A notion of non-negativity preserving relaxation for a mono-dimensional three velocities scheme with relative velocity

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Journal of Computational Science

j o u r n al ho me p a g e :w w w . e l s e v i e r . c o m / l o c a t e / j o c s

A notion of non-negativity preserving relaxation for a

mono-dimensional three velocities scheme with relative velocity

Franc¸ ois Dubois

^a,b

, Benjamin Graille

^b,∗

, S.V. Raghurama Rao

^c

aConservatoireNationaldesArtsetMétiers,LMSSClaboratory,Paris,France

bLaboratoiredeMathématiquesd’Orsay,Univ.Paris-Sud,CNRS,UniversitéParis-Saclay,91405Orsay,France

cDepartmentofAerospaceEngineering,IndianInstituteofScience,Bangalore560012,India

a r t i c l e i n f o

Articlehistory:

Received20November2019 Receivedinrevisedform19May2020 Accepted18June2020

Availableonline30June2020

MSC:

76M28 65M12 Keywords:

Non-negativitypreservingproperty Advectionprocess

Numericaloscillations

a b s t r a c t

In this contribution, we studya stabilitynotionfor afundamental linearone-dimensional lattice Boltzmannscheme,thisnotionbeingrelatedtothemaximumprinciple.Weseektocharacterizethe parametersoftheschemethatguaranteethepreservationofthenon-negativityoftheparticledistri- butionfunctions.Inthecontextoftherelativevelocityschemes,wederivenecessaryandsufficient conditionsforthenon-negativitypreservingproperty.Theseconditionsarethenexpressedinasimple waywhentherelativevelocityisreducedtozero.Forthegeneralcase,weproposesomesimplenec- essaryconditionsontherelaxationparametersandweputinevidencenumericallythenon-negativity preservingregions.Numericalexperimentsshowfinallythatnooscillationsoccurforthepropagationof anon-smoothprofileifthenon-negativitypreservingpropertyissatisfied.

©2020ElsevierB.V.Allrightsreserved.

1. Introduction

StudyingthestabilityoflatticeBoltzmannschemesisanon- trivialproblem.Classicallyforthispurpose,theschemeislinearized aroundaconstant stateanda vonNeumann-Fourieranalysisis performed.Forthisnotionofstability,werefertotheworkofSter- lingandChenin[21]wheresomestabilityresultsfora7-velocity hexagonallattice,a9-velocitysquarelattice,anda15-velocitycubic latticeareproposed;theworkofLallemandandLuo[17]fora9- velocitysquarelatticeschemeappliedtohydrodynamics;theone ofSieberetal.forathermalandthermalmodelswithalargernum- berofvelocitiesintwospacedimensions[20];theoneofGinzburg etal.[11]extendingtheFourieranalysistoawidevarietyofdiffer- enttwoandthree-dimensionallatticeBoltzmannschemes;theone ofKrivovichev[16]forsixwidelyusedbodyforceactionmodels;

theoneofWissocqetal.[24]forprojectinginformationcarriedby thelatticeBoltzmanneigenvectorsonthephysicalmodes.

Instabilitiesandtheirinterpretationintermsofbulkviscosity havebeenproposedbyDellar[4].Butnomathematicalanalysishas beenperformed.

∗Correspondingauthor.

E-mailaddress:[email protected](B.Graille).

A new way of improvingstability is proposed by Geier[9], whoproposedanewgeneralizedlatticeBoltzmannschemewith the approach of relative velocities and utilized it for hydrody- namicsapplications[6].Anattempttoanalyzethismethodfora two-dimensionalscalarlinearproblemhasbeenalsoproposed[7].

Interestingtentativeshavebeenproposedtoenforcestabilitycon- ditionsofmultiple-relaxationtimelatticeBoltzmannschemeswith raworcentralmomentswithvonNeumannanalysisandheuris- tic selectionofwave-number vectorsby Golbertetal.[12] and Chávez-Modenaetal.[3].

Evenifitisadifficulttask,itiswellknownthatFouriermethod (vonNeumannanalysis)isalinearapproachandinconsequenceis notrelevantforanalyzingnon-linearhyperbolicequations.

Totalvariationdiminishingschemes,developedforsuppress- ingoscillationsinhigher-orderCFDalgorithms[1,23,22],provide analternativenonlinear stabilityanalysistool foranalyzing the schemesfornonlinearwavepropagation.Theconvergenceofsuch schemesiswellestablished[18].Theunderlyingstabilitynotion concernsthemaximumprinciple.In brief,ifsomesolutionofa partialdifferentialequationispositiveontheboundary,itremains positivein allthedomainofstudy[13,10,15].ThelatticeBoltz- mannschemesdonotintrinsicallysatisfythepropertyofmaximum principleortheassociatednon-negativityconstraintasdetailedby Karimietal.[14].Thisnotioncanbeextendedtonon-linearcases andafirstattemptforlatticeBoltzmannschemeshasbeenpro- https://doi.org/10.1016/j.jocs.2020.101181

1877-7503/©2020ElsevierB.V.Allrightsreserved.

posedin[2],fortheD1Q2schemeusedtosimulatescalarnon-linear hyperbolicequations.

