Adeline Augier

François Dubois
email: francois.dubois@math.u-psud.fr.

Benjamin Graille

Pierre Lallemand

On rotational invariance of lattice Boltzmann schemes

Keywords: Acoustics, Taylor expansion method, linearized Navier-Stokes, isotropy PACS numbers : 02.60Cb (numerical simulation, solution of equations), 43.20.+g (General linear acoustics), 47.10.+g (Navier-Stokes equations)

We propose the derivation of acoustic-type isotropic partial differential equations that are equivalent to linear lattice Boltzmann schemes with a density scalar field and a momentum vector field as conserved moments. The corresponding linear equivalent partial differential equations are generated with a new "Berliner version" of the Taylor expansion method. The details of the implementation are presented. These ideas are applied for the D2Q9, D2Q13, D3Q19 and D3Q27 lattice Boltzmann schemes. Some limitations associated with necessary stability conditions are also presented.

1) Introduction

• Partial differential equations like Navier-Stokes equations are invariant by rotation and all space directions are equivalent. Due to the use of a given mesh, lattice Boltzmann schemes cannot be completely invariant by rotation. This difficulty was present in the early ages of lattice gas automata. The initial model of Hardy, de Pazzis and Pomeau [START_REF] Hardy | Time evolution of a two-dimensional model system. I. Invariant states and time correlation functions[END_REF] proposed very impressive qualitative results but the associated fluid tensor was not invariant by rotation. With a triangular mesh, the second model of Frisch, Hasslacher and Pomeau [START_REF] Frisch | Lattice-gas automata for the Navier-Stokes equation[END_REF] gives the correct physics. The lattice Boltzmann scheme with multiple relaxation times is the fruit of the work of Higuera and Jiménez [START_REF] Higuera | Boltzmann Approach to Lattice Gas Simulations[END_REF], Higuera, Succi and Benzi [START_REF] Higuera | Lattice Gas Dynamics with Enhanced Collisions[END_REF], Qian, d'Humières and Lallemand [START_REF] Qian | Lattice BGK Models for Navier-Stokes Equation[END_REF] and d'Humières [START_REF] Humières | Generalized Lattice-Boltzmann Equations[END_REF]. It uses in general square meshes and leads to isotropic physics for second order equivalent model as analyzed in [START_REF] Lallemand | Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability[END_REF]. The question of rotational invariance is still present in the lattice Boltzmann community and a detailed analysis of moment isotropy has been proposed by Chen and Orszag [START_REF] Chen | Moment isotropy and discrete rotational symmetry of twodimensional lattice vectors[END_REF].

• The invariance by rotation has to be kept as much as possible in order to respect the correct propagation of waves. In [START_REF] Dubois | Quartic Parameters for Acoustic Applications of Lattice Boltzmann Scheme[END_REF], using the Taylor expansion method proposed in [START_REF] Dubois | Towards higher order lattice Boltzmann schemes[END_REF] for general applications, we have developed a methodology to enforce a lattice Boltzmann scheme to simulate correctly the physical acoustic waves up to fourth order of accuracy. But unfortunately, stability is in general guaranteed only if the viscosities of the waves are much higher than authorized by common physics. In this contribution, we relax this constraint and suppose that the equivalent partial differential equation of the scheme are invariant under rotations. The objective of this contribution is to propose a methodology to fix the parameters of lattice Boltzmann schemes in order to ensure the invariance by rotation at a given order.

• The outline is the following. In Section 2, we consider the question of the algebraic form of linear high order "acoustic-type" partial differential equations that are invariant by two dimensional and three-dimensional rotations. In Section 3, we recall the essential properties concerning multiple relaxation times lattice Boltzmann schemes. The equivalent equation of a lattice Boltzmann scheme introduces naturally the notion of rotational invariance at a given order. In the following sections, we develop a methodology to force acoustic-type lattice Boltzmann models to be invariant under rotations. We consider the four lattice Boltzmann schemes D2Q9, D2Q13, D3Q19 and D3Q27 in the sections 4 to 7. A conclusion ends our contribution. The Appendix 1 presents with details the implementation of the "Berliner version" of our algorithm to derive explicitely the equivalent partial differential equations. Some long formulas associated of specific results for D2Q13 and D3Q27 lattice Boltzmann schemes are presented in Appendix 2.



On rotational invariance of lattice Boltzmann schemes

2) Invariance by rotation of acoustic-type equations

• With the help of group theory, and we refer the reader e.g. to Hermann Weyl [START_REF] Weyl | Elementary theory of invariants[END_REF] or Goodman and Wallach [START_REF] Goodman | Representations and invariants of the classical groups[END_REF], it is possible to write a priori the general form of systems of linear partial differential equations invariant by rotation. More precisely, if the unknown is composed by one scalar field ρ (invariant function under a rotation of the space) and one vector field J (a vector valued function that is transformed in a similar way to how than cartesian coordinates are transformed when a rotation is applied), a linear partial differential equation invariant by rotation is constrained in a strong manner. Using some fundamental aspects of group theory and in particular the Schur lemma (see e.g. Goodman and Wallach [START_REF] Goodman | Representations and invariants of the classical groups[END_REF]), it is possible to prove that general linear partial differential equations of acoustic type that are invariant by rotation admit the form described below.

• In the bidimentional case, we introduce the notation (1)

∇ ⊥ ≡ ∂ ∂x , ∂ ∂y ⊥ = ∂ ∂y , - ∂ ∂x , J ⊥ ≡ (J x , J y ) ⊥ = (J y , -J x ) .
Then acoustic type partial differential equations are of the form

           ∂ t ρ + k≥0 α k △ k ρ + β k △ k divJ + γ k △ k div(J ⊥ ) = 0 , ∂ t J + k≥0 δ k ∇△ k ρ + µ k △ k J + ζ k ∇div△ k J + ε k ∇ ⊥ △ k ρ + ν k △ k J ⊥ + η k ∇div△ k J ⊥ = 0 , (2)
where the real coefficients α k , β k , γ k , δ k , µ k , ζ k , ε k , ν k and η k are in finite number. The tridimensional case is essentially analogous. The "acoustic type" linear partial differential equations invariant by rotation take necessarily the form

(3)          ∂ t ρ + k≥0 α k △ k ρ + β k div△ k J = 0 ∂ t J + k≥0 δ k ∇△ k ρ + µ k △ k J + η k ∇div△ k J + ϕ k curl △ k J = 0 ,
with an analogous convention that the sums in the relations (3) contain only a finite number of such terms.

