Benjamin Graille

François Dubois

Tony Fevrier

Tony Février

THIRD ORDER EQUIVALENT EQUATION FOR THE RELATIVE VELOCITY LATTICE BOLTZMANN SCHEMES WITH ONE CONSERVATION LAW

come

Third order equivalent equation for the relative velocity

lattice Boltzmann schemes with one conservation law

aux schémas de d'Humières dans [START_REF] Dubois | Equivalent partial differential equations of a lattice Boltzmann scheme[END_REF][START_REF] Dubois | Third order equivalent equation of Lattice Boltzmann scheme[END_REF] et utilisée pour les schémas à vitesse relative à deux lois de conservation [START_REF] Dubois | Lattice Boltzmann schemes with relative velocities[END_REF]. Elle s'effectue grâce à un développement de Taylor de l'équation de transport [START_REF] Dubois | Third order equivalent equation of Lattice Boltzmann scheme[END_REF] à petit pas de temps ∆t. Pour cela, les distributions de particules sont supposées être les restrictions sur le réseau L de fonctions suffisamment régulières. Les paramètres de relaxation sont supposés constants et l'équilibre, provenant d'un vecteur de distributions f eq = (f eq 0 , . . . , f eq q-1 ) indépendant de u, vérifie (4). Afin d'alléger les notations, un indice latin dans les signes de sommation décrit implicitement les entiers de 0 à q-1, tandis qu'un indice grec décrit les entiers de 1 à d. Nous noterons par ailleurs ∆ = ∆t. Sous ces hypothèses, nous communiquons le résultat suivant.

Proposition L'équation de conservation sur la densité s'écrit au troisième ordre grâce à l'équation [START_REF] Dubois | Lattice Boltzmann schemes with relative velocities[END_REF].

La quantité de mouvement à l'équilibre q eq est donnée par q eq = q-1 j=0 v j f eq j , la dérivée particulaire

d j t par d j t = ∂ t + α v α j ∂ α , 0 j q -1, le défault de conservation θ k ( u) par θ k ( u) = j M ( u) kj d j t f eq j , 0 k q -1, et le paramètre de Hénon σ k [14] par σ k = 1/s k -1/2, pour s k = 0, 0 k q -1.
Ce résultat s'obtient en développant successivement l'équation de transport [START_REF] Dubois | Third order equivalent equation of Lattice Boltzmann scheme[END_REF] aux ordres de zéro à trois. La composition par la matrice des moments donne le développement équivalent sur les quantités macroscopiques. La clé pour monter en ordre est l'utilisation de lemmes de transition [START_REF] Février | Extension et analyse des schémas de Boltzmann sur réseau : les schémas à vitesse relative[END_REF]: ces lemmes sont les développements en ∆ à différents ordres des moments non conservés par la collision. Dans l'équation de conservation [START_REF] Dubois | Lattice Boltzmann schemes with relative velocities[END_REF], nous remarquons que les termes d'ordre inférieur ou égal à un sont indépendants de u grâce à l'hypothèse (4) sur l'équilibre. Nous rappelons que ce constat est aussi vrai dans le cas à deux lois de conservation [START_REF] Dubois | Lattice Boltzmann schemes with relative velocities[END_REF]. Le paramètre u apparaît dans le terme d'ordre deux : il est susceptible d'influer sur le comportement dispersif du schéma. Nous notons enfin l'importance du défaut de conservation : cette quantité, dépendante de l'équilibre, pilote les termes d'ordre un (diffusion) et deux (dispersion).

Introduction

The analysis of consistency for the lattice Boltzmann schemes uses some Chapman-Enskog type formal expansions. This method, based on a multiscale expansion at small Knudsen number [START_REF] Chapman | The Mathematical Theory of Non-Uniform Gases[END_REF], has been a useful tool for these schemes [START_REF] Humières | Generalized Lattice-Boltzmann equations[END_REF][START_REF] Chen | Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method[END_REF]. In this contribution, we choose an alternative method: the equivalent equations development proposed in [START_REF] Yanenko | The relation between the correctness of the first differential approximations and the stability of difference schemes for hyperbolic systems of equations[END_REF][START_REF] Lerat | Noncentered Schemes and Shock Propagation Problems[END_REF] for the study of finite difference schemes and adapted to the lattice Boltzmann framework in [START_REF] Dubois | Equivalent partial differential equations of a lattice Boltzmann scheme[END_REF]. The third order equivalent equations of a general d dimensions and q velocities-D d Q q -d'Humières scheme [START_REF] Humières | Generalized Lattice-Boltzmann equations[END_REF] for one and two conservation laws have been determined with this method [START_REF] Dubois | Third order equivalent equation of Lattice Boltzmann scheme[END_REF]. It has also been used to build high order schemes [START_REF] Dubois | Towards higher order lattice Boltzmann schemes[END_REF][START_REF] Dubois | Quartic parameters for acoustic applications of lattice Boltzmann scheme[END_REF], to enforce isotropy to an arbitrary order [START_REF] Augier | Linear Lattice Boltzmann Schemes for Acoustic: parameters choices and isotropy properties[END_REF][START_REF] Augier | On rotational invariance of lattice Boltzmann schemes[END_REF] and to study some vectorial lattice Boltzmann schemes for hyperbolic systems [START_REF] Graille | Approximation of mono-dimensional hyperbolic systems : A lattice Boltzmann scheme as a relaxation method[END_REF]. This method has already been extended to the relative velocity D d Q q scheme [START_REF] Dubois | Lattice Boltzmann schemes with relative velocities[END_REF][START_REF] Geier | Cascaded digital Lattice Boltzmann automata for high Reynolds number flow[END_REF] for two conservation laws. We adapt the work to one conservation law: the third order equivalent equation is derived. In a first section, we recall the characteristics of the relative velocity D d Q q scheme and exhibit the differences with the d'Humières scheme. The second section introduces the main result of the paper, the third order expansion on the conserved variable. The steps leading to this formal development are described in a third section. The influence of the relative velocity is studied: we particularly locate the terms involving it. We also check the consistency with the asymptotics obtained for the d'Humières scheme [START_REF] Dubois | Third order equivalent equation of Lattice Boltzmann scheme[END_REF].

1. The relative velocity D d Q q scheme

In this section, we recall the derivation for one conservation law of the relative velocity lattice Boltzmann schemes introduced in [START_REF] Dubois | Lattice Boltzmann schemes with relative velocities[END_REF] and inspired by the cascaded scheme [START_REF] Geier | Cascaded digital Lattice Boltzmann automata for high Reynolds number flow[END_REF]. We consider L, a cartesian lattice in d dimensions with a typical mesh size ∆x. The time step ∆t is determined by the acoustic scaling after the specification of the velocity scale λ ∈ R by the relation ∆t = ∆x/λ. For the scheme denoted by D d Q q , we introduce V = (v 0 , . . . , v q-1 ) a set of q velocities of R d . We assume that for each node x of L, and each v j in V , the point x + v j ∆t is also a node of the lattice L. The aim of the D d Q q scheme is to compute a particle distribution f = (f 0 , . . . , f q-1 ) on the lattice L at discrete values of time.

