Unexpected convergence of lattice Boltzmann schemes

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Computers and Fluids

journalhomepage:www.elsevier.com/locate/compfluid

Benchmark solutions

Unexpected convergence of lattice Boltzmann schemes ^R

Bruce M. Boghosian

^a

, Francois Dubois

^b,c,∗

, Benjamin Graille

^c

, Pierre Lallemand

^d

, Mohamed Mahdi Tekitek

^e

aDepartment of Mathematics, Tufts University, Bromfield-Pearson Hall, Medford, MA 02155, USA

bConservatoire National des Arts et Métiers, Laboratoire de Mécanique des Structures et des Systèmes Couplés, Paris F-75003, France

cDepartment of Mathematics, University Paris-Sud, Orsay Cedex F-91405, France

dBeijing Computational Science Research Center, Zhanggguancun Software Park II, Haidian District, Beijing, 10 0 094, China

eDepartment of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, Tunis 2092, Tunisia

a rt i c l e i n f o

Article history:

Received 6 November 2017 Revised 3 April 2018 Accepted 27 April 2018 Available online 28 April 2018 MSC:

76M28 Keywords:

Heat equation Damped acoustic Dispersion equation Taylor expansion method

a b s t r a c t

Inthiswork,westudynumericallytheconvergenceofthescalarD2Q9latticeBoltzmannschemewith multiplerelaxationtimeswhenthetimestepisproportionaltothe spacestepand tendstozero.We do thisbyacombinationoftheoryand numerical experiment.The classicalformalanalysis whenall therelaxationparametersarefixedand thetimesteptendstozeroshowsthatthenumericalsolution convergestosolutionsoftheheatequation,withaconstraintconnectingthediffusivity,thespacestep andthe coefficientofrelaxationofthemomentum.Ifthediffusivityisfixedandthe spacesteptends tozero,therelaxationparameterforthemomentumisverysmall,causing adiscrepancybetweenthe previousanalysisandthenumericalresults.Weproposeanewanalysisofthemethodforthisspecific situationofevanescent relaxation,based onthedispersionequationofthelattice Boltzmannscheme.

Anewasymptoticpartialdifferentialequation,thedampedacoustic system,isemergentas aresultof thisformalanalysis.ComplementarynumericalexperimentsestablishtheconvergenceofthescalarD2Q9 latticeBoltzmann schemewith multiple relaxationtimes and acoustic scaling inthis specificcase of evanescentrelaxationtowardsthenumericalsolutionofthedampedacousticsystem.

© 2018ElsevierLtd.Allrightsreserved.

1. Introduction

LatticeBoltzmannmodelsaresimplificationsofthecontinuum Boltzmannequationobtainedbydiscretizinginbothphysicalspace andvelocityspace.Thediscretevelocitiesv_iretainedtypicallycor- respond to lattice vectors of the discrete spatial lattice. That is, each lattice vertexx is linked to a finite number ofneighboring verticesbylatticevectorsv_i^t.Aparticledistributionfistherefore parameterizedbyitscomponentsineachofthediscretevelocities, the vertexxof thespatiallattice,andthediscrete timet.Atime stepofaclassicallatticeBoltzmannscheme[15]thencontainstwo steps:

(i)Arelaxationstepwherethedistributionfateachvertexxis locallymodifiedintoanewdistributionf^∗,and

R Contribution to be published in Computers and Fluids , edition 31 March 2018.

∗ Corresponding author at: Department of Mathematics, University Paris-Sud, Or- say Cedex F-91405, France.

E-mail addresses: [email protected] (B.M. Boghosian), [email protected] (F. Dubois), [email protected] (B. Graille), [email protected] (P. Lallemand), [email protected] (M.M.

Tekitek).

(ii) an advection step based on the method of characteristics as an exact time-integration operator. We employ the multiple- relaxation-timeapproach introducedby d’Humières [10],wherein thelocalmapping f−→ f^∗ isdescribedby adiagonaloperatorin aspaceofmoments.

In[6],wehavestudiedtheasymptoticexpansionofvariouslat- ticeBoltzmannschemeswithmultiple-relaxation timesfordiffer- entapplications.Weusedthe so-calledacoustic scaling,inwhich theratio

λ

≡^{x/tiskeptfixed.Wesupposedalsothattherelax-} ationoperatorremainsfixed.Inthismanner,wedemonstratedthe possibility ofapproximating diffusion processes describedby the heatequation.

The importance of using small values of relaxation parame- ters was recognized for linear viscoelastic fluids by Lallemand et al. [14]. Independently, unexpected results in simulations for advection-diffusionprocesses havebeen describedby Dellacherie in [4]. We have studied experimentally in [3] the curious con- vergence ofthe D1Q3 multiple-relaxation time lattice Boltzmann schemewithoneconservedvariablewhenusingtheacousticscal- inginonespatialdimension.Theasymptoticequationofthelattice Boltzmannschemeisnolongeranadvection-diffusionmodelbuta dampedacousticmodel.Inthiscontribution,weshowandanalyze https://doi.org/10.1016/j.compfluid.2018.04.029

0045-7930/© 2018 Elsevier Ltd. All rights reserved.