Inthiscontribution,weproposetoinvestigatethestabilityin themaximumsense,ofalinearmono-dimensionallatticeBoltz- mannschemewiththreevelocities.Moreprecisely,welooktoa non-negativityconstraintfortheparticledistributionfunctionsin thecontextofrelativevelocities.Theobjectiveisthedescription oftheparametersetsoftheschemethatallowtheparticledis- tributionfunctionstoremainnon-negative.Itisobviouslylinked withthenon-negativityoftheirequilibriumvaluesbutitonlycoin- cideswiththelatterpropertyifalltherelaxationparametersare takento1.WereferforinstancetoServan-Camasetal.[19]where thispropertyisinvestigatedforseveralschemesusedtosimulate advection–diffusionequation.

InSection2,wedescribetheschemeandtheunderlyingadvec- tionmodel. More precisely, thelocal relaxationstep is written asa linearoperator ontheparticledistributionfunctions.Ifall the coefficients of the underlying matrix are nonnegative, the non-negativityofthedistributionismaintainedduringthisstep.

Because the transport step is just a change of locus, the non- negativityismaintainedforthewholetimestepofthescheme.The questionisthentofindappropriateconditionstohandlethisprop- erty.InSection3,anecessaryandsufficientconditionisderived onthe parameters toensure that thescheme hasthestability property.InSection4,wecompletelydescribetheclassicalcase wheretherelative velocityisreduced tozero.In Section5,the generalcaseispresented,withananalyticalstudyfornecessary conditionsandanumericaloneforacompletedescriptionofthe stabilityzones.InSection6,thepresentednumericalexperiments show the correlation of the positivity constraint for a particle distribution and the presence of oscillations for discontinuous profiles.

2. Descriptionoftheframework 2.1. Descriptionofthescheme

In this contribution, we investigate a mono-dimensional 3 velocitieslinearlatticeBoltzmannschemewithrelativevelocity [6].Denotingxthespatialstep,tthetimestep,and=x/t theschemevelocity,thisschemecanbedescribedinageneralized d’Humière’sframework[5]:

(1)the3velocitiesc₁=−1,c₂=0,andc₃=1;

(2)the3associateddistributionsf1,f2,andf3; (3)the3moments,q(u),andε(u)givenby

=

1≤j≤3

f_j, q(u)=

1≤j≤3

(c_j−u)f_j, ε(u)

=3²

1≤j≤3

(c_j−u)²f_j−2²

1≤j≤3

f_j,

whereuisagivenscalarrepresentingtherelativevelocity;

(4)theequilibriumvalueofthe3moments

^eq=, q^eq(u)=(V−u), ε^eq(u)=²(3u²−6uV+˛), whereVand˛aregivenscalars(withoutlossofgenerality, weassumethatV>0);

(5)the2relaxationparameterssands suchthattherelaxation phasereads

q(u)=(1−s)q(u)+sq^eq(u), ε(u)=(1−s)ε(u)+sε^eq(u).

In this formalism, the moments are defined as polynomial functionsofdiscretevelocitiesandthediscretedistributionfunc- tions.Indeed,introducingtheonevariablepolynomialsP₁(X)=1, P₂(X)=X,andP₃(X)=²(3X²−2)relativetoanabstractindeter- minateX,thethreemomentsread

=

1≤j≤3

f_jP₁(c_j−u), q(u)=

1≤j≤3

f_jP₂(c_j−u), ε(u)

=

1≤j≤3

f_jP3(c_j−u).

Theequilibriumvaluesarechosensuchthattheequilibriumdis- tributionsdonotdependontherelativevelocityu.Indeed,wehave:

f_j^eq= 1 6

2+3c_jV+(3c²_j −2)˛

, 1≤j≤3.

Notethatthisschemecanbeused(seee.g.[8])tosimulateascalar transportequationwithconstantvelocityVgivenby

∂t+V∂x=0.

Thisadvectionequationisafirst-orderasymptoticlimitasthespace andtimestepstendtozerowithafixedratio≡x/t.Weare notinterestedinthiscontributioninthesecond-orderasymptotic expansionandrefertotherelativevelocityschemesofFévrieretal.

[6]forthesame.

Withthenotationx_k=kx,k∈Z,andtⁿ=nt,n ∈N,onetime stepoftheschemereads

f_j(tⁿ+t,x_k+c_jt)=f_j(tⁿ⁺¹,x_k+j)=f_j(tⁿ,x_k), 1≤j≤3.

