththe *A-matrix*abovecorrespondto*L*2_{-estimates}_{of}_{the}_{gradients,}_{we}_{intend}_{to}_{use}_{Sobolev}
embeddingtobound**v***L* _{and}_{v}*R* _{in}* _{L}*2

_{(}

_{(}

_{0}

_{,}

_{,}

_{T}_{;}

*∞*

_{L}_{(L}

_{(L}

*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

*2*

**Dv**

_{h}### =

*the*

**hv**TDT**Dv bounds***L*2-norm of the derivativeof

**v (interpreted**asapiecewiselinearfunction).

**Lemma3.1.***Foranygridfunction v,*

**v**

*T*

_{Av}_{>}

_{>}

_{c}_{}

_{Dv}_{}

2
*hforsomec*

*>*

*0.*

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

*O(*

*h*−1

_{)}

_{)}

_{and}

_{we}

_{assume}

_{that}

_{A is}_{N}_{×}

*ACholeskyfactorisationisgivenas*

_{N.}*UTU*

### =

*A.*Since

*A is*semi-deﬁnite,itwillresultinonezeroonthediagonalofthe upper-triangularmatrix

*U .*Furthermore,thebandedstructureof

*A is*preservedin

*U .*Since

**A1**### =

**0,**wemusthave

**U 1**### =

**0.**Hence,theelementinthelowerrightcornerhastobe0.

Sincetherowsof*U sum*tozero,wecanwrite*U*

### =

*R D,*where

*R is*an

*N*

### ×

*N matrix*withthelastrowbeingzero.(The lastcolumnisalsotakentobezerosinceitdoesnotcontributeanywayduetothebandstructure.)Furthermore,noother rowof

*R sum*to zero,since (most) rowsof

*A*

### =

*DTRTR D are*second derivative approximationsandthe two derivatives areaccountedforby

*DT*

_{and}

_{D.}_{Hence,}

_{R}T_{R is}_{symmetric}

_{positive}

_{semi-deﬁnite}

_{with}

_{one}

_{zero}

_{eigenvalue}

_{associated}

_{with}the last rowandcolumn. Moreover, we denotethe upper-left

*(*

*N*

### −

1*)*

*× (*

*N*

### −

1*)*

submatrix of *RTR as*

*M,*which isthen symmetricpositivedeﬁniteandhaselementsof

*O(*

*h*

*)*

.Furthermore,since *R is*banded,

*M*

### ≥

*chI for*some

*c*

*>*

0 where*I is*

the

*(*

*N*

### −

1*)*

2_{-identity}

_{matrix.}

_{Consequently,}

_{(}

_{(}

_{Dv}_{)}

_{)}

*T*

_{R}T_{R}_{(}

_{(}

_{Dv}_{)}

_{)}

_{≥}

_{c}_{}

_{Dv}_{}

2
*h*,sincethezerointhelast rowof *RTR coincide*with
thezeroin* Dv andM deﬁnes*anorm.

By(7) andLemma3.1,wehave**Dv**L,R

_{∈}

*2*

_{L}_{(}

_{(}

_{0}

_{,}

_{,}

_{T}_{;}

*2*

_{L}_{(L}

_{(L}

*,R*

*))*

which,togetherwiththe*L*2boundsin(7) on

**v**

*L,R*andSobolev embedding,implies

*vL*

_{i},R### ∈

*L*2

_{(}

_{(}

_{0}

_{,}

_{,}

_{T}_{;}

*∞*

_{L}_{(L}

_{(L}

*,R*

*))*

forall*i.*Withtheseaprioriestimates,andsincewealreadyknowfromthe stabilityproofthat

*(*

**Sv**L,R_{)}

_{)}

0

### ∈

*L*2

*(*

0*,*

*T*

*)*

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

**Remark.***Above,* *we* *used* *thebound* *on*

*(*

**Sv**L,R_{)}

_{)}

0 *obtained* *from* *the* *stabilityproof.* *However,* *since* *onlyh*

*(*

**Sv**L,R*)*

0 ### ∈

*L*2

*(*

0*,*

*T*

*)*

*wasneeded,* *we* *couldalsohaveobtainedthatdirectlyfrom* *theL*2

*(*

0*,*

*T*

### ;

*L*2

*(L*

*,R*

*))*

*bounds*

*on*

**Dv**L,R,whichgives### √

**hDv**L,R

### ∈

*L*2

_{(}

_{(}

_{0}

_{,}

_{,}

_{T}_{;}

*∞*

_{L}_{(L}

_{(L}

*,R*

*))*

*.Seeingthat*

*(*

**Sv**L,R*)*

0*arelinearcombinationsof*

*(*

**Dv**L,R*)i*

*,theL*2

*(*

0*,*

*T*

*)*

*boundson*

### √

*h*

*(*

**Sv**L_{)}

0_{)}

*and*

### √

*h*

*(*

**Sv**R_{)}

0
_{)}

*follow.*

Weendthissectionwithafewremarks.Thesamereasoningappliestoparabolicsystemssincetheytooyield*L*2

_{(}

_{(}

_{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 *H*1 inmultipledimensions. However,
in multipledimensionsthe boundednessrequirements are suitablymilder.Consider the 2D advection-diffusion equation,

*ut*

### +

*aux*

### +

*buy*

### =

*(*

*uxx*

### +

*uy y)*,withperiodicityinthey-direction(and

*N points).*

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

### =

1*...*

*N 1-D*problems,one foreach mode,

### ˆ

**v***L _{k},R* inthey-direction.They-derivativeisreducedtoalow-ordertermandcanbeignored.The

*H*1 embeddingimpliesan

*L*2

*(*

*L*∞

*)*

boundon**v**

### ˆ

*k*,forall

*k*

### =

1*...*

*N.*EachmodeconvergeswithoptimalrateandbyParseval’srelation,sodoes

**v**

*L,R*. Thisreasoningshowsthataboundon

_{0}

*T*

_{∂}*L,R*

*(*

*u*

*L,R*

_{)}

2_{)}

_{ds}_{dt is}_{suﬃcient}

_{in}

_{the}

_{periodic}

_{(in}

_{the}

_{extra}

_{dimensions)}

_{case,}andgivenbythe

*H*1 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.

**References**

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