Fig. 1. Particle distribution f jfor 0 ≤j ≤8 of the D2Q9 lattice Boltzmann scheme.

an analogous phenomenon for two spatial dimensions with the scalarD2Q9latticeBoltzmannscheme.Thedifficultyconcernsthe highlightingoftheconvergencewiththenumericalexperiments.

In Section 2, we recall some fundamentals relative to the D2Q9 lattice Boltzmann scheme for scalar conservation laws. In Section3, westudyconvergence ofthisschemefordiffusive and acousticscaling. Aformal analysisis proposed in Section 4,with thedispersion equationmethod,initially proposedin[16].Wees- tablish that with acoustic scaling, the convergence of the scalar D2Q9schemeisnot theheat equationbutanunexpectedmodel!

Finally,westudytheexperimentalconvergenceofthescalarD2Q9 schemeinseveralsituationsinSection5.

2. ScalarD2Q9latticeBoltzmannschemeforthermalproblems

The D2Q9lattice Boltzmannscheme usesa setofdiscrete ve- locitiesdescribedinFig.1.Adensitydistributionf_jisassociatedto eachvelocityv_j≡

λ

^ej,where

λ

≡ ^x_t is the fixednumerical lattice velocity.The firstthree momentsforthedensityandmomentum aredefinedaccordingto

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩ ρ

=

8 j=0

fj=m0,

Jx≡

ρ

^ux=8

j=0

λ

^e1jfj=m1,

Jy≡

ρ

^uy=8

j=0

λ

^e2jf_j=m2,

(1)

wherethee^α_j are the

α

^{thcartesiancomponents ofthevectorse}j

introducedpreviously.Wecompletethissetofmomentsandcon- structavectormofmomentsm_kaccordingto

m=M f, (2)

withaninvertiblefixedmatrixMusually[15]givenby

M=

⎛

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎜ ⎝

1 1 1 1 1 1 1 1 1

0

λ

⁰ −

λ

⁰

λ

−

λ

−

λ λ

0 0

λ

⁰ −

λ λ λ

−

λ

−

λ

−4

λ

² −

λ

² −

λ

² −

λ

² −

λ

^{2 2}

λ

^{2 2}

λ

^{2 2}

λ

^{2 2}

λ

²

0

λ

² −

λ

²

λ

² −

λ

^{2 0 0 0 0}

0 0 0 0 0

λ

² −

λ

²

λ

² −

λ

²

0 −2

λ

^{3 0 2}

λ

^{3 0}

λ

³ −

λ

³ −

λ

³

λ

³

0 0 −2

λ

^{3 0 2}

λ

³

λ

³

λ

³ −

λ

³ −

λ

³

4

λ

⁴ −2

λ

⁴ −2

λ

⁴ −2

λ

⁴ −2

λ

⁴

λ

⁴

λ

⁴

λ

⁴

λ

⁴

⎞

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎟ ⎠

.

ForscalarlatticeBoltzmannapplications,thedensity

ρ

^isthe“con-

servedvariable”.

• The particle distribution atequilibrium f^eq is a function only ofthisconservedvariable.ForthisthermalD2Q9latticeBoltz- mannscheme,thevectorofequilibriummomentsm^eq isgiven by

m^eq=

ρ

^,0,0,

αλ

²

ρ

^,0,0,0,0,

λ

⁴

βρ

t

. (3)

Inmostapplications,thecoefficients

α

^and

β

^{areusually takento}

be

α

^=−2,

β

^{=1. (4)}

The lattice Boltzmann scheme is comprised of two fundamental steps : Relaxation andadvection. During the relaxationstep, the conservedvariable

ρ

^{isnotmodified,andthe}non-conservedmo- mentsm₁tom₈ relaxtowardsan equilibriumvalue:m^eq_k =

ψ

k(

ρ

) fork≥1, wherethe

ψ

k arethe linearfunctionsofthe conserved momentgivenby (3).Thespecificationofthisstepalsoneedsre- laxationratess_k:Fork≥1suchthat

m^∗_k=m_k+s_k

m^eq_k −m_k

,

where thesuperscript ^∗ denotes themoment m_k after the relax- ationstep.Thetableofrelaxationparameterss_kchoseninoursim- ulationsisasfollows

[s]=

s_J,s_J,se,sx,sx,sq,sq,s_ε

. (5)

Weintroduce alsothe8×8diagonalmatrixSwhosediagonalel- ementsarethecomponentsofthevector[s].Inourcomputations, wetakethefollowingnumericalvalues

se=1.7,sx=1.1,sq=1.1,s_ε=1.7. (6) Onlytherelaxationcoefficient s_Jforthefirst ordermomentumis allowedtovaryinournumericalexperiments.