Notethattheschemedoesnotdependontherelativevelocity uinthecase wheretherelaxationparametersareidentical,i.e., s=s.Thetwomomentsq(u)andε(u)dependonubuttheparticle distributionfunctionsremainidenticalateach timeiteration.In thatcase,forallvaluesofu,theschemeyieldsthestandardBGK scheme.

2.2. Anotionofnon-negativitypreservingrelaxation

Inthiscontribution,weareconcernedwiththenon-negativity oftheparticledistributionfunctions.Weproposethefollowingdef- initionofstabilitythroughnon-negativitypreservingrelaxation.

Definition1(Non-negativitypreservingrelaxation). Therelaxation phaseissaidtobenon-negativitypreservingif

∀^{j f}j≥0⇒∀^{i f}_i≥0. (1)

Thispropertycanbeviewedasreferringtoaweakmaximum principleforschemes.Indeed,itis alwayspossible,byaddinga constant,toassumethatalltheparticledistributionfunctionsare initiallynon-negative.Iftheschemeensuresthatthispropertyof non-negativityremainsastimemarches,eachparticledistribution functionisthenbounded,astheirtotalsumisconserved.Moreover, asthetransportstepconsistssimplyinexchangingthepositionof theparticledistributionfunctions,wefocusontherelaxationstep.

Inotherwords,ifisconservedduringtherelaxationstep(and periodicboundaryconditionsareusedforcompactspaceset),the sumofoverallthespacepointsx_k,k ∈Z,isconstant.Thenthe particledistributionfunctionsverifyamaximumprinciple (∀k ∈Z,∀j f_j(0,x_k)≥0)⇒(∀k ∈Z,∀n ∈N,∀j0≤f_j(tⁿ,x_k)≤)¯ with

¯

=

k∈Z

j

f_j(0,x_k).

Notethatthepropertyofnon-negativitypreservingrelaxationis automaticallysatisfiediftheequilibriumvaluesofthedistribution functionsarenon-negativeandiftherelaxationparametersverify s=s∈(0,1].Thisresultdoesnotdependontherelativevelocity u(astheequilibriumvaluesofthedistributionfunctionsdonot dependonu).Wewillrecoverthisresultinthefollowing.

2.3. Matrixnotationfortherelaxationstep

Weuseamatrixnotationfortherelaxationstepasitcanbe readasamultiplicationbyamatrix.Asthisstepislocalinspace, weomitthedependencyontimeandspace.Wedefinethevector ofthedistributionfunctionsf

f=(f₁ f₂ f₃)^T.

Onerelaxationstepthenreads f=R(u)f,

wherethematrixR(u)isdefinedby R(u)=M⁻¹T(−u)(I+S(T(u)ET(−u)−I))T(u)M, with

M=

⎛

⎝

₋^{1 1}0 ^{1 2} −2^{2 2}

⎞

⎠

, T(u)=

⎛

⎝

₋¹u ⁰1 ⁰0 3²u² −6u 1

⎞

⎠

,

S=

⎛

⎝

⁰0 ⁰s ⁰0 0 0 s

⎞

⎠

, E=

⎛

⎝

V^{1 0}0 ⁰0

˛² 0 0

⎞

⎠

, I=

⎛

⎝

¹0 ⁰1 ⁰0 0 0 1

⎞

⎠

.

ThecoefficientsofthematrixMareobtainedbytherelations M_k,j=P_k(c_j),1≤k,j≤3andthoseofthematrixT(u)bythechange ofbasisformula:T(u)_k,listhecoefficientofthel^th-elementP_l(cj)in thedefinitionofP_k(c_j−u)accordingto

P_k(c_j−u)=

1≤l≤3

T(u)_k,lP_l(c_j), 1≤j≤3.

ThematrixMisthenthechangeofbasisthattransformsthevector fintothevectorm(0)=(,q(0),ε(0))^T:

m(0)=Mf, m(u)=T(u)Mf.

ThematrixT(u)canthenbeviewedasthechangeofbasismatrix fromtheclassicalmomentswithoutrelativevelocitytowardthe momentswithrelativevelocity.

2.4. Remarkonthechoiceofthemoments

Notethatthelastmomentε(u)thatischoseninthiscontribution isnottheenergybutamomentthatisorthogonaltothetwofirst ones,andq(u).Inthissection,weshowthatalltheresultsofthe contributionwouldbeidenticalbychoosingthelastmomentasthe energy:therelaxationmatrixR(u)wouldstillbethesame.