The scheme splits into two phases for each time iteration: first, the relaxation that is non linear and local in space, and second, the linear transport for which an exact characteristic method is used. In the framework of the relative velocity lattice Boltzmann schemes, the relaxation is written into a basis of moments depending on a velocity field u(x, t), a given function of space and time. In the following, we note the velocity field u because it appears only during the collision that is local in space. The matrix of moments M ( u), which is supposed to be invertible, is defined by

M ( u) kj = P k (v j -u), 0 k, j q-1, (1)
where (P 0 , . . . , P q-1 ) are some polynomials of R[X 1 , . . . , X d ]. We choose the first polynomials as follows:

P 0 = 1, P k = X k , 1 k d. The moments m( u) = (m 0 ( u), . . . , m q-1 ( u)) are then given by m( u) = M ( u) f , (2)
so that m k ( u) is the k th moment in the frame moving at the velocity u. The relaxation phase is diagonal in the shifted moments basis

m ⋆ k ( u) = m k ( u)+s k (m eq k ( u)-m k ( u)), 0 k q-1, (3)
where m eq k ( u) is the k th moment at equilibrium and s k , the relaxation parameter associated with the k th moment for 0 k q-1. The vector of the distribution functions at the equilibrium f eq is chosen independent of the velocity field u so that we have

m eq ( u) = M ( u)f eq = M ( u)M (0) -1 m eq (0). (4)
We choose to work in the case of one conservation law on a scalar variable named here "density" defined by

ρ = j f j . (5)
Consequently, the associated relaxation parameter s 0 is null. The particle distributions are then computed by

f ⋆ = M -1 ( u)m ⋆ ( u). (6)
The transport phase reads

f j (x, t + ∆t) = f ⋆ j (x -v j ∆t, t), 0 j q-1. (7)
Let us note that the d'Humières scheme is enclosed in this framework by taking u = 0.

Third order equivalent equation

A Taylor expansion method [START_REF] Dubois | Third order equivalent equation of Lattice Boltzmann scheme[END_REF] is performed to derive the third order equivalent equation of the relative velocity D d Q q scheme for one conservation law, with any general equilibrium. We explicit a formal development of the scheme when ∆t and ∆x go to zero with λ = ∆x/∆t and the relaxation parameters s fixed. We obtain in this way a set of partial differential equations that are consistent with the scheme at different orders.

In the following, we do not adapt the Boltzmann scheme to a particular partial differential equationat first or second order-but we put in evidence the various operators hidden inside the algorithm. The reasoning consists in a formal development of the transport phase (7) at small ∆t, assuming that all particle distributions are the restrictions on the discretized space of a sufficiently regular distribution function. The Taylor formula can thus be used as much as wanted.

The result we want to emphasize is given by the proposition 2.1: this is the third order equivalent equation for the relative velocity D d Q q scheme with one conservation law. We particularly prove that the second order equations are the same as the ones of the D d Q q d'Humières scheme [START_REF] Dubois | Third order equivalent equation of Lattice Boltzmann scheme[END_REF]: this independence results from our choice for the equilibrium (4). The relation ( 4) means that the approximated physics do not depend on the velocity field u up to the second order. The dependence only appears at the third order. In the following, we note ∆ = ∆t. Unless otherwise specified, a sum over a greek parameter goes from 1 to d and over a latin parameter from 0 to q -1.

Proposition 2.1. The third order conservation equation on the density reads:

∂ t ρ + β ∂ β ((q eq ) β ) -∆ β σ β ∂ β (θ β (0)) = -∆ 2 β,j,l 1 σ β σ l v β j ∂ β (d j t M ( u) -1 jl θ l ( u) + ∆ 2 12 β,γ,j v β j v γ j ∂ 2 βγ (d j t f eq j ) + ∆ 2 6 β ∂ 2 βt (θ β (0)) + O(∆ 3 ), ( 8
)
where the momentum equilibrium q eq is given by q eq = q-1 j=0 v j f eq j , the particular derivative d j t by

d j t = ∂ t + α v α j ∂ α , 0 j q -1, the conservation default θ k ( u) by θ k ( u) = j M (
u) kj d j t f eq j , 0 k q -1, and the Hénon's parameter σ k [START_REF] Hénon | Viscosity of a Lattice Gas[END_REF] by σ k = 1/s k -1/2, for s k = 0, 0 k q -1.

We observe that each term of order l in ∆ contains only space derivatives of order l + 1.

Principal ideas for the proof of the main result

The detailed calculations of the proof of the proposition 2.1 are available in [START_REF] Février | Extension et analyse des schémas de Boltzmann sur réseau : les schémas à vitesse relative[END_REF]. To obtain the equivalent equation up to any order p ∈ N ⋆ , a multidimensional Taylor formula relative to ∆ is applied to the transport phase [START_REF] Dubois | Third order equivalent equation of Lattice Boltzmann scheme[END_REF]. This expansion is then written in the moments basis after multiplying by M ( u) kj

0 j q-1, 0 l p ∆ l l! M ( u) kj ∂ l t f j = 0 j q-1, |a| p (-∆) |a| a! (v j ) a M ( u) kj ∂ a f ⋆ j + O(∆ p+1 ), 0 k q -1, (9)
where a = (a 1 , . . . , a

d ) ∈ N d , (v j ) a = d α=1 (v α j ) aα , ∂ a = ∂ a 1 • • • ∂ a d , a! = d α=1 a α ! and |a| = d α=1 a α .
This relation develops the different moments according to the choice of the moment index k. Taking k = 0 yields to the conservation of the density, and k > 1 to the expansions of the non conserved moments.

To obtain the third order equivalent equation on the density, we proceed order by order. The first step thus consists in deriving the zeroth order corresponding to p = 0 in (9). This gives the following proposition: its proof can be found in [START_REF] Dubois | Equivalent partial differential equations of a lattice Boltzmann scheme[END_REF].

Lemma 3.1 (The particles stay close to the equilibrium). The particle distributions can be expanded as

f j = f eq j + O(∆), f ⋆ j = f eq j + O(∆), 0 j q -1.
These identities are assumed to be formally derivable as much as wanted: this point is used to obtain the first order equivalent equation on ρ taking p = 1 and k = 0 in ( 9). This equation is the truncation of (8) at the first order.

The key to gain an order is to develop relatively to ∆ the non conserved moments. To get the second order equivalent equation on ρ, these moments must be expanded at the second order. We present these expansions in the third order case, the second one being obtained by truncation.

Lemma 3.2 (Third order transition lemma). The non conserved moments before and after the collision read,

m k ( u) = m eq k ( u) -∆ 1 2 + σ k ξ k ( u, ∆, σ) + O(∆ 3 ), 1 k q -1, (10)
m ⋆ k ( u) = m eq k ( u) + ∆ 1 2 -σ k ξ k ( u, ∆, σ) + O(∆ 3 ), 1 k q -1, (11)
with ξ k ( u, ∆, σ) = θ k ( u)-∆ j,l 1 σ l M ( u) kj d j t M ( u) -1 jl θ l ( u) .
Two relations are combined to obtain these identities: the expansion (9) at the desired order and the relaxation phase (3). They give immediately the second order truncation of the relations [START_REF] Dubois | Quartic parameters for acoustic applications of lattice Boltzmann scheme[END_REF][START_REF] Février | Extension et analyse des schémas de Boltzmann sur réseau : les schémas à vitesse relative[END_REF]. This second order version is the keypoint to get the complete third order [START_REF] Dubois | Quartic parameters for acoustic applications of lattice Boltzmann scheme[END_REF][START_REF] Février | Extension et analyse des schémas de Boltzmann sur réseau : les schémas à vitesse relative[END_REF].

To get the second order equivalent equation on the density, we choose p = 2 and k = 0 in [START_REF] Dubois | Towards higher order lattice Boltzmann schemes[END_REF]. The calculation necessitates the second order expansion of the post collision momentum that is given by ( 11) taking 1 k d.