Then using the matrix M⁻¹ the relaxation step becomes in f space:

f_i^∗

(

^x,t

)

^=M−i¹m^∗. (7)

Duringtheadvectionstepf_i(x_j)istransportedfromthenodex_jby the discretevelocity v_ito the node x_j+

v

i^t.Thus the evolution ofpopulationsf_ifor0≤i≤8atinternalnodexisdescribedby:

fi

(

x,t+

^t

)

=f_i^∗

(

^x−

v

i

^t,t

)

,0≤i≤8. (8)

•In[6],we haveanalyzed severallatticeBoltzmannmodelswith theTaylor-expansionmethod,includingthepresentonedefinedby Eqs.(2),(3),(5) and(8).Thehypothesis used wasthat therefer- encevelocity

λ

^{andtherelaxation coefficientss}J,se,sx,sq ands_ε remainconstantasthespatialstep^{xtendstozero.Thenthecon-} servedvariable

ρ

^{satisfies(atleastformally!)theheatequation:}

∂ρ ∂

^{t −}

κ ρ

=O

(

^x2

)

, (9)

wherethethermaldiffusivity

κ

^{isgivenbytherelation}

κ

^≡4+

α

6

σλ

^x,

σ

^≡

₁

s_J−1 2

. (10)

Thecoefficient

σ

^{isknownasthe“Hénon}parameter” inreference tothepioneeringworkofHénon[9].Observethatwhentherelax- ationcoefficientsJ andthemeshvelocity

λ

^{arefixed,thethermal}

diffusivitytends tozero asthespacestep ^{xtends tozero.This} latticeBoltzmannschemeisstableinthefluidcase(see [15]) un- derthecondition:

−4<

α

<2.

Forthescalarcase,thecondition

α

+4>0iscleartoassumethat the thermal diffusivity

κ

^{is positive (see (10)) and thecondition}

α

<2 corresponds to our experimental know how. Observe that

withthesechoices,thevalue oftherelaxationparameters_Jhasto befitwiththephysicaldiffusivity

κ

^{andthemeshsizexthrough} the relation(10)ifthespacestep andtime step arevarying pro- portionately.Inparticular,wehavetheexpansion

sJ=4+

α

6

κ λ

^x+O

⁽

^x2

⁾

⁽¹¹⁾

as^{xtendstozero.}

• Diffusive scaling can alsobe used andwe refer, e.g., to the workofJunketal.[11].Inthiscase,theratio

(

^x

)

²

^{t =}

λ

^x

remains fixed. This diffusive scaling is intensively used with the explicit finite difference methodfor solving the heat equation. It iswellknown[17]thatthetimestepmustbeproportionaltothe square of the spacial step in order for the method to be stable.

An asymptotic analysiscan be done forthis simplelattice Boltz- mannthermic model,as, e.g.,inourcontribution[7],andwe ob- tain againtheheatEq.(9)asthescalinglimit ofthemodel.With thisdiffusive scaling, theparameters

σ

^andsJ remain constant if the thermal diffusivity is given and the mesh size ^{x tends to} zero. Remark also that the convergence of the lattice Boltzmann schemewasrigorouslyprovedforthediffusivescalingforNavier–

Stokesflowsinperiodicandboundeddomainsin[12]andforone dimensionalconvection-diffusion-reactionequationsin[13]. 3. Firstnumericalexperiments

We studythe diffusion of a Gaussian profile in a square do- main.Inordertocontrol thecomputercost duringthenumerical experimentandtobecertainthatthenumericalexperimentisnot polluted by the boundary scheme, we impose periodic boundary conditions.We usetwo variants ofthe scalarD2Q9lattice Boltz- mannscheme:Diffusiveandacousticscaling.

• ScalarD2Q9numericalexperimentswithdiffusivescaling

Wesolvenumericallytheheatequation

∂ρ ∂

^{t −}

κ ρ

^{=0, (12)}

inthesquare=[−1,1]²,withperiodicboundaryconditions.The initialconditionisaGaussian:

ρ

0

(

^x,y

)

=exp

−x²+y² 0.09

,−1≤x,y≤1. (13)

Thecoefficients

α

^and

β

^oftheequilibriumare fixedaccordingto (4)andwekeepfixedtherelaxationcoefficientformomentum: sJ=3

2. (14)

Weusetheparticulardiffusivetimestep^t=^x2.Then

σ

≡ s¹_J −

1

2= ¹₆ andthediffusivityfollowstherelation

κ

=^σ₃^J and

κ

⁼ ₁₈^{1. (15)}

We have chosen an odd number ofmesh cells inthese numeri- cal experiments.With theconstraint^t=^x2, itis notpossible toobtain exactlythesameexactfinal time.We haveadaptedthe number of time steps in order to havevery close values forthe finaltimewiththedifferentmeshes.

• Comparisonwithfinite-differenceapproximation

Remarkthatthesolutionoftheheatequationonasquarewith an initial Gaussian and periodic boundary conditions has to our knowledgenoanalyticalsolution.Inconsequence,wecomparethe

Fig. 2. Two-dimensional heat Eq. (12) , κ= ₁₈¹. D2Q9 scheme with diffusive scaling (left) vs. explicit finite differences (right) ; mesh 111 ×111, time = 0.19479.