Weconsidertwoschemeswithtwodifferentchoicesofpoly- nomials:themomentsofthefirstschemearedefinedby(P1,P2, P₃)whilethemomentsofthesecondschemeby( ˆP₁,Pˆ₂,Pˆ₃).The firstmomentisthesameinbothschemestobeabletosimulate thesametransportequation.Wethenhave ˆP₁=P₁.WedefineC thechangeofbasismatrixassociatedtothetranformationMinto^M:

^M=CM.

ThefirstlineofCisthen(1,0,0).

Proposition2. Weassumethattheequilibriumvaluesofthedistri- butionfunctionsarethesame,thatis^E=CEC⁻¹,andthattherelaxation parametersarethesame,thatis^S=S.Then,wehave^R(u)=R(u)for all(s,s)iff

Pˆ₂ ∈Span(P₁,P₂), Pˆ₃ ∈Span(P₁,P₃), in R[X]/X(X−1)(X+1).

Inapracticalway,theexactchoiceofthemomentshasnoinflu- enceontheprecisecomputation ofthematrixR(u)even inthe relativevelocityframework.Ifforexample,weuseononehand thematrixproposedpreviouslyinthiscontributionMandonthe otherhandthefavoritemomentmatrixofoneofus,^M,with

M=

⎛

⎝

₋^{1 1}0 ^{1 2} −2^{2 2}

⎞

⎠

, ^M=

⎛

⎝

₋^{1 1}0 ^{1 2}/2 0 ²/2

⎞

⎠

,

wejustchangethedefinitionofthethirdmomentnamed“energy”

inthiscontribution.Weobservethat

C=

⎛

⎝

¹0 ⁰1 ⁰0 ²/3 0 1/6

⎞

⎠

.

TheequilibriummatricesEand^E

E=

⎛

⎝

V^{1 0}0 ⁰0

˛² 0 0

⎞

⎠

, ^E=

⎛

⎜ ⎝

1 0 0

V 0 0

²˛+2

6 0 0

⎞

⎟ ⎠

arelinkedby^E=CEC⁻¹ inordertomaintainidenticaltheequi- libriumdistributionfunctions.Thenwemaintainunchangedthe relaxationcoefficientsbecauseS=^S.

Proof. First, weimmediatelyobtainthefollowingrelationsby identifyingthecoefficientsof^MandofM:

Pˆ_k(c_j)=

1≤l≤3

C_k,lP_l(c_j), 1≤j≤3.

Wededucethat

^T(u)=CT(u)C⁻¹. Wehave

^R(u) =^M⁻¹^T(−u)(I+^S(^T(u)^E^T(−u)−I))^T(u)^M

=M⁻¹T(−u)C⁻¹(I+S(CT(u)ET(−u)C⁻¹−I))CT(u)M

=M⁻¹T(−u)(I+C⁻¹SC(T(u)ET(−u)−I))T(u)M.

Then

^R(u)−R(u) =M⁻¹T(−u)C⁻¹(SC−CS)T(u)(E−I)M.

AsthematricesM,T(u),T(−u),andCareinvertible,thecondition

^R(u)=R(u)isequivalentto(SC−CS)T(u)(E−I)=0.Denoting

C=

⎛

⎝

1 0 0

c21 c22 c23

c₃₁ c₃₂ c₃₃

⎞

⎠

,

astraightforwardcalculationyields

(SC−CS)T(u)(E−I)=(s−s)

⎛

⎝

0 0 0

c23²(˛−6V) c236u −c23

c₃₂V −c₃₂ 0

⎞

⎠

.

Thentheproperty^R(u)=R(u)forallvaluesofsandsisequivalent toc₂₃=c₃₂=0,thatendstheproof.䊐

2.5. PositivityoftheR(u)matrix

ThevelocityVbeingfixed,weproposetogiveafulldescription ofthesets

^V,u= {(s,s,˛)∈R³ such that R(u) is a non−negative matrix}, u∈R. Therelaxationphaseisnon-negativitypreservinginthesenseofthe

Definition1,forgivenVandu,ifandonlyiftheparameters(s,s,˛) arein^V,u.Indeed,thenon-negativityofthematrixR(u)imposes thatallthedistributionsf_˛,˛∈{−,0,+},remainnon-negativeif theyaresoattheinitialtime.Thesesetsarefirstdescribedbya setofnineinequalitiesthatcanbejoinedintojustone.Numeri- calillustrationsarethengiventovisualizeitinthecharacteristic casesincludingsinglerelaxationtime,multiplerelaxationtimeand relativevelocityscheme.

Inthiscontribution,weassumethatV≥0withoutlossofgen- eralityas

^−V,−u=^V,u.

Thislastpropertyisobviousaftersomealgebraorafterremark- ingthat thetransformationjust exchangesthef₁ andf₃ values (correspondingtothevelocities±).