3) Lattice Boltzmann schemes with multiple relaxation times

• Each iteration of a lattice Boltzmann scheme is composed by two steps: relaxation and propagation. The relaxation is local in space: the particle distribution f (x) ∈ IR q for x a node of the lattice L, is transformed into a "relaxed" ' distribution f * (x) that is non linear in general. In this contribution, we restrict to linear functions

IR q ∋ f -→ f * ∈ IR q .
As usual with the d'Humières scheme [START_REF] Humières | Generalized Lattice-Boltzmann Equations[END_REF], we introduce an invertible matrix M with q lines and q columns. The moments m are obtained from the particle distribution thanks to the associated transformation (4)

m k = q-1 j=0 M k j f j , 0 ≤ k ≤ q -1 .  A.
Augier, F. Dubois, B. Graille and P. Lallemand

Then we consider the conserved moments W ∈ IR N :

(5)

W i = m i , 0 ≤ i ≤ N -1 .
For the usual acoustic equations for d space dimensions, we have N = d + 1. The first moment is the density and the next ones are composed by the d components of the physical momentum. Then we define a conserved value m eq k for the non-equilibrium moments m k for k ≥ N. With the help of "Gaussian" functions G k (•), we obtain:

(6) m eq k = G k (W ) , N ≤ k ≤ q -1 .
In the present contribution, we suppose that this equilibrium value is a linear function of the conserved variables. In other terms, the Gaussian functions are linear:

(7) G N +ℓ (W ) = n-1 i=1 E ℓi W i , ℓ ≥ 0
for some equilibrium coefficients E ℓi for ℓ ≥ 0 and 0 ≤ i ≤ N -1.

• The relaxed moments m * k are linear functions of m k and m eq k :

(8) m * k = m k + s k m eq k -m k , k ≥ N .
For a stable scheme, we have

(9) 0 < s k < 2 .
We remark that if s k = 0, the corresponding moment is conserved. In some particular cases, the value s k = 2 can also be used (see e.g. [START_REF] Dubois | Unpublished pedagogical experiments with the D1Q3 lattice Boltzmann scheme for the heat equation[END_REF][START_REF] Dubois | Stable lattice Boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves[END_REF]). The conserved moments are not affected by the relaxation:

m * i = m i = W i , 0 ≤ i ≤ N -1 .
From the moments m * ℓ for 0 ≤ ℓ ≤ q -1 we deduce the particle distribution f * j by resolution of the linear system M f * = m * .

• The propagation step couples the node x ∈ L with his neighbours xv j ∆t for 0 ≤ j ≤ q -1. The time iteration of the scheme can be written as

(10) f j (x, t + ∆t) = f * j (x -v j ∆t, t) , 0 ≤ j ≤ q -1 , x ∈ L.
• From the knowledge of the previous algorithm, it is possible to derive a set of equivalent partial differential equations for the conserved variables. If the Gaussian functions G k are linear, this set of equations takes the form

(11) ∂W ∂t -α 1 W -∆t α 2 W -• • • -∆t j-1 α j W -• • • -= 0 ,
where α j is for j ≥ 1 is a space derivation operator of order j. We refer the reader to [START_REF] Dubois | Equivalent partial differential equations of a lattice Boltzmann scheme[END_REF] for the presentation of our approach in the general case. In this contribution, we have developed an explicit algebraic linear version of the algorithm detailed in Appendix 1. Moreover, we consider that the lattice Boltzmann scheme is invariant by rotation at order ℓ if the equivalent partial differential equation [START_REF] Goodman | Representations and invariants of the classical groups[END_REF] by truncation at the order ℓ is invariant by rotation. For acoustic-type models, the partial differential equation [START_REF] Hardy | Time evolution of a two-dimensional model system. I. Invariant states and time correlation functions[END_REF] has to be identical to (2) or (3) for dimension 2 or 3. In the following, we determine the equivalent partial differential equations for classical lattice Bolzmann schemes in the general linear case. Then we fit the equilibrium and relaxation parameters of the scheme in order to enforce rotational invariances at all orders between 1 and 4.

(12) ∂W ∂t - ℓ j=1 ∆t j-1 α j W = 0  obtained from

4) D2Q9

The isotropy of the lattice Boltzmann scheme D2Q9 for the acoustic equations has been studied in detail in [START_REF] Augier | Isotropy conditions for Lattice Boltzmann schemes. Application to D2Q9[END_REF][START_REF] Augier | Linear lattice Boltzmann schemes for Acoustic: parameter choices and isotropy properties[END_REF]. We give here only a summary of our results.

• The matrix M for the D2Q9 lattice Boltzmann model is of the form (13)

M kj = p k (v j ) , 0 ≤ j , k ≤ q -1
with polynomials p k detailed in the contribution [START_REF] Lallemand | Some results on energy-conserving lattice Boltzmann models[END_REF]. With this choice, the moments are named according to the following notations:

                 0 1 λ 0 1, 2 X, Y λ 1 3 ε λ 2 4, 5 XX, XY λ 2 6, 7 q x , q y λ 3 8 ε 2 λ 4 . (14)
We have recalled in [START_REF] Higuera | Boltzmann Approach to Lattice Gas Simulations[END_REF] the degrees of homogeneity of each moment m k relative to the reference numerical velocity λ ≡ ∆x ∆t . • At first order, the invariance by rotation (2) takes the form [START_REF] Higuera | Lattice Gas Dynamics with Enhanced Collisions[END_REF] ∂ t ρ + divJ = O(∆t) ∂ t J + c 2 0 ∇ρ = O(∆t) if the next moments of degree 2 follow the simple equilibrium: [START_REF] Humières | Generalized Lattice-Boltzmann Equations[END_REF] ε eq = α λ 2 ρ , XX eq = XY eq = 0 .