Fig. 3. Two-dimensional heat Eq. (12) , κ= ₁₈¹. D2Q9 scheme with diffusive scaling (left) vs. explicit finite differences (right) ; mesh 223 × 223, time = 0.16473.

Table 1

D2Q9 numerical experiments with diffusive scaling, s Jgiven by (14) and diffusivity κby (15) .

Number of cells 13 ×13 27 ×27 55 ×55 111 ×111 223 ×223 Nb. of time steps D2Q9 8 36 128 600 2048 Final time 0.18935 0.19753 0.17741 0.19479 0.16473

solution obtained by the lattice Boltzmann scheme withthe re- sultcomputedwithtwo-dimensionalfinitedifferences,centeredin spaceandexplicitintime.The degreesof freedomare locatedat half-integerpositions,exactly asdone withthelattice Boltzmann scheme:

ρ

_iⁿ₊¹

2,j+¹2≈

ρ

i+1 2

x,

j+1 2

x,n

^t

.

WefinitedifferencetheheatEq.(12)inthefollowingway:

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

1

^t

ρ

_iⁿ₊⁺¹¹

2,j+¹2−

ρ

_iⁿ₊¹

2,j+¹2

−

κ

1^x2

ρ

_iⁿ₊³

2,j+¹2

−2

ρ

_iⁿ₊¹

2,j+¹2

+

ρ

ⁿ_i−¹

2,j+¹2

+ 1

^y2

ρ

_iⁿ₊¹

2,j+³2−2

ρ

_iⁿ₊¹

2,j+¹2+

ρ

ⁿ_i+¹

2,j−¹2

=0.

Weuseexactly thesamegrid inspaceforboth schemesandex- actly the sametime step(and in consequencethe same number oftimesteps).Theparameters forbothschemesare comparedin Table1.

Theresultsfollowwhatisexpected.Theapproximatesolutions ofbothschemesare verysimilar asobserved inFigs.2and3for 111×111 and223×223 meshes. The difference betweenthetwo schemesatthefinaltime ispresentedinFig.4.Theorderofcon- vergenceofthisresidualisapproximatelyoforder4.Sincethefi- nitedifferencemethodisofsecond orderaccuracy[17],thisindi- catesthatthelatticeBoltzmannmethodapproachestheheatequa- tionwithsecond-orderaccuracy.

Table 2

D2Q9 numerical experiments with acoustic scaling. The diffusivity κ= ₁₈¹ is imposed in all the sim- ulations.

Number of cells 13 ×13 27 ×27 55 ×55 111 ×111 223 ×223

D2Q9 s Jparameter 1.5 1.182 0.830 0.52 0.298

Nb. of time steps D2Q9 8 16 32 64 128

Nb. of time steps, finite differences 8 32 128 512 2048

Final time 0.18935 0.18234 0.17902 0.17741 0.17661

Fig. 4. Two-dimensional heat Eq. (12) , κ⁼ 18¹. Difference of the numerical results computed with the D2Q9 scheme with diffusive scaling and explicit finite differ- ences at the final times presented in Table 1 . The order of convergence for this residual in the L ^∞norm is equal to 3.41 and in the L ²norm it is 3.96.

Fig. 5. Two-dimensional heat Eq. (12) , κ= ₁₈¹. D2Q9 lattice Boltzmann scheme with acoustic scaling (left) vs. explicit finite differences (right) ; results at time = 0.17741 for a 111 × 111 mesh.

• ScalarD2Q9numericalexperimentswithacousticscaling

We still wish to solve the heat Eq. (12) in the square = [−1,1]² withperiodic boundary conditions. The initial condition isagain givenby a Gaussian profile (13).The given diffusivity is imposed by the value (15). We adopt an acoustic scaling with ^t=^{xfortheD2Q9latticeBoltzmann}simulations.Wecompare the results with explicit finite differences; in this case, we take ^tࣃ^{x2 and the time step is chosen in order to obtain exactly} thesamefinaltimethanwiththelatticeBoltzmannmethod.

• ThenumericalresultspresentedinFigs.5and6forthetwo meshesof111×111and223×223seemcorrect.Butaquantitative examinationoftheresults(Fig.7) showsthat afteraconvergence similartotheoneobtainedfordiffusivescaling(seeFig.4),aper- sistentdifferenceappears.Thisqualitativebehaviorisverysimilar towhathasbeenobservedin[3]inonespatialdimension.

Fig. 6. Two-dimensional heat Eq. (12) , κ= ₁₈¹. D2Q9 lattice Boltzmann scheme with acoustic scaling (left) vs. explicit finite differences (right) ; results at time = 0.1766 for a 223 ×223 mesh.

Fig. 7. Two-dimensional heat Eq. (12) , κ= ₁₈¹, D2q9 scheme with acoustic scaling vs. explicit finite differences at the final times presented in Table 2 . There is no nu- merical evidence of coherence between the two methods when the mesh is refined.

Anewanalysisoftheschemeisnecessary toexplainthislack ofconvergencetowardstheexpecteddiffusivemodel.