3. Positivityoftheiterativematrix

ThenineinequalitiesobtainedfromthematrixR(u)canbecom- binedneatlyintooneformula.

Theinequalitiesare

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩

R_0,0 =Vsu−Vs

2 −Vsu+˛s

6 +su− s

2−su−s

6+10, R0,1 =Vsu−Vs

2 −Vsu+˛s 6 +s

3 0, R_0,2 =Vsu−Vs

2 −Vsu+˛s

6 −su+ s

2+su−s 60, R_1,0 =−2Vsu+2Vsu−˛s

3 −2su+2su+s 3 0, R1,1 =−2Vsu+2Vsu−˛s

3 −2s

3 +10, R_1,2 =−2Vsu+2Vsu−˛s

3 +2su−2su+s 3 0, R_2,0 =Vsu+Vs

2 −Vsu+˛s

6 +su+ s

2−su−s 60, R2,1 =Vsu+Vs

2 −Vsu+˛s 6 +s

3 0, R_2,2 =Vsu+Vs

2 −Vsu+˛s

6 −su− s

2+su−s

6+10.

(2)

Weprovenowthatthepreviousnineinequalitiescanbewritten inamuchmorelucidway.

Proposition 3. We introduce the reduced parameters u and accordingto

u=2u(s−s), =s

6(1−˛)−u(s−s)V. (3) ThentheninepreviousinequalitiesR_i,j≥0displayedin(2)areequiv- alentto

max(s−1,|u|)≤2 ≤min(2−s−|u−sV|,s−|u+sV|,s−|sV|).(4) Proof. ConsiderfirstthetwoinequalitiesassociatedwithR0,0and R_2,2:

⎧ ⎪

⎨

⎪ ⎩

Vsu−Vsu+˛s 6 − s

2−s

6+1Vs

2 −su+su Vsu−Vsu+˛s

6 − s 2−s

6+1−Vs

2 +su−su.

Theycanbesynthesizedinthefollowingform:

^u2−Vs 2

^≤1−2^s−(s

6(1−˛)+u(s−s)V)=1− s 2− andwecanwritethisrelationas

2 ≤2−s−|u−sV|. (5)

Writenowtheinequalities(2)associatedwithR_0,1andR_2,1:

⎧ ⎪

⎨

⎪ ⎩

Vsu−Vsu+˛s 6 +s

3 Vs 2 Vsu−Vsu+˛s

6 +s 3−Vs

2. Inotherwords,|^Vs₂|≤− +^s₂.Then

2 ≤s−|sV|. (6)

Wenowfocusontheinequalities(2)associatedwithR0,2andR2,0:

⎧ ⎪

⎨

⎪ ⎩

Vsu−Vsu+˛s 6 + s

2−s 6 Vs

2 +su−su Vsu−Vsu+˛s

6 + s 2−s

6 −Vs

2 −su+su

Wehave|^Vs₂ +su−su|≤Vsu−Vsu+^˛s₆+^s₂−^s₆ =₂^s− .Incon- sequence,

2 ≤s−|u+sV|. (7)

ConsideringtheinequalitieswithR1,0andR1,2,wehave

⎧ ⎪

⎨

⎪ ⎩

−Vsu+Vsu−˛s 6 +s

6 su−su

−Vsu+Vsu−˛s 6 +s

6 −su+su

and|¹₂u|≤−Vsu+Vsu−^˛s₆+^s₆ = .Inconsequence,

|u|≤2 . (8)

ThelastinequalityR1,1≥0canbewrittenas−Vsu+Vsu−^˛s₆−

s

3 +¹₂ 0and this inequalityis equivalentto −^s₂+¹₂ 0. In otherterms,

s−1≤2 . (9)

4. Theparticularcaseu=0

Inthissection,wesupposethattherelativevelocityuisreduced to zero. Then the necessary and sufficient conditions for non- negativitypreservingrelaxationcanbewrittenas

max(s−1,0)≤ s

3(1−˛)≤min(2−s−|sV|,s−|sV|,s−|sV|).(10) Proposition4. Tofixtheideas,wesupposethattheadvectionveloc- ityVispositive:

V≥0. (11)

ThecaseV≤0followsdirectly.Whenu=0,thereducedstabilitycon- ditions

max(s−1,0)≤min(2−s−|sV|,s−|sV|,s−|sV|) (12)

Fig.1. Necessaryandsufficientstabilityregionsdescribedbytheinequalities(12) foranullrelativevelocityu.IllustrationproposedforV=²₃.

areequivalenttothefollowingconditionsfortherelaxationparame- ters

⎧

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎩

0≤s,s≤2 s≥sV s≤ 2

1+V

s≤min(3−(1+V)s,1+(1−V)s)

(13)

joinedwithanaturalCouranttypeconditionforexplicitschemes

V≤1 (14)

fortheadvectionvelocity.