Then the sound velocity c 0 in the equation ( 15) satisfies

(17) c 2 0 = λ 2 6 (4 + α) .
• At second order, the invariance by rotation (2) is realized under specific conditions for the third order moments q ≡ (q x , q y ). This equilibrium condition is defined with the help of Hénon's [START_REF] Hénon | Viscosity of a Lattice Gas[END_REF] parameters σ k defined from the coefficients s k according to [START_REF] Lallemand | Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability[END_REF]

σ k ≡ 1 s k - 1 2 when k ≥ 3 .
The stability condition (9) can be written as We have [START_REF] Leriche | Stokes eigenmodes in cubic domain: primitive variable and Lattice Boltzmann formulations[END_REF] q eq = σ 4 -4 σ 5 σ 4 + 2 σ 5 λ 2 J .

We obtain with these conditions the following equivalent partial differential equations ( 21)

∂ t ρ + divJ = O(∆t 2 ) ∂ t J + c 2 0 ∇ρ -µ △J -ζ ∇ divJ = O(∆t 2
) . The physical viscosities µ and ζ can be determined according to

(22) µ = σ 4 σ 5 σ 4 + 2 σ 5 λ ∆x , ζ = σ 3 (2 σ 4 -2 σ 5 -α σ 4 -2 α σ 5 ) 6 (σ 4 + 2 σ 5 ) λ ∆x .
We observe that the classical isotropy condition σ 4 = σ 5 for the second order moments XX and XY is not necessary for the relaxation at this second order step.

• At third order, the system (2) takes the particular form ( 23)

∂ t ρ + divJ + ξ △ divJ = O(∆t 3 ) ∂ t J + c 2 0 ∇ρ -µ △J -ζ ∇ divJ + χ ∇△ρ = O(∆t 3 ) . This is possible if the complementary relations (24) σ 4 = σ 5 , ε eq 2 = - λ 4 ρ 2 3 α + 4
hold. Then the heat flux at equibrium has an expression [START_REF] Leriche | Stokes eigenmodes in cubic domain: primitive variable and Lattice Boltzmann formulations[END_REF] that can be simply written as (25) q eq = -λ 2 J .

The coefficients in the equations ( 23) are given by the expressions (26)

     µ = 1 3 σ 4 λ ∆x , ζ = - 1 6 σ 3 α λ ∆x , ξ = 1 72 (α -2) ∆x 2 , χ = 1 216 (α + 4) 2 + 6 α σ 2 3 -α -12 σ 2 4 λ 2 ∆x 2 .
We remark that the dissipation of the acoustic waves γ ≡ µ+ζ

2

(see e.g. Landau and Lifshitz [START_REF] Landau | Fluid Mechanics[END_REF]) is given according to

(27) γ = λ ∆x 12 2σ 4 -α σ 3 .
• The invariance by rotation at fourth order of the D2Q9 lattice Boltzmann scheme does not give completely satisfactory results, as observed previously in [START_REF] Augier | Linear lattice Boltzmann schemes for Acoustic: parameter choices and isotropy properties[END_REF]. In order to get equivalent equations at order 4 of the type (28)

∂ t ρ + divJ + ξ △ divJ + η △ 2 ρ = O(∆t 4 ) ∂ t J + c 2 0 ∇ρ -µ △J -ζ ∇ divJ + χ ∇△ρ + µ 4 △ 2 J + ζ 4 ∇ div △ J = O(∆t 4
), it is necessary to fix some relaxation parameters:

(29) σ 3 = σ 4 = σ 8 , σ 6 = σ 7 = 1 6 σ 4 .



The two viscosities µ and ζ are now dependent and the dissipation of the acoustic waves introduced previously in ( 27) admits now the expressions

(30) µ = 1 3 σ 4 λ ∆x , ζ = - 1 6 σ 4 α λ ∆x , γ = λ σ 4 ∆x 12 (2 -α) .
Observe that the dissipation γ is positive under usual conditions. If the conditions (29) are satisfied, we can specify the coefficients of the fourth order terms in the equations ( 28):

(31)

       η = λ ∆x 3 432 (α + 4) (α -2) , µ 4 = λ ∆x 3 108 (12 σ 2 4 -1) , ζ 4 = λ ∆x 3 216 σ 4 12 -α -2 α 2 + 12 σ 2 4 (α 2 -α -4) .
• In [START_REF] Augier | Linear lattice Boltzmann schemes for Acoustic: parameter choices and isotropy properties[END_REF], we have conducted a set of numerical experiments that make more explicit the isotropy qualities of four different variants of the D2Q9 lattice Boltzmann scheme for the numerical simulation of acoustic waves. The orders of isotropy precision numerically computed are in coherence with the level of accuracy presented in this section.

5) D2Q13

• Four more velocities are added to the D2Q9 scheme to construct D2Q13. The details can be found e.g. in [START_REF] Lallemand | Some results on energy-conserving lattice Boltzmann models[END_REF]. Nine moments are analogous to those proposed in ( 14) for D2Q9 and four moments (r x , r y , ε 3 , XX e ) are new:

(32)

                               0 1 λ 0 1, 2 X, Y λ 1 3 ε λ 2 4, 5 XX, XY λ 2 6, 7 q x , q y λ 3 8, 9 r x , r y λ 5 10 ε 2 λ 4 11 ε 3 λ 6 12 XX e λ 4 .
• At first order, the invariance by rotation (2) takes again the form [START_REF] Higuera | Lattice Gas Dynamics with Enhanced Collisions[END_REF]. The conditions ( 16) are essentially unchanged, except that the sound velocity is now evaluated according to the relation

(33) c 2 0 = λ 2 26 (28 + α) ≡ c 2 s λ 2 .
• At second order, the equivalent partial differential equations take the isotropic form (21) when we have :

(34) q eq = ϕ λ 2 J , r eq = 1 12

20 σ 5 -85 σ 4 -49 ϕ σ 4 -14 ϕ σ 5 σ 4 + σ 5 λ 4 J .