4. Dispersionequationforanevanescentrelaxation

Inthissection, weproposeafirst-orderanalysiswhenthelast relaxationcoefficientsin (6)remainfixed orwhen therelaxation coefficients_J forthemomentum Jfollowsthechoicepresentedin Eq.(11),idest

sJ= 4+

α

6

λ

²

κ

^{t +O}

⁽

^t2

⁾

^{. (16)}

• Fixedrelaxations

Wewrite the relation(8)interms ofthemoments mdefined in(2):

m_k

(

^x,t+

^t

)

=

j

M_{k j}M⁻_j¹m^∗

(

^x−

v

j

^t,t

)

. (17)

Before doinga Taylorexpansionatorder 1,weintroduce thefol- lowing“momentumvelocity” operatormatrix^{definedaccording} to

k ≡ −

j α

Mk j

v

^α_jM⁻_j¹

∂

∂

^xα. (18)

For the D2Q9 scheme, this matrix can be explicitly calculated [5]andwehave

(19)

Wesplitthemomentvectorintotwoblocks:

m=

W Y

(20)

withW=

ρ

^{inourscalarexampleandYa columnvector with8}

components.Wedecompose alsotheoperatormatrix^intofour blocksthatrespectthedecomposition(20):

≡

A B

C D

. (21)

In our case, A is a scalar 1×1 matrix, B has one line and 8 columns,Ciscomposedby8linesand1columnandDisa8×8 squarematrixasshownintheright-handsideofrelation(19).We can alsointroduce a constantmatrix E with8linesandone col- umnsuchthattherelation(3)canbewrittenintheform

Y^eq≡Em. (22)

Therelation(17)isexpandedatfirstorder:

m+

^t

∂

^tm+O

(

^t2

)

=m^∗+

^t

^m∗+O

(

^t2

)

⁽²³⁾

anddueto(22),wehave m^∗=

I 0 SE I−S

m. (24)

Therelation(23)canbewrittenintheform Lm≡m^∗−m+

^t

−

∂

^tm+

^m∗

=O

(

^t2

)

, (25) with

L≡

0 0 SE −S

+

^t

−

∂

^{t 0}

0 −

∂

^t

+

A B

C D

I 0 SE I−S

. (26)

The dispersion relation associated with the relation (25) can be writteninasimpleway:

detL=0. (27)

We expand this determinant in order to eliminate the non- conserved moments Y. Moreover, due to the right-hand side of Eq.(25),wecanneglectallthetermsofsecondorthirdorderrel- ative to^{t.We write the expression(26) ofthe matrix Linthe} form

L=

^t

(

−

∂

^t+A+BSE

)

^tB

(

^I−S

)

SE+

^t

(

^C+DSE

)

−S+

^t

(

−

∂

t+D

(

^I−S

))

.

WeapplyGaussian eliminationinordertomakeexplicitthecon- dition(27).WemultiplythismatrixatleftbytheregularmatrixK definedby

K=

I

^tB

(

^I−S

)

^S−1

0 I

. (28)

Thenwehave,aftersomelinesofalgebra, KL=

I

^tB

(

^I−S

)

^S−1 0 I

×

^t

(

−

∂

^t+A+BSE

)

^tB

(

^I−S

)

SE+

^t

(

^C+DSE

)

−S+

^t

(

−

∂

t+D

(

^I−S

))

=

^t

(

−

∂

^t+A+BSE

)

+

^tB

(

^I−S

)

^{S−1SE O}

(

^t2

)

SE+O

(

^t

)

^−S+O

(

^t

)

andwehavethefollowingtriangularformfortheproductKL: KL=

^t

(

−

∂

^t+A+BE

)

^O

(

^t2

)

SE+O

(

^t

)

−S+O

(

^t

)

.

Thentherelation(27) isequivalentatfirstorderto thefollowing setoffirstorderpartialdifferentialequations:

(

−

∂

^t+A+BE

)

^W=O

(

^t

)

, (29) recoveringthefirststepoftheBerlinalgorithmpresentedinAugier etal.[2].Forthescalardiffusionproblem,thisequation expresses simplythat

∂

t

ρ

=O

(

^t

)

.

Thisresultisconsistentwiththesecond-order analysispresented attherelationin(9).

Whenweusediffusivescaling,thisdispersionequationcanbe adaptedinordertorecovertheheatequation atzeroorderofac- curacy.It isthenequivalenttotheTaylorexpansionmethodwith thediffusivescaling,asusedin[7].

• Evanescentrelaxations

When^{tandxtendtozerowiththeacousticscaling,these} twoinfinitesimalsareofthesameorder.Theexpansion(16)ofthe relaxationcoefficient s_J implies that theprevious asymptoticcal- culus has to be mademore precise. The coefficient s_J is now at first order proportional to the time step ^{t. We decompose the} non-conservedmomentsYintotwofamilies:The quasi-conserved momentsUidestthetwocomponentsofthemomentumJinthe scalarcase– andtheothertrulynon-conservedmomentsZ: Y=

U Z

. (30)

The 8-componentvector Yis split into a first vectorU∈R² and asecond oneZ with6components.Inother words,thefamilyof momentsissplitintothreecomponents:

m=

_W

U Z

.