To get the complete third order equivalent equation on ρ, the same reasoning is used an order further. The required third order development of the post collision momentum is obtained using [START_REF] Février | Extension et analyse des schémas de Boltzmann sur réseau : les schémas à vitesse relative[END_REF] with 1 k d. The second order truncation of ( 8) is then useful to characterize this development up to the third order, that leads to the proposition 2.1.

Let us now recover the equation on the density for the d'Humières scheme [START_REF] Dubois | Third order equivalent equation of Lattice Boltzmann scheme[END_REF] corresponding to u = 0. We focus on the second order terms, the first order being independent of u. The quantity

- ∆ 2 6 β ∂ 2 βt (θ β (0)),
is already present in [START_REF] Dubois | Third order equivalent equation of Lattice Boltzmann scheme[END_REF]. The second term, independent of the relaxation parameters, reads β,γ,j v β j v γ j ∂ 2 βγ (d j t f eq j ) =

β,γ,l ∂ 2 βγ (Λ l βγ (0)θ l (0)).

There is an only term depending on σ in [START_REF] Dubois | Lattice Boltzmann schemes with relative velocities[END_REF]. Splitting the time and space derivatives of the particular derivative d j t , it reads β,j,l 1

σ β σ l v β j ∂ β (d j t M (0) -1 jl θ l (0) = β,l 1 σ β σ l ∂ 2 βt (θ β (0)) + β,γ,l 1 σ β σ l ∂ 2 βγ (Λ l βγ (0)θ l (0)),
where Λ l βγ (0) is the momentum velocity tensor defined by Λ l βγ (0) = j v β j v γ j M (0) -1 jl , 1 β, γ d, 0 l q -1. We have thus recovered all the terms exhibited in [START_REF] Dubois | Third order equivalent equation of Lattice Boltzmann scheme[END_REF].

(François Dubois) CNAM Paris, département de mathématiques, Univ Paris Sud, Laboratoire de mathématiques, UMR 8628, Orsay, F-91405 E-mail address: Francois.Dubois@math.u-psud.fr (Tony Février) Univ Paris Sud, Laboratoire de mathématiques, UMR 8628, Orsay, F-91405, Orsay, F-91405 E-mail address: Tony.Février@math.u-psud.fr (Benjamin Graille) Univ Paris Sud, Laboratoire de mathématiques, UMR 8628, Orsay, F-91405, CNRS, Orsay, F-91405 E-mail address: Benjamin.Graille@math.u-psud.fr


François Dubois

Tony Février

Benjamin Graille

ON THE STABILITY OF A RELATIVE VELOCITY LATTICE BOLTZMANN SCHEME FOR COMPRESSIBLE NAVIER-STOKES EQUATIONS

This paper studies the stability properties of a two dimensional relative velocity scheme for the Navier-Stokes equations. This scheme inspired by the cascaded scheme has the particularity to relax in a frame moving with a velocity field function of space and time. Its stability is studied first in a linear context then on the non linear test case of the Kelvin-Helmholtz instability. The link with the choice of the moments is put in evidence. The set of moments of the cascaded scheme improves the stability of the d'Humières scheme for small viscosities. On the contrary, a relative velocity scheme with the usual set of moments deteriorates the stability.

Introduction

The lattice Boltzmann schemes have been successfully used for the simulation of the compressible Navier-Stokes equations in two or three dimensions [START_REF] Benzi | The Lattice Boltzmann equation: theory and applications[END_REF][START_REF] Chen | Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method[END_REF][START_REF] Humières | Lattice BGK models for Navier-Stokes equation[END_REF][START_REF] Humières | Generalized Lattice-Boltzmann equations[END_REF]. This method aims to mimic the microscopic behaviour in order to simulate some macroscopic problems. The algorithm consists in evaluating some particle distributions. The particles, moving from node to node of a lattice, undergo a phase of collision and a phase of transport. Different collision operators have been proposed for the simulation of the Navier-Stokes equations. The simplest one is the single relaxation time operator [START_REF] Bhatnagar | The Lattice Boltzmann equation: theory and applications[END_REF][START_REF] Benzi | The Lattice Boltzmann equation: theory and applications[END_REF][START_REF] Chen | Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method[END_REF][START_REF] Humières | Lattice BGK models for Navier-Stokes equation[END_REF]] also called BGK. An alternative called the multiple relaxation times (MRT) operator [START_REF] Humières | Generalized Lattice-Boltzmann equations[END_REF][START_REF] Lallemand | Theory of the Lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance and stability[END_REF] has been proposed. During the collision, some moments, linear combinations of the particle distributions, relax towards the equilibrium with a priori different velocities. It contains the particularity to offer more degrees of freedom to fix the different parameters as the viscosities. The multiple relaxation times approach is thus more flexible than the BGK. Both schemes have been well studied particularly in terms of stability [START_REF] Lallemand | Theory of the Lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance and stability[END_REF]. They still encounter some instability features as the viscosities tend to zero that limits high Reynolds number simulations. In 2006, a cascaded scheme improving the stability for low viscosities has been presented [START_REF] Geier | Cascaded digital Lattice Boltzmann automata for high Reynolds number flow[END_REF]. Its relaxation occurs in a frame moving with the fluid velocity. To understand the positive features of this scheme, a general notion of relative velocity schemes was defined [START_REF] Dubois | Lattice Boltzmann schemes with relative velocities[END_REF]. Their relaxation is made for a set of moments depending on a velocity field function of space and time that is the velocity fluid for the cascaded scheme [START_REF] Dubois | Lattice Boltzmann schemes with relative velocities[END_REF] and zero for the d'Humières scheme [START_REF] Humières | Generalized Lattice-Boltzmann equations[END_REF]. These relative velocity schemes are not restricted to the simulation of the Navier-Stokes equations: they are defined for an arbitrary number of conservation laws. Their consistency has already been studied for one and two conservation laws [START_REF] Dubois | Lattice Boltzmann schemes with relative velocities[END_REF][START_REF] Février | Extension et analyse des schémas de Boltzmann sur réseau : les schémas à vitesse relative[END_REF][START_REF] Dubois | Third order equivalent equation for the relative velocity lattice Boltzmann schemes with one conservation law[END_REF] but the same reasoning holds for an arbitrary number of conservation laws. The purpose of this contribution is to present some numerical stability results of the two dimensional nine velocities (D 2 Q 9 ) relative velocity scheme for the compressible Navier-Stokes equations. We want to characterize the influence of the relative velocity and the link with the moments choice: the polynomials defining the moments of the cascaded scheme are different from the usual ones and may act on the stability. In a first part, we recall the basis of the relative velocity schemes. We then present the relative velocity D 2 Q 9 we are interested in. The second part exhibits the results of stability, first in a linear context (L 2 von Neumann notion) and then for a non linear test case, the Kelvin-Helmholtz instability. It puts in evidence the link between the relative velocity, the choice of the polynomials defining the moments and the stability.

Description of the scheme

We first introduce the relative velocity scheme for an arbitrary number of dimensions and velocities. We then particularize it to the case of two dimensions and nine velocities.