A. Augier, F. Dubois, B. Graille and P. Lallemand

Then the isotropy coefficients in [START_REF] Higuera | Lattice Gas Dynamics with Enhanced Collisions[END_REF] have the following expressions:

(35) µ = λ ∆x 2 σ 4 σ 5 σ 4 + σ 5 (3 + ϕ) , ζ = λ ∆x 26 σ 3 (11 + 13 ϕ -α) , σ k > 0 when k ≥ 3 .
• The invariance by rotation at third order of the mass equation is realized if we impose a unique value for the relaxation coefficients of the second order moments XX and XY introduced in (32):

(36)

σ 4 = σ 5 .
Moreover, the attenuation of sound waves γ does not depend at first order on the advective velocity if (36) is satisfied. For the invariance by rotation of the momentum equation, we must impose also the following equilibrium values for the moments m 10 ≡ ε 2 , m 11 ≡ ε 3 and m 12 ≡ XX e :

(37)

     ε eq 2 = -5 α + 77 26 ϕ α + 1078 13 ϕ λ 4 ρ ε eq 3 = α 48 - 137 12 - 135 208 α ϕ - 945 52 ϕ λ 4 ρ ,
XX eq e = 0 . Then the equivalent equations at order 3 of the D2Q13 lattice Boltzmann scheme are still given by the equations [START_REF] Weyl | Elementary theory of invariants[END_REF]. The equilibrium condition (34) is now written as (38)

r eq = - 1 24 (65 + 63 ϕ) λ 4 J
and the coefficients in the equations ( 23) can be clarified:

(39)

                   µ = 1 4 σ 5 (3 + ϕ) λ ∆x , ζ = 1 26 σ 4 (11 + 13 ϕ -α) λ ∆x , ξ = 1 624 (2 α -39 ϕ -61) ∆x 2 , χ = 1 8112 (28 + α) 61 + 39 ϕ + 12 α σ 2 3 -2 α -78 ϕ σ 2 4 -156 ϕ σ 2 3 -234 σ 2 4 -132 σ 2 3 λ 2 ∆x 2 .
• The invariance by rotation at fourth order is satisfied if we add to the previous conditions (36) (37) and (38) the new ones:

(40)

                             q eq = - 7 5 λ 2 J , σ 6 = σ 7 = 1 12 σ 4 , σ 8 = σ 9 = 5 24
155 -a a -308

1 σ 4 +
                           ξ = 5 α -16 1560 ∆x 2 , η = α + 28 40560 36σ 3 + 5σ 3 α -52σ 4 λ ∆x 3 µ = 2 5 σ 4 λ ∆x , ζ = - 1 130 σ 3 (36 + 5 α) λ ∆x χ = 1 20280 (28 + α) 16 -5 α + 216 σ 2 3 -312 σ 2 4 + 30 α σ 2 3 λ 2 ∆x 2 µ 4 = σ 4 λ ∆x 3 300 σ 3 (a -308) 4483σ 4 -5099 σ 3 -23 a σ 4 + 25 a σ 3 -14784 σ 3 σ 2 4 + 48 a σ 3 σ 2 4 .
The algebraic expression of the coefficient ζ 4 is quite long. With Hénon's coefficients σ j defined according to [START_REF] Lallemand | Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability[END_REF], the related moments numbered by the relations (32), the equilibrium of the energy [START_REF] Humières | Generalized Lattice-Boltzmann Equations[END_REF] parametrized by α, and the parameter a introduced at (40), the coefficient ζ 4 for the fourth order term in (28) can be evaluated according to:

                                                 ζ 4 = σ 4 λ ∆x 3 56581200 σ 3 (89a -

6) D3Q19

• For the scheme D3Q19, we have 4 conservation laws and a total of 19 moments. We refer e.g. to [START_REF] Dubois | Towards higher order lattice Boltzmann schemes[END_REF] for an algebraic expression of the polynomials p k in ( 13):

(44)

                               0 1 λ 0 1, 2, 3 X, Y, Z λ 1 4 ε λ 2 5, 6 XX, W W λ 2 7, 8, 9
XY, Y Z, ZX λ 2 10, 11, 12 q x , q y , q z λ 3 13 ε 2 λ 4 14, 15

XX e , W W e λ 4 16, 17, 18 antisymmetric of order 3 λ 3 .

The results summarized in this Section have been essentially considered (quickly) in the previous contribution [START_REF] Dubois | Quartic Parameters for Acoustic Applications of Lattice Boltzmann Scheme[END_REF]. They have been also used by Leriche, Lallemand and Labrosse [START_REF] Leriche | Stokes eigenmodes in cubic domain: primitive variable and Lattice Boltzmann formulations[END_REF] for the numerical determination of the eigenmodes of the Stokes problem in a cubic cavity.

• We write the four equivalent partial differential equations at first order with the method explained in Appendix 1. Then we impose that the associated modes are isotropic, i.e. contain only partial differential operators that are invariant by rotation. In Fourier space, the coefficients of the associated determinant must contain only powers of the wave vector. The associated equations are highly nonlinear relative to the coefficients of the equilibrium matrix introduced in [START_REF] Dubois | Unpublished pedagogical experiments with the D1Q3 lattice Boltzmann scheme for the heat equation[END_REF]. We have obtained a family of parameters by enforcing the linearity of the solution of the isotropic equations. With this constraint, we have to impose a relation for the "energy" moment m 4 ≡ ε at equilibrium:

(45) ε eq = α λ 2 ρ .
Moreover, the equilibrium values for the moments m 5 to m 9 of degree two introduced in (44) are equal to zero:

(46) XX eq = W W eq = XY eq = Y Z eq = ZX eq = 0 .