Then the8×8relaxation matrixS can be decomposedintotwo blocks:

S=

^tS+O

(

^t2

)

⁰

0 SZ

. (31)

Thetopleftblockintherighthandsideof(31)tendstozeroasthe meshisrefined.TheequilibriumvectorEisnaturallysplitintothe quasi-conservedcomponentE_U andthe trulyrelaxingcomponent EZ:

E=

EU

EZ

. (32)

We have: SE=

^{tS 0}

0 S_Z

E_U E_Z

=

^tSEU

S_ZE_Z

and the relation (24)takestheform

m^∗=

_{I 0 0}

^{tSEU I−}

^{tS 0} SZEZ 0 I−SZ

m. (33)

Then the momentum velocity operator matrix ^{is split into 9} blocks:

=

_{A A}

2 B1

A₃ A₄ B₂ C1 C2 D4

. (34)

Thisblockstructure (34)isexplicitlygivenforourthermal D2Q9 intheform

Then L=

_{0 0 0}

^{tSEU −}

^{tS 0} SZEZ 0 −SZ

−

^t

∂

^{t 0 0}

0

∂

t 0 0 0

∂

^t

+

^t

_{A A}

2 B1

A3 A4 B2

C1 C2 D4

_{I 0 0}

^tSEU I−

^{tS 0} SZEZ 0 I−SZ

.

This expression can be expanded tofirst order in^{t without} anychangeintheresultoftheGaussianelimination.Thenwecan neglectthe terms of order one in^{t inthe last product oftwo} matrices.Weobtain

_{A A}

2 B₁ A3 A4 B2

C1 C2 D4

_{I 0 0}

0 I 0

SZEZ 0 I−SZ

=

_A+B

1S_ZE_Z A₂ B₁

(

^I−SZ

)

A3+B2SZEZ A4 B2

(

^I−SZ

)

C1+D4SZEZ C2 D4

(

^I−SZ

)

, and,uptoorderO(^t),wehave

L=

_t

₍

₋

∂

t+A+B1SZEZ

)

^tA2

^tB1

(

^I−SZ

)

^t

(

^SEU+A₃+B₂S_ZE_Z

)

^t

(

−S−

∂

^t+A₄

)

^tB2

(

^I−S_Z

)

SZEZ+

^t

(

^C1+D4SZEZ

)

^tC2 −SZ+

^t

(

−

∂

^t+D4

(

I−SZ

)

.

(35) WiththemethodofGaussianeliminationusedpreviously,wemul- tiplythematrixLobtainedin(35)ontheleftbythefollowingma- trix

K=

_{I 0}

_tB

1

(

^I−SZ

)

^S_Z⁻¹ 0 I

^tB2

(

^I−SZ

)

^SZ⁻¹

0 0 I

whosedeterminantis equal to1. After some elementaryalgebra, weobtain

KL=

_t

₍

₋

∂

^t+A+B1EZ

)

^tA2 O

(

^t2

)

^t

(

^SEU+A3+B2EZ

)

^t

(

−S−

∂

^t+A4

)

^O

(

^t2

)

SZEZ+O

(

^t

)

^tC2 −SZ+O

(

^t

)

.

Ononehand,detK=1andontheotherhand,thelastcolumnof thematrixKLiscomposedofnegligibletermsexceptforthelast

one. Thenwe havethe condition(27) ifandonlyifthe determi- nant ofthe 2×2 upperblock matrix isnull. In other terms,this matrixhasanontrivialkernelatorderonerelativeto ^tandwe have

−

∂

t+A+B1EZ A2

SEU+A3+B2EZ −

∂

t−S+A4

W U

=O

(

^t

)

^{. (36)}

Then the equivalent partial differential equations are written as a system involving theconserved variable W andthe quasi con- servedmomentsU:

∂

^{tW =}

(

^A+B1EZ

)

^W+A2U+O

(

^t

)

∂

^tU+SU=

(

^A3+B2EZ+SEU

)

^W+A4U+O

(

^t

)

. (37) Thisresultgeneralizesthe firstanalysisdone in[3]fortheD1Q3 scheme. When we replace the block matrices introduced in the relations (31), (32) and (34) by their D2Q9 values, we estab- lishthat withthe acousticscaling, the scalarD2Q9 lattice Boltz- mannscheme withacoustic scaling admitsthe following asymp- toticdampedacousticmodel

⎧ ⎪

⎨

⎪ ⎩

∂ρ ∂

^{t +divJ=O}

⁽

^x

⁾

∂

^Jα

∂

^{t +c20}

∂ρ

∂

^xα +gJ_α=O

(

^x

)

,1≤

α

^{≤2, (38)}

withasoundvelocityc₀ andadampingcoefficientggivenbythe relations

c²₀=

λ

²

6

(

⁴+

α )

, g= c²₀

κ

^{. (39)}

The above is a very interesting analysis, and clearly the correct two-dimensional analog ofthe earlier resultfor D1Q3. We point outthatitisequivalenttoadampedwaveequation.