1.1. The relative velocity D d Q q scheme This section presents the derivation of the relative velocity lattice Boltzmann schemes introduced in [START_REF] Dubois | Lattice Boltzmann schemes with relative velocities[END_REF] and inspired by the cascaded scheme [START_REF] Geier | Cascaded digital Lattice Boltzmann automata for high Reynolds number flow[END_REF]. Let L be a cartesian lattice in d dimensions with a typical mesh size ∆x. The time step ∆t is linked to the space step by the acoustic scaling ∆t = ∆x/λ for λ ∈ R the velocity scale. We introduce V = (v 0 , . . . , v q-1 ) a set of q velocities of R d . This defines the scheme called D d Q q . We assume that for each node x of the lattice L, and each v j in V , the point x + v j ∆t is still a node of L. The D d Q q scheme computes a particle distribution f = (f 0 , . . . , f q-1 ) on the lattice L at discrete values of time. An iteration of the scheme consists in two phases: the relaxation that is non linear and local in space, and the linear transport solved exactly by a characteristic method.

The relaxation phase reads more easily in a moments basis using the d'Humières framework [START_REF] Humières | Generalized Lattice-Boltzmann equations[END_REF]. A velocity field u(x, t) that depends on space and time being given, we define the matrix of moments M ( u) by

M ( u) kj = P k (v j -u), 0 k, j q-1,
where (P 0 , . . . , P q-1 ) are some polynomials of R[X 1 , . . . , X d ]. This matrix of moments, supposed to be invertible, defines the moments m( u) = (m 0 ( u), . . . , m q-1 ( u)) by the relation

(1) m( u) = M ( u) f ,
where m k ( u) is the k th moment.

The relative velocity schemes use a diagonal relaxation phase in the shifted moments basis

(2) m k ( u) = m k ( u)+s k (m eq k ( u)-m k ( u)), 0 k q-1,
where m eq k ( u) is the k th moment at equilibrium and s k , the relaxation parameter associated with the k th moment for 0 k q-1. Some of these moments are conserved by the relaxation: they are associated with relaxation parameters equal to zero. The equilibrium derives from a vector of distribution functions at the equilibrium f eq independent of the velocity field u, only dependent of the conserved moments.

(3)

m eq ( u) = M ( u)f eq .
The inverse of the matrix of moments is used to return to the distributions

(4) f = M -1 ( u)m ( u).
The transport phase spreads the particle distributions on the neighbouring nodes

f j (x, t + ∆t) = f j (x -v j ∆t, t), 0 j q-1.
This framework embeddes the d'Humières scheme for u equal to 0 and the cascaded scheme for u equal to the fluid velocity and a particular set of moments [START_REF] Dubois | Lattice Boltzmann schemes with relative velocities[END_REF]. In the following, when u is specified, we call the associated relative velocity scheme the scheme relative to u.

1.2.

The study framework: the relative velocity D 2 Q 9 scheme The purpose of this section is to introduce the scheme whose stability properties are investigated: the relative velocity D 2 Q 9 scheme with two conservation laws on the density and the momentum

ρ = j f j , q α = j v α j f j , 1 α d.
for the compressible Navier-Stokes equations. We expose its features and the different degrees of freedom used to check its stability. We put a particular attention on the definition of the moments.

For this two-dimensional scheme, nine velocities are involved: they are defined by v = {(0, 0), (λ, 0), (0, λ), (-λ, 0), (0, -λ), (λ, λ), (-λ, λ), (-λ, -λ), (λ, -λ)}, with λ ∈ R the velocity scale. These velocities are also represented on the figure 1.

We need to deal with the set of the moments and the equilibrium to completely characterize the scheme. Historically the D 2 Q 9 scheme has been mainly used with the following set of moments

(5) 1, X, Y, X 2 + Y 2 , X 2 -Y 2 , XY, X(X 2 + Y 2 ), Y (X 2 + Y 2 ), (X 2 + Y 2 ) 2 ,
or its orthogonalized analogue for the simulation of the Navier-Stokes equations [START_REF] Lallemand | Theory of the Lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance and stability[END_REF].

They have been chosen because of their physical meaning: they involve the density, the momentum, the energy, the diagonal and off-diagonal components of the stress tensor, the heat flux and the square of the energy. Nevertheless, the D 2 Q 9 cascaded scheme [START_REF] Geier | Cascaded digital Lattice Boltzmann automata for high Reynolds number flow[END_REF], that seems to improve the stability at low viscosities, has brought to light an other set of moments given by ( 6)

1, X, Y, X 2 + Y 2 , X 2 -Y 2 , XY, XY 2 , Y X 2 , X 2 Y 2 .
This scheme has been written as a relative velocity scheme for the moments (6) and the fluid velocity [START_REF] Dubois | Lattice Boltzmann schemes with relative velocities[END_REF]. The relaxation of ( 5) and ( 6) are equivalent in the d'Humières framework corresponding to u = 0 (section 2.2). However, this is not true any more when u is different from 0. This point naturally leads to the question: does the set of moments have an influence on the stability properties of the relative velocity scheme? Giving some experimental rudiments of an answer is the purpose of this study.

That's why we introduce two sets of moments tuned by a parameter α ∈ R:

(7) 1, X, Y, X 2 +Y 2 , X 2 -Y 2 , XY, X(αX 2 +Y 2 ), Y (X 2 +αY 2 ), α 2 (X 4 +Y 4 )+X 2 Y 2 , and (8) 1, X, Y, X 2 + Y 2 , X 2 -Y 2 , XY, XY 2 + α(X 2 + Y 2 ), Y X 2 + α(X 2 + Y 2 ), X 2 Y 2 .
The moments [START_REF] Dubois | Equivalent partial differential equations of a lattice Boltzmann scheme[END_REF] generalize those of the D 2 Q 9 cascaded scheme corresponding to α = 0 [START_REF] Dubois | Lattice Boltzmann schemes with relative velocities[END_REF] given by ( 6) and the ones associated with α = 1 defined by [START_REF] Humières | Generalized Lattice-Boltzmann equations[END_REF]. The introduction of α results from the will not to restrict the study to two sets of moments. This allows also to understand the impact on the stability of the X 3 component when α moves from 0 to 1. The choice of the moments (8), even if it seems strange because it mixes some second and third order polynomials, improves the understanding of the differences of stability between ( 5) and ( 6) for the relative velocity D 2 Q 9 scheme. Taking α = 0 also recovers the cascaded moments.

The equilibrium may also have an influence on the stability. That's why, denoting u = q/ρ the fluid velocity, we introduce

(9) f eq j (ρ, u) = ρω j 1 + u.v j c 2 0 + (u.v j ) 2 2c 4 0 - |u| 2 2c 2 0 , 0 j 8, and (10)
f eq j (ρ, u) = ρω j 1 + u.v j c 2 0 + (u.v j ) 2 2c 4 0 - |u| 2 2c 2 0 + (u.v j ) 3 6c 6 0 - |u| 2 (u.v j ) 2c 4 0 + d j (u x ) 2 (u y ) 2 c 4 0 , 0 j 8,
where

d 0 = -1/4, d j = 1/2, j = 1, . . . , 4, d j = -1, j = 5, . . . , 8
, respectively corresponding to the second order truncated equilibrium [START_REF] Humières | Lattice BGK models for Navier-Stokes equation[END_REF] and to the product equilibrium used for the D 2 Q 9 cascaded scheme [START_REF] Geier | Ab initio derivation of the cascaded Lattice Boltzmann Automaton[END_REF]. The product equilibrium corresponds to the fourth order truncation of the maxwellian equilibrium. Both equilibria allow to simulate the compressible Navier-Stokes equations whatever the velocity field u. Indeed, the second order equivalent equations of the relative velocity schemes are independent of u [START_REF] Dubois | Lattice Boltzmann schemes with relative velocities[END_REF] and the Navier-Stokes equations are recovered by the D 2 Q 9 d'Humières scheme ( u = 0) at the second order for small Mach numbers [START_REF] Dubois | Equivalent partial differential equations of a lattice Boltzmann scheme[END_REF]. Let's note that the simulations of the section 3 have been also made for the incompressible analogue of ( 9) and ( 10) used in [START_REF] Lallemand | Theory of the Lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance and stability[END_REF]: same trends as those presented in the section 3 are obtained.