Ofcourse,theconditions(10)havestilltobeimposedforthe equilibriumparameter˛whenthepairs,sisgiven.Inparticular,

˛≤1 (15)

and s≤ 3

˛+2. (16)

Fig.1illustrates thenecessaryand sufficient stabilityregion describedbytheinequalities(12)inthecaseV=2=3.

Proof. Wefirstobservethat0≤max(s−1,0)≤|sV|≤min(2−s, s).

Then0≤s≤2.Secondly,wehave|sV|≤sandbecauseboths andVarepositive,wehave0≤sV≤s.WehavealsosV≤sand(14) isestablished.Moreover,s≥0and|sV|≤2−simpliess≤₁₊²_V.Due tothepositivityoftheparameters,wededucefrom(10)anew setofinequalities:s−1≤max(s−1,0)≤min(2−s−sV,s−sV).

Thens≤min(3−(1+V)s,1+(1−V)s).Moreover,s≤3−(1+V)s≤2 becauseV≥0.

Conversely, if the relations (13) and (14) are satisfied, we have s−1≤2−s−s V, s−1≤s−s V and s−1≤s−s V. Then s−1≤min(2−s−s V, s−s V, s−s V).Moreover,0≤2−s−s V, 0≤s−sVbecauseV≤1and 0≤s−sV.Thus0≤min(2−s−sV, s−sV,s−sV).Finallytheinequalities(12)areestablishedandthe propositionisproven.䊐

5. Thegeneralcase

Weanalysethegeneralcaseofnon-zerouinthissection,with suitableillustrationsofthestabilityregionforvariousrangesofthe parameters.

5.1. Necessaryconditionsforstability

Inthissubsection,weprovethefollowingproposition.

Proposition5. Wesupposethatthenecessaryandsufficientnon- negativitypreservingrelaxationconditions(4)aresatisfiedas

max(s−1,|u|)≤2 ≤min(2−s−|u−sV|,s−|u+sV|,s−|sV|) withthenotations(3):

u=2u(s−s), = s

6(1−˛)−u(s−s)V.

WesupposealsothattheadvectionvelocityVisnon-negative.Then, thepoint(s,s)satisfiesthefollowinginequalities

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩

0≤sV≤s≤2, 0≤sV≤1, 0≤s≤2,

s≤min(2−sV,s+1,3−s), s≤ 2

1+V.

(17)

Withthesenecessarystabilityconditions,theparameteruhasbeen eliminated.Wehavealso,necessarily

V≤1 (18)

and

|u|≤ 1

2. (19)

Proof. Westartfromtheinequalities(4).Thenwehave 0≤|u|≤max(s−1,|u|)≤min(2−s−|u−sV|,s−|u +sV|,s−|sV|)≤s−|sV|

ands≥|sV|≥sV.

We have the triangular inequality |sV|≤|u−sV|+|u|. Then from the general stability conditions (4), we deduce |u−sV|+

|u|≤2−sand |s V|≤2−s.In asimilar way,|sV|≤|u+sV|+|u|,

|u+sV|+|u|≤sfrom(4)andfinally|sV|≤2−s.Weputtogether thetwoinequalitiesandwehave0≤|sV|≤min(s,2−s).

Inconsequence,wehave0≤s≤2andthethirdpointisproven.

SincewemadethechoiceofV≥0thens≥0.Thefirstinequalityof thetwofirstpointsareestablished.FromsV≤swehaveV≤1and therelation(18)istrue.Moreover,sV≤2−sandthelastinequality of(17)istrue.

Consider now the inequalities s−1≤2 ≤min(2−s−|u− sV|,s−|u+sV|).We deduce s−1+|u−sV|≤2−s and s−1+

|u+sV|≤s.Duetothepositvityoftheabsolutevalues,wehave alsos≤3−sands≤s+1.Apartofthefifthinequalityof(17)is proven.

Finally,duetothetriangularinequality,sV=|sV|≤ ¹₂|u−sV|+

1 2|u+sV|.

Weadd thisinequalitywiththetwofollowingones:s−1+

|u−sV|≤2−s and s−1+|u+sV|≤s. Thens−1+sV≤¹₂(2− s+s)=1ands≤2−sV.Thefifthinequalityof(17)iscompletely established.BecausesV≥0,wehavealsos≤2andthefirstinequal- ityof(17)isalsoestablished.䊐

WeillustratethezonesofnecessarystabilityinFig.2forfive particularvelocities:V=0,¹₄,¹₃,¹₂and1.

Fig.2.Necessarystabilityregionsdescribedbytheinequalities(17)forV=0,V=¹₄[firstline,fromlefttoright],V=¹₃,V=¹₂[secondline,fromlefttoright],andforV=²₃, V=1[thirdline,fromlefttoright].