5. ScalarD2Q9schemeconvergingtowardsdampedacoustic

Wehave nowtwo partialdifferentialequations withwhichto comparethenumericalsolutionobtainedwiththescalarD2Q9lat- ticeBoltzmannscheme: Theinitialheat Eq.(12)andthedamped acoustic system (38). We first consider numerical experiments done in Section 3 and compare our previous results with this new model. We also study in detail the eigenmodes of the sys- tem (38)and propose a simple numerical experimentwith a si- nusoidal analytic solution.The evolution of an initial Gaussian is againperformed,withtwodiffusioncoefficientsvaryingbyoneor- derofmagnitude.

• Dampedacousticsasalimitingmodelforthepreviousnumeri- calexperiments?

We wish to approximate the system of damped acoustic Eqs.(38).Thesoundvelocityisgivenby(39).Withthechoice(4), we obtain the classical value c₀= ^√λ₃. The imposed diffusivity

κ

andtherelation(39)fixthevalueg=6forthezero-orderdamp- inginthemomentumequationof(38).Thegeometryisthesquare =[−1,1]²withperiodicboundaryconditions.Theinitialdensity isstill givenbya Gaussianprofile(13). Becausethemomentum J atequilibriumisidenticallynull,wehavetakenthisspecificvalue asinitialconditionofourlattice Boltzmannsimulations.Wesup- poseinconsequencethat theinitialconditionforthemomentum issimplyJ(^x,t=0)=0.

We adoptacoustic scaling with ^t=^{xfor the D2Q9 lattice} Boltzmann simulations. Forthe acoustic system(38), we useex- plicit finite differences withstaggered grids, hereafter named as

“HaWAY” method and described with some details in the Ap- pendix.Inthiscase,theacoustictimestep^taforfinite-difference

Table 3

Numerical experiments with the D2Q9 lattice Boltzmann test case studied in Section 3 com- pared with the damped acoustic model (38) simulated with the HaWAY method.

Number of cells 13 ×13 27 ×27 55 × 55 111 ×111 223 ×223 D2Q9 s Jparameter 1.5 1.182 0.830 0.52 0.298

Nb. of time steps d2q9 8 16 32 64 128

Idem, finite differences 32 64 128 256 512

Final time 0.18935 0.18234 0.17902 0.17741 0.17661

Fig. 8. Numerical results for the damped acoustic model with the experimental plan proposed in Table 3 .

Fig. 9. Two-dimensional wave with wave vector k = 2 π(1 , 1). Initial condition.

simulations is proportional to the spatial step ^{x, with a sta-} bility constraint. The corresponding experimentsare describedin Table3.

The resultsobtained withthisnewexperimentare very simi- lar tothe one obtainedin Section 3.In particular, the numerical resultscomputedwiththedampedacoustic modelareveryclose totheonespresentedinFigs.5and6.Whenwelooktothecon- vergence withquitefinegrids (Fig.8),thesignalisbetterthan in Fig.7butthisexperimentisstillnotentirelyconvincing.

• Wavesforthedampedacousticmodel Wesearchmodesofthetype

ρ

=

ρ

0exp

(

−

γ

^t+ik·x

)

J=J0exp

(

−

γ

^t+ik·x

)

⁽⁴⁰⁾

Fig. 10. Two-dimensional wave with wave vector k = 2 π(1 , 1), g ࣃ 6, 2| k | c 0ࣃ 5.924.

Autocorrelation of density.

Fig. 11. Two-dimensional wave with wave vector k = 2 π(1 , 1), g ࣃ 6, 2| k | c 0ࣃ 5.924.

Convergence towards the damped acoustic model (38) .

forthedampedacousticmodel38–(39).Thenwehavetosolvethe followingill-posedlinearsystem:

−

γ ρ

^{0+ik·J0=0, ik·}

ρ

⁰⁺

(

^g−

γ )

^J0=0. (41) Afirstsolutionisatransversestationarywave with

γ

=g,

ρ

0=0 andk·J₀=0.We donot considerthismode inthiscontribution.

Thentheothermodessatisfythefollowingdispersionrelation

γ

^2−g

γ

⁺

|

^k

|

^2c20=0. (42) Thisequationhascomplexpropagativerootswhen

g<2

|

^k

|

^c0, (43)

Fig. 12. Two-dimensional wave with wave vector k = 2 π(1 , 1), g ࣃ 5.6470. Autocor- relation of density with 2| k | c 0ࣃ 5.924 for various meshes.

Fig. 13. Two-dimensional wave with vector k = 2 π(1 , 1), g ࣃ 5.6470. Convergence towards the damped acoustic model (38) with 2| k | c 0ࣃ 5.924. The order of conver- gence is 1.27 for the L ²norm and 1.30 in norm L ^∞.

i.e.,whenthediffusivity

κ

^{issufficientlylargemeasuredinascale}

systembasedonthesoundvelocityandwavenumber:

κ

> c0

2

|

^k

|

^.