We choose to work with several two relaxation times schemes (TRT) to understand the role of each polynomial of the moments: the one given by (11) s = (s e , s ν , s ν , s e , s e , s e ), called TRT 1 and the one given by ( 12) s = (s e , s e , s e , s p , s p , s e ), called TRT 2 where s e , s ν , s p ∈ R. Note that the TRT 1 and the TRT 2 differ from the TRT schemes defined in [START_REF] Ginzburg | Optimal Stability of Advection-Diffusion Lattice Boltzmann Models with Two Relaxation Times for Positive/Negative Equilibrium[END_REF] and based on the symmetry of the lattice. If all the relaxation parameters are identical, we recover the BGK scheme [START_REF] Humières | Generalized Lattice-Boltzmann equations[END_REF].

Four degrees of freedom are tunable in this section: the moments, the vector of the relaxation parameters s, the velocity field u and the equilibrium. The link between these parameters and their influence on the stability is studied in the following.

Experimental study of linear stability

In this section, we study the linear von Neumann L 2 stability of the relative velocity D 2 Q 9 scheme defined in the section 1.2. The influence of the moments according to the choice of the velocity field u is the keypoint of the section. Our first interest goes to the moments ( 6) and ( 5), respectively corresponding to α = 0 and α = 1 in ( 7), because they are usually chosen by the community [START_REF] Humières | Generalized Lattice-Boltzmann equations[END_REF][START_REF] Lallemand | Theory of the Lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance and stability[END_REF][START_REF] Geier | Cascaded digital Lattice Boltzmann automata for high Reynolds number flow[END_REF][START_REF] Wang | Lattice Boltzmann simulations of thermal convective flows in two dimensions[END_REF]. The first subsection compares those two sets according to the velocity field parameter. We show that taking u = u, the velocity of the fluid, improves the stability if the moments (6) are chosen, deteriorates it if the set ( 5) is taken. The second subsection answers the following question: what is the better choice of moments (of α) in terms of stability? A range of α is proposed and α = 0 is showed to be the most stable choice.

Methodology: the von Neumann stability

The study of the relative velocity D 2 Q 9 scheme is based on the L 2 von Neumann stability. This notion being adapted to linear contexts, we linearize the equilibria ( 9) and ( 10) around a velocity

V = |V |e iθ ∈ R 2 , θ ∈ R.
Thus there exists a matrix E so that f eq = Ef . Using (1,2,3,4) the linearized relaxation phase of the relative velocity schemes reads

f = (I + M ( u) -1 DM ( u)(E -I))f ,
where D = diag(s) is the diagonal matrix of the relaxation parameters. This expression holds for each node x of the lattice, the relaxation being local in space. One can deduce the expression of the distribution after an iteration thanks to the transport phase

f j (x, t + ∆t) = [(I + M ( u) -1 DM ( u)(E -I))f ] j (x -v j ∆t, t), x ∈ L, t ∈ R.
In the Fourier space, the transport operator becomes local in space and is represented by the diagonal matrix A whose diagonal components are given by e i∆tk.v j , 0 j 8. We can then define the amplification matrix

L( u) = L( u, V , k, s, α) = A(I + M ( u) -1 DM ( u)(E -I)), for k, V , u ∈ R 2 , s ∈ R 9 , α ∈ R, V ∈ R 2 characterizing
a time iteration of the scheme in the Fourier space

f (k, t + ∆t) = L( u) f (k, t), t ∈ R,
where f is the Fourier transform of f . We want to determine the quantity [START_REF] Ginzburg | Optimal Stability of Advection-Diffusion Lattice Boltzmann Models with Two Relaxation Times for Positive/Negative Equilibrium[END_REF] max{|V |, max

k∈R 2 r(L( u)) 1},
for some parameters s, u, α, a direction of linearization θ ∈ R and r(L( u)) the spectral radius of L( u). It characterizes the set of the linearization velocities V for which the scheme verifies the necessary condition of L 2 stability max

k∈R 2 r(L( u)) 1.

2.2.

Comparison between the d'Humières scheme and the scheme relative to the linearization velocity We show that the schemes relative to u = V can improve or deteriorate the linear stability compared to the d'Humières scheme. The stability behaviour depends strongly on the choice of the moments.

We compare the schemes relative to u = 0 and u = V for the two sets of moments ( 5) and ( 6): we have α = 0, 1 in [START_REF] Dubois | Equivalent partial differential equations of a lattice Boltzmann scheme[END_REF]. We here restrict to the second order truncated equilibrium [START_REF] Dubois | Third order equivalent equation for the relative velocity lattice Boltzmann schemes with one conservation law[END_REF] Table 2. Highest stable V = (V x , 0) in λ units for the scheme relative to u = V with α = 0 of equilibrium [START_REF] Dubois | Third order equivalent equation for the relative velocity lattice Boltzmann schemes with one conservation law[END_REF]. s e = 2 -2 -m and s ν = 2 -2 -n . velocity V (13) for a linearization direction θ equal to 0. We choose to deal with the TRT 1 [START_REF] Geier | Ab initio derivation of the cascaded Lattice Boltzmann Automaton[END_REF] for s e = 2 -2 -m and s ν = 2 -2 -n where m, n, ∈ N, 0 m, n 7.

The parameters s e and s ν respectively tune the bulk and the shear viscosities of the Navier-Stokes equations. This choice of parameters allows to study the zero viscosity limit by increasing m or/and n. The table 1 deals with the d'Humières scheme for both sets of moments, the values for those two sets being identical. The tables 2 and 3 give analogous results for the scheme relative to u = V : they correspond respectively to the moments with α = 0 (6) and α = 1 (5).

We notice the importance of the choice of the moments for the schemes relative to u = V : stability areas are the biggest for α = 0 (6) (table 2) and the smallest for α = 1 (5) (table 3) whatever the choice of s. The d'Humières scheme ( u = 0, table 1) has smaller stability areas than the scheme relative to u = V with α = 0 and bigger than the one with α = 1. Table 3. Highest stable V = (V x , 0) in λ units for the scheme relative to u = V with α = 1 of equilibrium [START_REF] Dubois | Third order equivalent equation for the relative velocity lattice Boltzmann schemes with one conservation law[END_REF]. s e = 2 -2 -m and

s ν = 2 -2 -n .
The scheme relative to u = V with α = 0 provides the most important gain compared to the d'Humières scheme when s e or s ν is close to 2 and the other is far from 2 (for one small and one large viscosity). Instead these areas are the most deteriorated when α is equal to 1. When s e and s ν are close, the scheme presents stability areas nearly independent of u. The case s e = s ν corresponds to the BGK scheme: the velocity field u does not play any role since (3) is verified.

Finally, for the d'Humières scheme, the results are independent of the choice of the moments. Indeed, the third order moment X(X 2 + Y 2 ) is equivalent to λ 2 X + XY 2 on the velocity network [START_REF] Février | Extension et analyse des schémas de Boltzmann sur réseau : les schémas à vitesse relative[END_REF]: its relaxation is then equivalent to the relaxation of XY 2 , X being a conserved component. The same reasoning holds for the symmetrical moment. The moment

(X 2 + Y 2 ) 2 is equal to X 2 Y 2 + λ 2 (X 2 + Y 2 )
on the velocity network. Its relaxation is equivalent to relax X 2 Y 2 because X 2 + Y 2 and X 2 Y 2 are both in the eigenspace related to s e for the TRT 1 .