Fig.3.Numericalstudyofnecessaryandsufficientstabilityregions.

Fig.4. Smoothprofilewithacontinuousderivative.TheparametersfortheD1Q3aretunedinordertohave(left)ornot(right)thenon-negativityproperty.

Fig.5.Continuousprofile.TheparametersfortheD1Q3aretunedinordertohave(left)ornot(right)thenon-negativityproperty.

5.2. Numericalstudyofnecessaryandsufficientconditionsfor stability

Wenowillustratethenecessaryandsufficientstabilityregions forV=0,V=¹₄,V=¹₃,V= ¹₂,V=²₃ andV=1forvariousranges ofuinFig.3.Eachfigurerepresentstheprojectionoftheset^V,u ontothetwo-dimensionalplane(s,s)∈R²givenbytheexternal inequalitiesof(4):

max(s−1,|u|)≤min(2−s−|u−sV|,s−|u+sV|,s−|sV|).

Indeed,if(s,s)verifies(4),thereexistsalwaysavalueof˛suchthat (s,s,˛)∈^V,u.Thefigureswereobtainedbyfollowingthestraight edgesofthedomain,whichcanbewrittenasanintersectionof half-planes.

ForeachvalueofV,thenecessaryandsufficientstabilityregions givenbytherelations(4)aredisplayedforseveralvaluesofthe relativevelocity:u∈{−2V,−V,0,V/2,V,2V}.Indottedline,the necessarystabilityregionoftheProposition5isaddedforcompar- ison.Theparticularvalueu=0isenhancedbyfillingtheregionin gray.

Someanalysiscanbedrawnfromthesefigures:

•thestabilityregionchangeswiththerelativevelocity;

•themaximalvalueofthefirstrelaxationparametersisobtained (notonly)foru=0;

•thestabilityregionisnotclearlymorefavorable(largerorinclud- inggreatervalueofs)foru=V;

•thesegmentcorrespondingtos=s∈(0,1]isalwaysinthesta- bilityregion.

Toconcludethissection,thisnotionofstabilityallowsalarge setofvaluesfortherelaxationparameters.Iftheschemeisusedto

simulatethehyperbolicadvectionequationwithoutsecond-order operators,wecantrytominimizethenumericaldiffusionwhile maintainingthis stabilityproperty.Thistaskiscomplicatedand outofthescopeofthepaperasthenumericalsecond-orderterm readsasanon-linearformulathatlinksalltheparameterss,˛,and V.

6. Numericalillustrations

Inthissection,weillustratethestabilitypropertywithnumer- ical simulations involving the D1Q3 model, with and without relativevelocity,usedtosimulatethelinearadvectionequation.

Thestability isdemonstratedfortheparameterschosenaccord- ingtotheanalysispresentedintheprevioussections.Oscillations areseenwhenevertheparametersgobeyondthestabilitylimits presented,ashighlightedintheresults.

Theparameterschosenforthesimulationsarethefollowing:

V u s s ˛

Left(stable) 0.25 0.0 1.6 1.3 0.3076923076923076 0.25 0.25 1.6 1.3 −0.17548076923076938 Right(unstable) 0.25 0.0 1.9 1.4 0.14285714285714302

0.25 0.25 1.9 1.4 −0.10491071428571441 Moreover,allthesimulationsareperformedforthesamespace range[0,1]withperiodicboundaryconditions,withthesamespace stepx=1/128,thesametimestept=x,suchthat=1,until thefinaltimet=1.

Fortheleftfigures,theparametersarechoseninordertosatisfy thestability property.We observenumericallyalsoamaximum principle(theconservedmomentremainsintheinterval[0,1]

correspondingtotheinitialcondition),evenifit isnotformally thestabilitynotionthatweinvestigate.Fortherightfigures,the parametersarechoseninordertobreakthestabilityproperty.

Fig.6.Discontinuousprofile.TheparametersfortheD1Q3aretunedinordertohave(left)ornot(right)thenon-negativityproperty.

Theinitialconditionsarebuiltfromthepolynomialfunctions ₀(X)=1,₁(X)=X,and₂(X)=X(3−X²)/2by

_k(x)=

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎩

1 2

1+_k

2x−3

if ≤x≤2,

1 2

1+_k

5−2x

if2 ≤x≤3,

0 otherwise,

with=0.125,for0≤k≤2.ForFig.4,wechoose(0,x)=₂(x)to haveasmoothinitialcondition,forFig.5,(0,x)=₁(x)tohavea continuousinitialcondition,andforFig.6,(0,x)=0(x)tohavea discontinuousinitialcondition.