When the conditions (45) and ( 46) are realized, the isotropic equivalent system is given by the system of first order acoustic equations [START_REF] Higuera | Lattice Gas Dynamics with Enhanced Collisions[END_REF]. Moreover, the sound velocity c 0 satisfies (47)

c 2 0 = α + 30 57 λ 2 ≡ c s λ 2 .
• Invariance by rotation at second order is realized if we impose on one hand (48) q eq = 2 3 σ 5 -4 σ 7 σ 5 + 2 σ 7 λ 2 J and on the other hand (49)

σ 5 = σ 6 , σ 7 = σ 8 = σ 9 .



On rotational invariance of lattice Boltzmann schemes

Then the equivalent partial differential equations of the D3Q19 lattice Boltzmann scheme take the form [START_REF] Qian | Lattice BGK Models for Navier-Stokes Equation[END_REF]. The associated coefficients are given according to (50)

     µ = σ 5 σ 7 σ 5 + 2 σ 7 λ ∆x ζ = λ ∆x 57 (σ 5 + 2 σ 7 )
27 σ 4 σ 5 + 19 σ 5 σ 7 -22 σ 4 σ 7 -α σ 4 σ 5 -2 α σ 7 σ 4 α .

• At third order, if we impose the previous relations (45) (46) (48) and ( 49), id est an equilibrium for the "energy square" ε 2 given below, a null value for m eq 14 and m eq 15 and a supplementary condition for the relaxation coefficients, id est (51) ε eq 2 = 42 + 9 α 9 λ 2 ρ , XX eq e = W W eq e = 0 , σ 5 = σ 7 , the equivalent equations of the D3Q19 lattice Boltzmann scheme are exactly given by [START_REF] Weyl | Elementary theory of invariants[END_REF]. We observe that the relation (48) takes now the form (52)

q eq = - 2 3 λ 2 J
and the coefficients associated to the equations ( 23) can be deduced through an elementary process:

(53) • We have found also a variant of the previous relations to enforce third order isotropy. We can replace the relations (51) by the following ones, with an undefined parameter β:

     µ = 1 3 σ 5 λ ∆x , ζ = λ ∆x 171 5 σ 4 + 19 σ 5 -3 α σ 4 , ξ = ∆x 2 684 (α -27) , χ = λ 2
(55)    ε eq 2 = β λ 2 ρ , XX eq e = W W eq e = 0 , σ 5 = σ 7 , σ 10 = σ 11 = σ 12 = 1 12 σ 5 , σ 16 = 1 8 σ 5 , σ 17 = 1 4 σ 5 , σ 18 = 1 12 σ 5 .
The relations (52), ( 53) and (54) are not changed, except that the coefficient χ in the second line of ( 23) is now given by (56) • The invariance by rotation at fourth order has also been considered. But due to the low number of remaining parameters, the family of Boltzmann schemes that we have obtained impose constraints between physical parameters. We must have in particular Then the fourth order isotropic equivalent equations (28) are satisfied and the associated coefficients can be clarified:

         χ = λ 2
(59)

     ξ = α -27 684 ∆x 2 , η = (α + 30) (α -27) 38988 σ 5 λ ∆x 3 , µ 4 = 12 σ 2 5 -1 108 σ 5 λ ∆x 3 ζ 4 = σ 5 λ ∆x 3 38988 2062 + 45 α -4 α 2 -5304 σ 2 5 -612 α σ 2 5 + 24 α 2 σ 2 5 .

7) D3Q27

• For the schemes D3Q27, we refer e.g. to [START_REF] Dubois | Quartic Parameters for Acoustic Applications of Lattice Boltzmann Scheme[END_REF] for an algebraic expression of the polynomials p k of the relation [START_REF] Hénon | Viscosity of a Lattice Gas[END_REF]. The moments follow now the nomenclature (60)

                                                 0 1 λ 0 1, 2, 3 X, Y, Z λ 1 4 ε λ 2 5, 6 XX, W W λ 2 7, 8, 9
XY, Y Z, ZX λ 2 10, 11, 12 q x , q y , q z λ 3 • At first order, we follow the same methodology as the one presented for the previous schemes. We keep the relation (45) for the momentum m 4 ≡ ε at equilibrium. As for the D3Q19 scheme, the equilibrium values for the moments m 5 to m 9 of degree two introduced in (60) are equal to zero and the relation (46) still holds. Then the first order isotropic equivalent system is given by [START_REF] Higuera | Lattice Gas Dynamics with Enhanced Collisions[END_REF]. Observe that the sound velocity c 0 now satisfies (61)

c 2 0 = α + 2 3 λ 2 ≡ c s λ 2 .
• At second order the equilibrium values have to be constrained: the "heat flux" at equilibrium is given by the relation (62) q eq = 2 σ 5 -4 σ 7 σ 5 + 2 σ 7 λ 2 J  On rotational invariance of lattice Boltzmann schemes and a null value for the equilibrium of the third order moments is imposed:

(63) m eq 23 = m eq 24 = m eq 25 = m eq 26 = 0 . Moreover, the relations (49) between Hénon's parameters of second order moments still have to be imposed. Then the second order equivalent equations are isotropic and the coefficients of ( 21) follow the non-traditional relations

(64) µ = σ 5 σ 7 σ 5 + 2 σ 7 λ ∆x , ζ b ≡ 3 ζ -µ = λ σ 4 ∆x σ 5 + 2 σ 7 σ 5 -2 σ 7 -α σ 5 -2 α σ 5 .
• At third order, we have two options as for the D3Q19 scheme. If we suppose that the heat flux at equilibrium and only one time relaxation are fixed, id est (65) q eq = -2 λ 2 J , ε eq 2 = -(2 + 3 α) λ 2 ρ , XX eq e = W W eq e = XY eq e = Y Z eq e = ZX eq e = 0 , σ 5 = σ 7 , the third order equivalent equations of the D3Q27 lattice Boltzmann scheme are given by the expressions [START_REF] Weyl | Elementary theory of invariants[END_REF]. The coefficients in these equations are simple to evaluate with a software of formal calculus:

(66)      µ = 1 3 σ 5 λ ∆x , ζ b = - 1 3 σ 4 (1 + 3 α) λ ∆x , ξ = 1 36 (α -1) ∆x 2 , χ = 1 54 (α + 2) 1 + α + 6 α σ 2 4 -2 σ 2 4 + 8 σ 2 5 λ 2 ∆x 2 .
• The second solution for third order isotropy does not specify completely the "square of the energy" ε 2 but fixes an important number of relaxation times:

(67)

       q eq = -2 λ 2 J , ε eq 2 = β λ 2 ρ , σ 5 = σ 7 , σ 10 = σ 11 = σ 12 = 1 12 σ 5 , σ 23 = 1 12 σ 5 , σ 24 = 1 4 σ 5 , σ 25 = 1 8 σ 5 ,
XX eq e = W W eq e = XY eq e = Y Z eq e = ZX eq e = 0 . The parameters µ, ζ, ζ b and ξ are still given by the relations (66). But the value of the parameter χ is modified:

(68)      χ = 1 162 σ 5 4 σ 5 + 2 σ 4 + 3 α σ 4 -6 α σ 5 -3 α 2 σ 5 + 18 α 2 σ 2 4 σ 5 + 42 α σ 2 4 σ 5 +12 σ 2 4 σ 5 -24 α σ 3 5 -48 σ 3 5 + β σ 4 -β σ 5 λ 2 ∆x 2 .
• The search of an isotropic form of the fourth order equivalent partial differential equations like (28) leads to a nonlinear system of 33 equations. We have obtained a first solution with the following particular parameters:

(69) r eq = 2 λ 4 J , σ 10 = σ 11 = σ 12 , σ 18 = σ 19 , σ 23 = σ 24 = σ 25 .

It is possible to fix the other parameters of the scheme with the ratio ψ of Hénon's parameters associated with the moments XX e and XX. With the notations proposed in (60), we set (70) ψ ≡ σ 18 σ 5 .

When we impose the following relations between the coefficients of relaxations (all defined through their associated Hénon's parameter introduced at the relation ( 20)),

                                     σ 4 = σ 5 3 ψ 3 -4 ψ 2 -13 ψ + 32 3 ψ 3 -22 ψ 2 + 23 ψ + 14 , σ 10 = σ 11 = σ 12 = 1 12 σ 5 (3 ψ -7) (ψ -4) 3 ψ 2 -11 ψ + 14 , σ 16 = σ 5 6 ψ 5 -24 ψ 4 + 100 ψ 3 -267 ψ 2 + 506 ψ -364 (4 ψ -7) (3 ψ 3 -22 ψ 2 + 23 ψ + 14) , σ 18 = σ 19 ≡ ψ σ 5 , σ 23 = σ 24 = σ 25 = 1 12 σ 5 3 ψ 2 -7 ψ + 16 3 ψ 2 -11 ψ + 14 , σ 26 = 1 18 σ 5 21 ψ 2 -73 ψ + 100 3 ψ 2 -11 ψ + 14 , (71)
the equivalent partial differential equations of the D3Q27 lattice Boltzmann scheme are isotropic at fourth order of accuracy. We can make more explicit graphically the previous result. We observe in Figure 1 that the fundamental stability property σ 16 > 0 can be maintained only if 0 < ψ < 1.5. With this restriction, we see in Figure 1 again that Hénon's parameters σ 10 , σ 16 and σ 18 remain positive only if Due to the expressions (66) of the shear and the bulk viscosities, the inequality (72) imposes signifiant restrictions for the physical parameters µ and ζ b . The coefficients η and µ 4 associated with the fourth order equation (28) can be evaluated easily:

(72) 1 < σ 4 σ 5 < 2.
(73)            η = σ 5 λ ∆x 3 108 N η 14 + 23 ψ -22 ψ 2 + 3 ψ 3 , N η = (α + 2) (32 α -8 -13 α ψ -35 ψ -4 α ψ 2 + 28 ψ 2 + 3 α ψ 3 -3 ψ 3 ) µ 4 = σ 5 λ ∆x 3 108 132 ψ σ 2 5 -ψ + 36 ψ 2 σ 2 5 + 168 σ 2 6 + 3 ψ 2 -8 3 ψ 2 -11 ψ + 14 .
The expression of ζ 4 is quite long and is reported in the relation (114) of Appendix 2.

8) Conclusion

• In this contribution, we have presented the "Berliner version" of the Taylor expansion method in the linear case. This is done with explicit algebra and allows a huge reduction of computer time for formal analysis. We have also considered in all generality acoustic type partial differential equations that are rotationally invariant at an arbitrary order.

• The generalization of a methodology of group theory for discrete invariance groups of a lattice Boltzmann scheme remains still under question, in the spirit of the previous study of Rubinstein and Luo [START_REF] Rubinstein | Theory of the lattice Boltzmann equation: symmetry properties of discrete velocity sets[END_REF].

• Concerning the fundamental examples considered in this contribution, the D2Q9 scheme can be invariant by rotation at third order. At fourth order, physical parameters have to be strongly correlated. The D2Q13 scheme is invariant by rotation at fourth order for an ad hoc fitting of the parameters. We have not explored all the possible solutions of the strongly nonlinear set of equations that is necessary to solve in order to fit the fourth order isotropy. Numerical experiments have to confirm our theoretical considerations. The D3Q19 lattice Boltzmann scheme admits two sets of coefficients in order to impose rotational invariance at third order. Particular physics has to be imposed to satisfy fourth order isotropy. The D3Q27 scheme is rotationally invariant at fourth order for a parameterized set of parameters. Our analysis imposes restrictions for the physical parameters to guarantee the stability. A complementary numerical experiment will be welcome ! Appendix 1. Formal expansion in the linear case