Inthatcase,theeigenvalue

γ

^takestheform

γ

⁼₂^g∓i

ω

^,

ω

⁼

|

^k

|

^2c20−g²

4. (44)

Theeigenvectorsarefinallygivenaccordingto

⎧ ⎨

⎩

ρ

=

ρ

0exp

−g 2t

exp

i

(

^k·x±

ω

^t

)

J=i k

|

^k

|

²

ρ

0

−g 2∓i

ω

exp

−g 2t

exp

i

(

^k·x±

ω

^t

)

. (45)

Weconsiderapureanalyticaltestcaseasthenextexperiment.

• Atwo-dimensionalsinusoidalwave

We keep the value

κ

= 18¹ 0.05555 of the diffusivity intro- ducedin(15). We usethe traditionalvalue c₀= ^√1₃ andthe dis- sipationcoefficient g (see (39)) is still equalto g=6.We change thedomain and consider[0, 2

π

^{]2 with the initial condition}

ρ

=

cos

(²

π

(^x+y)

andJ=0.Then k=2

π

(¹,1) ^{andthe right-hand} side of (43) is 2|k|c0ࣃ5.924. In this case, the damped acoustic model(38)exhibitsanon-propagativemode.

Theinitial conditionispresentedinFig.9.Theautocorrelation ofdensity

(

^t

)

≡

ρ (

x,t

) ρ (

x,0

)

^dx

| ρ (

^x,0

) |

^2dx

is typical ofa diffusion process asshownin Fig.10. The conver- genceforsimpledyadicmeshesispresentedinFig.11.

Asecond numericalexperimenthasbeenconducted. We keep the same domain [0, 2

π

^{]2 with the same initial condition}

ρ

= cos

(²

π

(^x+y)

. Then k=2

π

(¹,1) ^{and the right-hand side of} (43)is equal to2|k|c₀ࣃ5.924. We change thevalue of thediffu- sivity

κ

^{introducedin(15)to}

κ

=288¹⁷ 0.05903.Wekeepthetra- ditionalvaluec₀= ^√1₃.Thenthedissipationcoefficientg(see(39)) isnowgࣃ5.6470.Thenthedampedacousticmodel(38)exhibitsa propagativemodeinthiscase.Theautocorrelationfunctionispre- sentedinFig.12.TheconvergencecurveisdepictedinFig.13.We observe that this convergence is not regular. An extra-fine mesh withdimensions1024×1024hasbeennecessaryinordertocon- firmtheorderofaccuracy.

• ComplementaryexperimentsforaninitialGaussian

We havecomparedthe scalarD2Q9lattice Boltzmannscheme with acoustic scaling with numerical solutions of the heat Eq. (12) as presented in Section 3 and with HaWAY simulations of the damped acoustic system (38) in Section 4. We consider again the first geometry studied in this contribution, id est the square =[−1,1]² withperiodic boundaryconditions.An initial Gaussianprofile(13)isgivenatt=0.Twonumericalexperiments havebeenconsidered:Aquiteviscousonewithimposeddiffusivity

κ

=0.15andanotheronewith

κ

= 0.015.Thenumericalparame- tersaredisplayedinTable4.

The resultsfor the firsttest case with

κ

=0.15 are presented in Figs. 14–17. In Fig.14, a qualitative view of the numericalre- sultonagivenmeshshowsthatthescalarD2Q9schemeandthe HaWAYschemefordampedacousticareclosertoeachotherthan theyaretothesolutionoftheheatequation.Thethreeprofilesof densityareshowninFig. 15andacomparisonofautocorrelation functionsinFig.16.Evenonarelativelycoarsemesh,theconclu- sion is the same andour new asymptoticanalysis of the acous- tic system(38)is consistent withthe numericalresults. Lastbut notleast,boththeerrorbetweenD2Q9andthermicsononehand, andthatbetweenD2Q9anddampedacoustics ontheotherhand are displayedinFig.17.The errorbetweenthe latticeBoltzmann schemeandthedampedacousticresultstendstozerowhereasthe errorbetweenD2Q9andthethermicmodelremainsstationary.

Thesecond numericalexperiment with

κ

=0.015ispresented inFigs.18–20.Attime =2onarelatively coarsemesh,thethree numericalsolutionscan notbe distinguishedasshowninFig.18. Itisalsothecasefortheautocorrelationfunctionaspresentedin Fig. 19. The numerical convergence is delicate for this test case.

Duringonedecadeofmeshrefinement,thethreemethodspresent very close results as shownin Fig. 20. Two additional computa- tionson895×895and1791×1791refinedmesheshavebeennec- essarytodemonstratetheconvergenceofthescalarD2Q9scheme towards the damped acoustic system. Observe that the most re- finedmeshcontainsmorethan3millionscells!

6. Conclusion

We have first considered the scalar D2Q9 lattice Boltzmann scheme with diffusive scaling. Our experiments confirm numeri-