Iftheprofileissmooth(Fig.4),wehaveobservedthatnonumer- icaloscillationsoccurevenifthenon-negativitypropertyofthe matrixisnotsatisfied.Iftheprofileisjustcontinuous(Fig.5),small negativevaluesofthemacroscopicquantityareobservedwhenour non-negativitypropertyisnotsatisfied.Lastbutnotleast,classi- caloscillationsarevisiblefordiscontinuousprofiles(Fig.6)ifour non-negativitypropertyisnotsatisfied.Theseoscillationsareelim- inatedwhenthenon-negativitypropertyofthematrixisrealized.

7. Conclusion

Inthiscontribution,wehaveinvestigatedastabilitypropertyfor aclassicalmono-dimensionallinearthreevelocitieslatticeBoltz- mannschemewithrelativevelocity.Thispropertyensures that non-negativityoftheinitialparticledistributionfunctionscontin- uestoremainthesameintime. We thengivea necessaryand sufficientconditiontodescribethestabilityregion.Thecasewith- outrelativevelocityiscompletelydescribedandsimplernecessary conditionsaregivenforthegeneralcase.Wefinallyproposesome numericalsimulationsthatillustratethestabilityproperty:evenif thestabilitynotionthatweinvestigateisnotexactlyaconstraint ofconvexity,anumericalmaximumprincipleis observedifthe parametersareinsidethestabilityregionwhereasnumericaloscil- lationsappear(in particularforthenon-smoothprofiles)ifthe parametersareoutside.

Moreover,relativevelocitiesmodifythestabilityarrayinanon- trivialmanner.Forinstance,intuitionmighthavesuggestedthat thestabilityregionfor therelativevelocity equaltotheadvec- tionvelocitycontainsalltheothersbutitisreallynotthecase.

Foragivenadvectionvelocity,relativevelocitiescannotbeused toincrease thevalue ofthe firstrelaxation parameter,theone involvedinthenumericaldiffusionoperator.

Inorder tofocusonfundamental aspects,wehave assumed periodicboundaryconditionsinthiscontribution.Ofcourse,the

possibilityofincludingmorerealisticboundaryconditionsandthe associatedsourcetermsisanimportanttaskforafuturework.

Thenon-negativityoftherelaxationmatrixcouldbeextended tononlinear schemes.The theoreticalstudy willthen bemuch moretechnicalandhasnotbeenperformed.Nevertheless,numer- icalexperimentsfortheBurgersequationshowthatthebehavior oftheD1Q3scheme,andinparticulartheappearanceorabsence ofoscillations,isanalogoustothelinearcase.

Authors’contribution

ASG credited for code development, simulations; datacura- tion; formal analysis; validation; visualization; roles/writing – original draft; writing – review & editing. DVP contributed for conceptualization;codedevelopment,formalanalysis;validation;

supervision;roles/writing–originaldraft;writing–review&edit- ing.

Conflictofinterest

Nonedeclared.

Acknowledgements

Thisworkwassupportedby thepublicgrant“Lattice Boltz- mannMethodsandExtensions”givenbytheIndo-FrenchCentrefor AppliedMathematics(IFCAM),affiliatedtotheIndo-FrenchCentre forthePromotionofAdvancedResearch(CEFIPRA,India)andthe CentreNationaldelaRechercheScientifique(CNRS,France).

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Franc¸ois Dubois (61 years old) University Professor, MemberofStructuralMechanicsandCoupledSystems Laboratory,ConservatoireNationaldesArtsetMétiers, Paris(since1994),AssociatedtoMathematicsLaboratory, UniversitéParis-Saclay(since2003).

B.Graille(43yearsold)isanAssociateProfessor(Maître deconférenceshabilitéàdirigerdesrecherches)inthe LaboratoryofmathematicsoftheUniversityParis-Saclay, France,since2005.Hisresearchinterestsincludedevel- opingandinvestigatingthelatticeBoltzmannmethods, developinghydrodynamicsmodelsfromthekineticthe- ory.

S.V. RaghuramaRaois anAssociateProfessor inthe DepartmentofAerospaceEngineering,IndianInstituteof Science,Bangalore.HiseducationincludesB.E.(Mechan- icalEngineering)fromNationalInstituteofTechnology, Surat, M.Sc.(Engg.)& Ph.D.degreesfromIndian Insti- tuteofScience,postdoctoralfellowshipsfromUniversité PierreetMarieCurie,Paris,FranceandFraunhoferInstitut fürTechno-undWirtschaftsmathematik,Kaiserslautern, Germany,beforebecomingafacultymemberatIndian InstituteofScience.Hisresearchinterestsincludedevel- opingCFD algorithms,kinetictheorybasednumerical methods, Lattice Boltzmannmethods, meshless algo- rithms,aerodynamicshapeoptimizationandalgorithms forturbulencesimulations.