• We present in this Appendix the "Berliner version" [START_REF] Dubois | Méthode générale de calcul de l'équation équivalente pour un schéma de Boltzmann sur réseau dans le cas linéaire[END_REF] of the algorithm proposed in all generality in our contribution [START_REF] Dubois | Towards higher order lattice Boltzmann schemes[END_REF]. We suppose having defined a lattice Boltzmann scheme "DdQq" with d space dimensions and q discrete velocities at each vertex. The invertible matrix M between the particules and the moments is given:

(74)

m k = q-1 j=0 M kj f j ≡ M•f k , 0 ≤ k ≤ q -1 .
The lattice Boltzmann scheme generates N conservation laws: the first moments

m k ≡ W k , 0 ≤ k ≤ N -1
are conserved during the collision step :

(75)

m * = m k = W k .
The q -N "slave" moments Y with (76)

Y ℓ ≡ m N +ℓ , 0 ≤ ℓ ≤ q -N -1
relax towards an equilibrium value Y eq ℓ . This equilibrium value is supposed to be a linear function of the state W . We introduce a constant rectangular matrix E with Nq lines and N columns to represent this linear function:

(77) Y eq ℓ = N -1 k=0 E ℓk W k , 0 ≤ ℓ ≤ q -N -1 .
The relaxation step is obtained through the usual algorithm [START_REF] Humières | Generalized Lattice-Boltzmann Equations[END_REF] that decouples the moments:

(78)

Y * ℓ = Y ℓ + s ℓ (Y eq ℓ -Y ℓ ) , s ℓ > 0 , 0 ≤ ℓ ≤ q -N -1 .
Observe that the numbering of the "s" coefficients used in (78) differ just a little from the one used for the equation ( 8) and the four examples considered previously. With a matricial notation, the relaxation can be written as:

(79) m * = J 0 • m with a matrix J 0 of order q decomposed by blocks according to (80)

J 0 =   I N 0 S • E I q-N -S  
and a diagonal matrix S of order q -N defined by S ≡ diag s 0 , s 1 , . . . , s q-N -1 . The discrete advection step follows the method of characteristics:

(81)

f j (x, t + ∆t) = f * j (x -v j ∆t, t) , 0 ≤ j ≤ q -1 .
• With the d'Humières's lattice Boltzmann scheme [START_REF] Humières | Generalized Lattice-Boltzmann Equations[END_REF] previously defined, we can proceed to a formal Taylor expansion:



On rotational invariance of lattice Boltzmann schemes

m k (t + ∆t) = j M kj f * j (x -v j ∆t) = jℓ M kj M -1 jℓ m * ℓ (x -v j ∆t) = jℓ M kj M -1 jℓ ∞ n=0 ∆t n n ! - d α=1 v α j ∂ α n m * ℓ = ∞ n=0 ∆t n n ! jℓp M kj M -1 jℓ - d α=1 v α j ∂ α n (J 0 ) ℓp m p .
We introduce a derivation matrix of order n ≥ 0, defined by blocks of space differential operators of order n:

(82)   A n B n C n D n   k p ≡ 1 n ! j ℓ M k j M -1 j ℓ - d α=1 v α j ∂ α n (J 0 ) ℓp , n ≥ 0 .
We observe that in the relation (82), the blocks A n and D n are square matrices of order N and q-N respectively. The matrices B n and C n are rectangular of order N ×(q-N) and (q -N) × N respectively. We remark also that at order zero, the matrices A 0 , B 0 , C 0 and D 0 are known:

(83)

  A 0 B 0 C 0 D 0   = J 0 =   I N 0 S • E I q-N -S   .
The previous Taylor expansion can now be written under a matricial form:

(84)   W Y   (x, t + ∆t) = ∞ n=0 ∆t n   A n B n C n D n   •   W Y   (x, t) .
• At order zero relative to ∆t we have:

W Y (x, t) + O(∆t) = J 0 • W Y + O(∆t) = W S • E • W + (I -S) • Y + O(∆t)
and the non-conserved moments are close to the equilibrium:

(85) Y (x, t) = E • W (x, t) + O(∆t) .
• We make now the hypothesis of a general form for the expansion of the nonconserved moments:

(86) Y (x, t) = E + n≥1 ∆t n β n • W (x, t)
and the hypothesis of a formal linear partial differential system of arbitrary order for the conserved variables W :

(87) ∂W ∂t = ℓ ≥ 0 ∆t ℓ α ℓ+1 • W (x, t) ,
where α ℓ and β n are space differential operators of order ℓ and n respectively. We develop the first equation of (84) up to first order:

W + ∆t ∂W ∂t + O(∆t 2 ) = W + ∆t A 1 W + B 1 Y + O(∆t 2 ) = W + ∆t A 1 W + B 1 E W + O(∆t 2 )
due to (85). Then (88)

∂W ∂t = A 1 + B 1 E • W + O(∆t)
and the relation ( 87) is satisfied at order one, with (89)

α 1 = A 1 + B 1 E .
The "Euler equations" are emerging ! We have an analogous calculus for the second equation of (84) :

Y + ∆t ∂Y ∂t + O(∆t 2 ) = S E W + (I -S) Y + ∆t C 1 W + D 1 E W + O(∆t 2 ) .
We clarify the time derivative ∂ t Y at order zero by differentiating (formally !) the relation (85) relative to time:

∂Y ∂t = E ∂W ∂t + O(∆t) = E α 1 W + O(∆t) .
We introduce this expression inside the previous calculus. Then:

S Y + ∆t E α 1 W + O(∆t 2 ) = S E W + ∆t C 1 W + D 1 E W + O(∆t 2 ) .
Consequently we have established the expansion of the nonconserved moments at order one:

(90)

Y = E W + ∆t S -1 C 1 + D 1 E -E α 1 W + O(∆t 2 ) with (91) β 1 = S -1 C 1 + D 1 E -E α 1 . Now, we have formally ∂ 2 W ∂t 2 = ∂ ∂t α 1 W + O(∆t) = α 1 ∂W ∂t + O(∆t) = α 1 α 1 W + O(∆t) = α 2 1 W + O(∆t) and we recognize the "wave equation" (92) ∂ 2 W ∂t 2 -α 2 1 W = O(∆t) .
• We can derive a formal expansion at order two. We go one step further in the Taylor expansion of equation (84) :

W + ∆t ∂W ∂t + 1 2 ∆t 2 α 2 1 W + O(∆t 3 ) = = W + ∆t A 1 W + B 1 Y + ∆t 2 A 2 W + B 2 Y + O(∆t 3 ) = W + ∆t A 1 W +B 1 (E W +∆t β 1 W ) + ∆t 2 A 2 W +B 2 E W + O(∆t 3 )
and dividing by ∆t, we obtain a "Navier-Stokes type" second order equivalent equation:

∂W ∂t = α 1 W + ∆t B 1 β 1 + A 2 + B 2 E - 1 2 α 2 1 W + O(∆t 2 ) .