All these trends are independent of the direction of linearization θ: the results are similar for θ = π/8, π/4, π/3.

We now do the same job for the product equilibrium [START_REF] Février | Extension et analyse des schémas de Boltzmann sur réseau : les schémas à vitesse relative[END_REF] restricting the study to the choice α = 0. Note that the combination of these moments and this equilibrium corresponds to the D 2 Q 9 cascaded scheme [START_REF] Geier | Ab initio derivation of the cascaded Lattice Boltzmann Automaton[END_REF][START_REF] Dubois | Lattice Boltzmann schemes with relative velocities[END_REF]. The TRT 1 (11), tuned by s e and s ν , and the direction θ = 0 are chosen. The table 4 is about the d'Humières scheme, the table 5 corresponds to the scheme relative to u = V .

The results are analogous to the ones associated with the equilibrium [START_REF] Dubois | Third order equivalent equation for the relative velocity lattice Boltzmann schemes with one conservation law[END_REF]. The scheme relative to u = V has bigger linear stability areas than the d'Humières scheme ( u = 0) when α = 0. The gain is more important when the relaxation parameters are far from each other. The velocity field impact is lightened when s e and s ν are close. 4. Highest stable V = (V x , 0) in λ units for the d'Humières scheme ( u = 0), with α = 0, of equilibrium [START_REF] Février | Extension et analyse des schémas de Boltzmann sur réseau : les schémas à vitesse relative[END_REF]. Table 5. Highest stable V = (V x , 0) in λ units for the scheme relative to u = V with α = 0 of equilibrium [START_REF] Février | Extension et analyse des schémas de Boltzmann sur réseau : les schémas à vitesse relative[END_REF]. s e = 2 -2 -m and s ν = 2 -2 -n .

s e = 2 -2 -m and s ν = 2 -2 -n . n m 0
We finally assess the influence of the equilibrium on the linear stability. Whatever the choice of u and s, the equilibrium (10) provides bigger stability areas than the truncated equilibrium [START_REF] Dubois | Third order equivalent equation for the relative velocity lattice Boltzmann schemes with one conservation law[END_REF]. Particularly, the BGK scheme associated with ( 10) is more stable than the one corresponding to [START_REF] Dubois | Third order equivalent equation for the relative velocity lattice Boltzmann schemes with one conservation law[END_REF].

As a conclusion, the most important fact of the study is the following: the scheme relative to u = V for α = 0 is more stable than for α = 1. Instead choosing a scheme relative to u = V with a "inappropriate" set of moments can deteriorate the stability.

Influence of the choice of the moments on the stability

The previous section has studied the stability of the relative velocity and d'Humières schemes for two choices of α. This parameter seems to be crucial for the relative velocity schemes. The purpose of this section is to see more precisely its influence on the stability. It studies the stability properties of the schemes relative to u = 0 and u = V for a bigger range of α. We show numerically and justify that α = 0 We are interested in the stability of the relative velocity schemes for both sets of moments ( 7) and ( 8): the discussion carries on the choice of the parameter α ∈ R characterizing these moments. Both equilibria leading to the same trends, we focus on the truncated one (9) linearized around V = (V x , 0) ∈ R 2 . Two sets of relaxation parameters s are used: the TRT 1 [START_REF] Geier | Ab initio derivation of the cascaded Lattice Boltzmann Automaton[END_REF] and the TRT 2 [START_REF] Geier | Cascaded digital Lattice Boltzmann automata for high Reynolds number flow[END_REF] where s e = 2 -2 -m and s ν = s p = 2 -2 -n with (m, n) = (0, 3), (3, 0), (0, 7), (7, 0), [START_REF] Dubois | Equivalent partial differential equations of a lattice Boltzmann scheme[END_REF][START_REF] Dubois | Equivalent partial differential equations of a lattice Boltzmann scheme[END_REF]. The quantity ( 13) is drawn as a function of α in [-1, 1]. A negative value of (13) means that the scheme is unstable for all V .

We first focus on the d'Humières scheme corresponding to u = 0. The figures 2 and 3 represent respectively the draws associated with the moments ( 7) and ( 8). On each figure, the left draw is associated with the TRT 1 and the right one with the TRT 2 .

For the moments [START_REF] Dubois | Equivalent partial differential equations of a lattice Boltzmann scheme[END_REF], the draws are independent of α whatever the TRT chosen and s. For the moments [START_REF] Dubois | Lattice Boltzmann schemes with relative velocities[END_REF], the draw corresponding to the TRT 1 is independent of α unlike the TRT 2 . The figure associated with the TRT 2 induces to choose α = 0: it corresponds to the maximum of the curve and the stability area decreases as |α| increases. As expected, the draw for m = n = 7 corresponding to a BGK scheme is constant in α. We notice that α = 0 belongs to the set of α maximizing the stability whatever the draw. We can exhibit the origin of the dependence or independence on α. Let's consider the moments [START_REF] Dubois | Equivalent partial differential equations of a lattice Boltzmann scheme[END_REF]. For the d'Humières scheme, the relaxation of these moments is independent of α. The last three moments of (7) are

αX 3 + XY 2 , αY 3 + X 2 Y, α 2 (X 4 + Y 4 ) + X 2 Y 2 .
Knowing that X 3 = λ 2 X on the velocity set [START_REF] Février | Extension et analyse des schémas de Boltzmann sur réseau : les schémas à vitesse relative[END_REF], the scheme is unchanged if we replace them by

λ 2 αX + XY 2 , λ 2 αY + X 2 Y, λ 2 α 2 (X 2 + Y 2 ) + X 2 Y 2 .
Relaxing the moments ( 7) is then equivalent to relax the same moments for α = 0. Indeed, X and Y are associated with some conserved moments and X 2 + Y 2 has the same relaxation parameter s e as the fourth order moment. It is thus consistent for this draws to be independent of α.

We now focus on the moments (8): the parameter α appears only in the third order moments. For the TRT 1 , X 2 + Y 2 and the third order polynomials are relaxed with the same relaxation parameter s e . Choosing

XY 2 + α(X 2 + Y 2 ), X 2 Y + α(X 2 + Y 2 ),
is then equivalent to choose XY 2 , X 2 Y, and the scheme does not depend on α as the left draw of the figure 3 shows it. For the TRT 2 , X 2 +Y 2 and the third order moments are relaxed with different relaxation We now do the same job for the scheme relative to u = V . The figure 4 is associated with the moments [START_REF] Dubois | Equivalent partial differential equations of a lattice Boltzmann scheme[END_REF] and the figure 5 with the moments [START_REF] Dubois | Lattice Boltzmann schemes with relative velocities[END_REF].

The stability of the scheme relative to u = V depends on α whatever the moments (figure 4). The maximum is reached for α = 0 whatever the choice of s. For the moments (8), the stability of the TRT 1 is not linked to α (figure 5 on the left side). Instead, this parameter is influential for the TRT 2 (figure 5 on the right side): α = 0 still corresponds to the optimum.

We interpret the figure 4 corresponding to the moments [START_REF] Dubois | Equivalent partial differential equations of a lattice Boltzmann scheme[END_REF]. Because X 3 = λ 2 X and Y 3 = λ 2 Y on the velocity set, relaxing the relative moments associated with ( 7) is equivalent to relax 1, X, Y, X 2 + Y 2 , X 2 -Y 2 , XY, P 6 ( u, α), P 7 ( u, α), P 8 ( u, α), where

P 6 ( u, α) = XY 2 + α(-3 u x X 2 + (λ 2 -3( u x ) 2 )X + u x (λ 2 -( u x ) 2 )), (14)
P 7 ( u, α) = X 2 Y + α(-3 u y Y 2 + (λ 2 -3( u y ) 2 )Y + u y (λ 2 -( u y ) 2 )), (15)
P 8 ( u, α) = X 2 Y 2 + α 2 (λ 2 + 6( u x ) 2 )X 2 + (λ 2 + 6( u y ) 2 )Y 2 + 2 u x (-λ 2 + 4( u x ) 2 )X + 2 u y (-λ 2 + 4( u y ) 2 )Y -3( u x ) 2 (λ 2 -( u x ) 2 ) -3( u y ) 2 (λ 2 -( u y ) 2 ) .
Let's observe the equivalent class of the third order moment given by [START_REF] Lallemand | Theory of the Lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance and stability[END_REF]. The dependence on α of the stability comes from the term -3α u X 2 . Indeed, relaxing ( 14) is equivalent to relax XY 2 -3α u x X 2 since the moments corresponding to the polynomials 1 and X are conserved by the collision. On the contrary, the moment associated with X 2 is not conserved. For the TRT 1 , it is a linear combination of the moments X 2 + Y 2 and X 2 -Y 2 associated with different relaxation parameters s e and s ν . For the TRT 2 , it is associated with s e whereas P 6 corresponds to s p .

These remarks justify the introduction of the moments [START_REF] Dubois | Lattice Boltzmann schemes with relative velocities[END_REF] to study the influence of the non conserved components X 2 and Y 2 . The figure 5 implying the moments [START_REF] Dubois | Lattice Boltzmann schemes with relative velocities[END_REF] gives similar results as its analogous for u = 0 (figure 3): the same interpretation is still valid. Note that for α = 0, the areas are bigger with u = V (figure 5) than with u = 0 (figure 3). This confirms the observations of the section 2.2. We obtain results consistent with the linear stability study. First, choosing a scheme relative to u = u has a positive effect if α = 0, negative if α = 1. We must choose the moments of the D 2 Q 9 cascaded scheme to improve the stability. This improvement occurs whatever the equilibrium and the mesh: the stability limit s ν = 2 is stable (table 7) and high Mach numbers are reached for this scheme (table 6). Second, the d'Humières scheme is independent of α as for the linear stability study. Its stability area is smaller than the scheme relative to u = u when α = 0, greater when α = 1.

Finally, the equilibrium does not influence a lot the stability unlike the linear case. The obtained values are close whatever the choice of the equilibrium. It is important to note that the table 6 exhibits a convergence of all the schemes as ∆x decreases. .24 α = 0, u = 0, equilibrium (9) 0.18 0.13 0.12 0.12 0.12 0.12 0.12 α = 0, u = u, equilibrium (9) 0.96 0.82 0.62 0.49 0.43 0.39 0.39 α = 0, u = 0, equilibrium (10) 0.18 0.13 0.12 0.12 0.12 0.12 0.12 α = 0, u = u, equilibrium (10) 0.92 0.80 0.62 0.50 0.43 0.39 0.39 α = 1, u = 0, equilibrium (9) 0.18 0.13 0.12 0.12 0.12 0.12 0.12 α = 1, u = u, equilibrium ( 9 Table 7. Maximum of the Reynolds number stable for Ma = 0.09 according to the mesh.

α = 0, u = u, equilibrium (9) s ν = 2 s ν = 2 s ν = 2 s ν = 2 α = 0, u = 0, equilibrium (10
We now characterize the behaviour of the scheme when the diffusion is weak (when the relaxation parameters are close to 2). The table 8 presents the maximal Ma stable for decreasing bulk viscosity µ. The domain is constituted of 128 2 points and ν is still equal to 10 -4 .

This table is also consistent with the linear stability study. When s e and s ν are far from each other, the linear case (the tables 2 and 5) presents an important gain for the scheme relative to u = u for α = 0 whatever the equilibrium. These results are confirmed by the first columns of the table 8 corresponding to take a big bulk viscosity. When µ tends to 0, the stability areas for the different choices of u are expected to be close at fixed equilibrium: this case corresponds to close parameters s e and s ν , regime where the linear stability results are homogeneous in u. This behaviour is confirmed by the table 8: indeed, the four cases associated with the equilibrium (9) have the same stability areas when the bulk viscosity is smaller than 10 -4 . Similarly, the two cases associated with the equilibrium [START_REF] Février | Extension et analyse des schémas de Boltzmann sur réseau : les schémas à vitesse relative[END_REF] have close stability areas for these viscosities.

The table 9 deals with the influence of the velocity field u on the stability: other choices than 0 and u are considered. We determine the maximal Mach number stable for different u according to the choice of the moments. We study the two choices α = 0 and α = 1 for a mesh of 128 2 points.

This table is an evidence of the importance of the moments for the relative velocity schemes. Taking a velocity different from 0 provides stability improvements only for α = 0. These moments improve the numerical stability for u = u compared to the d'Humières scheme whatever the equilibrium. Instead, choosing u = 0 for the moments (5) deteriorates the stability of the scheme. The most stable choice for α = 1 corresponds to the d'Humières scheme.

Conclusion

We have studied the numerical stability of the relative velocity D 2 Q 9 scheme with two conservation laws. A linear stability study was presented and strenghtened by a non linear test case for the compressible Navier-Stokes equations: the Kelvin-Helmholtz instability. The main conclusion of the article is the following: the relative velocity schemes improve or deteriorate the stability of the d'Humières schemes and it depends strongly on the choice of the moments. An improvement occurs if the moments of the cascaded scheme are chosen whatever the equilibrium. It is bigger when one viscosity is very small and the other is important. The usual set of moments and its (François Dubois) CNAM Paris, département de mathématiques, Univ Paris Sud, Laboratoire de mathématiques, UMR 8628, Orsay, F-91405 E-mail address: Francois.Dubois@math.u-psud.fr (Tony Février) Univ Paris Sud, Laboratoire de mathématiques, UMR 8628, Orsay, F-91405, Orsay, F-91405 E-mail address: Tony.Février@math.u-psud.fr (Benjamin Graille) Univ Paris Sud, Laboratoire de mathématiques, UMR 8628, Orsay, F-91405, CNRS, Orsay, F-91405 E-mail address: Benjamin.Graille@math.u-psud.fr

Figure 1 .

 1 Figure 1. The D 2 Q 9 velocities.

Table

Figure 2 .

 2 Figure 2. Draw of V x as a function of α for the d'Humières scheme with the moments (7). Left: TRT 1 , s e = 2 -2 -m and s ν = 2 -2 -n . Right: TRT 2 , s e = 2 -2 -m and s p = 2 -2 -n .

Figure 3 .

 3 Figure 3. Draw of V x as a function of α for the d'Humières scheme with the moments (8). Left: TRT 1 , s e = 2 -2 -m and s ν = 2 -2 -n . Right: TRT 2 , s e = 2 -2 -m and s p = 2 -2 -n .

Figure 4 .

 4 Figure 4. Draw of V x as a function of α for the scheme relative to u = V with the moments (7). Left: TRT 1 , s e = 2 -2 -m and s ν = 2 -2 -n . Right: TRT 2 , s e = 2 -2 -m and s p = 2 -2 -n .

Figure 5 .

 5 Figure 5. Draw of V x as a function of α for the scheme relative to u = V with the moments (8). Left: TRT 1 , s e = 2 -2 -m and s ν = 2 -2 -n . Right: TRT 2 , s e = 2 -2 -m and s p = 2 -2 -n .

Figure 6 .Figure 7 .

 67 Figure 6. Vorticity draw at t=0.6.

  u = u, equilibrium[START_REF] Février | Extension et analyse des schémas de Boltzmann sur réseau : les schémas à vitesse relative[END_REF] s ν = 2 s ν = 2 s ν = 2 s ν = 2 α = 1, u = 0, equilibrium[START_REF] Dubois | Third order equivalent equation for the relative velocity lattice Boltzmann schemes with one conservation law[END_REF]

Table 1 .

 1 linearized around V . The variable of comparison is the largest stable Highest stable V = (V x , 0) in λ units for the d'Humières scheme ( u = 0), with α = 0 or α = 1, of equilibrium[START_REF] Dubois | Third order equivalent equation for the relative velocity lattice Boltzmann schemes with one conservation law[END_REF]. s e = 2-2 -m and s ν = 2 -2 -n .

	n	m	0	1	2	3	4	5	6	7
	0		0.42	0.41	0.34	0.26	0.20	0.15	0.11	0.08
	1		0.42	0.41	0.36	0.30	0.23	0.18	0.13	0.09
	2		0.31	0.34	0.34	0.32	0.28	0.23	0.17	0.13
	3		0.21	0.28	0.32	0.30	0.25	0.22	0.18	0.15
	4		0.14	0.21	0.28	0.26	0.22	0.18	0.16	0.13
	5		0.10	0.16	0.22	0.23	0.20	0.17	0.13	0.11
	6		0.07	0.12	0.17	0.20	0.18	0.16	0.12	0.11
	7		0.05	0.08	0.12	0.17	0.16	0.15	0.11	0.11
	n	m	0	1	2	3	4	5	6	7
	0		0.42	0.42	0.40	0.37	0.33	0.28	0.24	0.21
	1		0.42	0.41	0.37	0.34	0.33	0.30	0.27	0.23
	2		0.42	0.36	0.34	0.33	0.29	0.25	0.22	0.19
	3		0.36	0.34	0.33	0.30	0.25	0.21	0.18	0.16
	4		0.32	0.32	0.30	0.26	0.22	0.18	0.16	0.13
	5		0.29	0.30	0.27	0.24	0.21	0.17	0.13	0.11
	6		0.26	0.28	0.24	0.21	0.18	0.16	0.12	0.11
	7		0.23	0.26	0.22	0.19	0.16	0.15	0.11	0.11

Table 6 .

 6 Maximum of Ma stable according to the mesh. The two last columns indicates a convergence of each scheme as ∆x decreases.

	) 0.09	0.07	0.06	0.05	0.05	0.05	0.05
	Space step ∆x	1/16	1/32	1/64		1/128
	Corresponding s e	0.44		0.25	0.13		0.07
	Corresponding s ν α = 0, u = 0, equilibrium (9)	1.98 s ν = 2	1.96 21.10 3	1.93 17.10 3		1.86 17.10 3

Table 8 .

 8 Maximum of Ma stable according to µ.

	Viscosity µ			10 -2	0.5.10 -2 10 -3	10 -4		10 -5
	Corresponding s e			0.23	0.41	1.13	1.86		1.98
	Corresponding s ν			1.86	1.86	1.86	1.86		1.86
	α = 0, u = 0, equilibrium (9)		0.22	0.29	0.43	0.38		0.32
	α = 0, u = u, equilibrium (9)		0.73	0.76	0.68	0.63		0.60
	α = 0, u = 0, equilibrium (10)		0.22	0.30	0.45	0.38		0.32
	α = 0, u = u, equilibrium (10)		0.72	0.76	0.76	0.63		0.61
	α = 1, u = 0, equilibrium (9)		0.22	0.29	0.43	0.38		0.32
	α = 1, u = u, equilibrium (9)		0.10	0.14	0.32	0.38		0.32
	u	0	0.2u	0.4u	0.6u	0.8u	u	1.2u	1.4u
	α = 0, equilibrium (9)	0.12	0.15	0.21	0.34	0.60	0.49	0.42	0.33
	α = 1, equilibrium (9)	0.12	0.11	0.09	0.07	0.06	0.05	0.05	0.04
	α = 0, equilibrium (10) 0.12	0.15	0.21	0.34	0.60	0.50	0.42	0.33

Table 9 .

 9 Maximum of Ma stable according to u.

Stability for the Kelvin-Helmholtz instability

The purpose of this section is to confirm on a non linear test case the previous linear stability results: this test case is the Kelvin-Helmhotz instability [START_REF] Minion | Performance of under-resolved two-dimensional incompressible flow simulations[END_REF][START_REF] Dellar | Bulk and shear viscosities in lattice Boltzmann equations[END_REF].

We compare six versions of the relative velocity D 2 Q 9 scheme to study the influence of the moments, of the velocity field u and of the equilibrium. We consider the scheme associated with α = 0 relative to u = 0 and u = u (the fluid velocity) for the equilibria ( 9) and [START_REF] Février | Extension et analyse des schémas de Boltzmann sur réseau : les schémas à vitesse relative[END_REF]. We compare it to the choice α = 1 for the relative velocities 0 and u with the equilibrium [START_REF] Dubois | Third order equivalent equation for the relative velocity lattice Boltzmann schemes with one conservation law[END_REF]. We choose not to consider the product equilibrium [START_REF] Février | Extension et analyse des schémas de Boltzmann sur réseau : les schémas à vitesse relative[END_REF] for α = 1, this equilibrium being introduced for the moments of the cascaded scheme [START_REF] Geier | Ab initio derivation of the cascaded Lattice Boltzmann Automaton[END_REF]. We work with the TRT 1 defined by ( 11): unless otherwise specified, s e et s ν are fixed by

where σ e = 1/s e -1/2 and σ ν = 1/s ν -1/2, so that the viscosities µ and ν are set to 0.0366 and 10 -4 .

We test the stability of the scheme by increasing the velocity U defining the initial shear layers

u y (x, y, 0) = U δ sin(2π(x + 1 4 )), (x, y) ∈ [0, 1] 2 . This velocity U is chosen as Ma/ √ 3 for Ma∈ R the Mach number. The parameters k and δ controlling the width of the shear layers and the magnitude of the initial data are set to 80 and 0.05.

We first validate the vorticity draws obtained in [START_REF] Ricot | Performance of under-resolved two-dimensional incompressible flow simulations[END_REF][START_REF] Dellar | Bulk and shear viscosities in lattice Boltzmann equations[END_REF][START_REF] Minion | Performance of under-resolved two-dimensional incompressible flow simulations[END_REF] using the scheme relative to the fluid velocity u for the second order truncated equilibrium [START_REF] Dubois | Third order equivalent equation for the relative velocity lattice Boltzmann schemes with one conservation law[END_REF]. This vorticity is defined by

For this simulation, the domain is constituted of 128 × 128 points, the Mach number is fixed at 0.04 (λ is chosen as in [START_REF] Dellar | Bulk and shear viscosities in lattice Boltzmann equations[END_REF] so that U = 1). The figures 6 and 7 are the vorticity plots at time t = 0.6 and t = 1.

We now present a stability analysis depending on the different parameters for λ = 1.

We expect to confirm the linear stability results.