On rotational invariance of lattice Boltzmann schemes

With the notations introduced in (87), we have made explicit the partial differential equations for the conserved variables at the order two:

∂W ∂t = α 1 W + ∆t α 2 W + O(∆t 2 ) with (93) α 2 = A 2 + B 2 E + B 1 β 1 - 1 2 α 2 1 .
We remark that this Taylor expansion method can be viewed as a "numerical Chapman Enskog expansion" relative to a specific numerical parameter ∆t instead of a small physical relaxation time step. For the moments Y out of equilibrium, we expand the first order derivative of Y relative to time with a formal derivation of the relation (90):

∂Y ∂t = ∂ ∂t E W + ∆t β 1 W + O(∆t 2 ) = E α 1 W + ∆t α 2 W + ∆t β 1 α 1 W + O(∆t 2 ) = E α 1 + ∆t E α 2 + β 1 α 1 W + O(∆t 2 ) . Then (94) ∂Y ∂t = E α 1 + ∆t E α 2 + β 1 α 1 W + O(∆t 2 ) .
Analogously for the second order time derivative:

SY = S E W -∆t E α 1 + ∆t (E α 2 + β 1 α 1 ) W - ∆t 2 2 E α 2 1 W + ∆t C 1 W + D 1 E + ∆t β 1 W + ∆t 2 C 2 W + D 2 E W ) + O(∆t 3 ) Y = E W + ∆t S -1 C 1 + D 1 E -E α 1 W + ∆t 2 S -1 C 2 + D 2 E + D 1 β 1 -E α 2 -β 1 α 1 - 1 2 E α 2 1 W + O(∆t 3
) . It is exactly the expansion (87) at second order :

Y = E W + ∆t β 1 W + ∆t 2 β 2 W + O(∆t 2 ) with (96) β 2 = S -1 C 2 + D 2 E + D 1 β 1 -E α 2 -β 1 α 1 - 1 2 E α 2 1
• For the general case, we proceed by induction. We suppose that the developments (86) and (87) are correct up to the order k, that is: The term ∂ j t W = ∞ ℓ=1 ∆t ℓ-1 α ℓ j on the left hand side of (98) can be evaluated by taking the formal power of the equation (87) at the order j. We define the coefficients Γ j m according to:

(99)

∞ ℓ=1 ∆t ℓ-1 α ℓ j ≡ ∞ ℓ=0
∆t ℓ Γ j j+ℓ , j ≥ 0 .

They can be evaluated without difficulty from the coefficients α ℓ , taking care of the noncommutativity of the product of two matrices. We report the corresponding terms and we identify the coefficients in factor of ∆t k between the two sides of the equation ( 98), with the help of the induction hypothesis (97). We deduce:

(100)

α k+1 = A k+1 + k+1 j=1 B j β k+1-j - k+1 j=2 1 j! Γ j k+1 .
We do the same operation with the second relation of ( 84 As in the previous case, we suppose that we have evaluated formally the temporal derivative • The explicitation of the coefficients Γ j j+ℓ and K j k+1 of the matricial formal series is now easy, due to the relations (99) and (102). We specify the coefficients Γ ℓ j+ℓ obtained

∂ j t Y = ∂ j t E +

( 19 )

 19 σ k > 0 , for k ≥ N .  A. Augier, F. Dubois, B. Graille and P. Lallemand

20680) 525433428 a σ 3 σ 4 + 576972000 α 2 σ 2 3 + 18001526400 α σ 3 3 σ 4 + 18001526400 α σ 3 σ 3 4 + 65975 a 2 α σ 3 σ 4 + 170558856 a σ 2 4 -334055628 a σ 2 3 -858312 a 2 σ 2 4 -159243217380 σ 3 σ 4 + 75143778660 σ 2 3 + 18001526400 α σ 2 3 σ 2 4 -77472720 a α σ 3 3 σ 4 -77472720 a α σ 2 3 σ 2 4 -77472720 a α σ 3 σ 3 4 + 504042739200 σ 3 σ 3 4 + 129610990080 σ 3 3 σ 4 + 129610990080 σ 2 3 σ 2 4 + 858312 a 2 σ 3 σ 4 + 22940190 a α σ 3 σ 4 + 17841109925 α σ 2 3 -65975 a 2 α σ 2 4 + 13110175 a α σ 2 4 -3461832000 α 2 σ 4 3 -85853433600 α σ 4 3 -2483100 a α 2 σ 2 3 -78263065 a α σ 2 3 -438683351040 σ 4 3 + 369485280 a α σ 4 3 + 14898600 a α 2 σ 4 3 + 1887950592 a σ 4 3 -557803584 a σ 2 3 σ 2 4 -2169236160 a σ 3 σ 3 4 -557803584 a σ 3 3 σ 4 -

 434444344344444444434433333333444 8032585925 α σ 3 σ 4 . 

∆x 2 19494 α + 30 27 + 6 α σ 2 4 -10 σ 2 4 -

 44 152 σ 2 5 -α . The bulk viscosity ζ b ≡ 3 ζµ (see e.g. Landau and Lifshitz [19]) is essentially function of the relaxation parameter associated to the energy ε :

∆x 2 409374 σ 5 16212 σ 5 + 126 α 2 σ 2 4 σ 5 + 3570 α σ 2 4 σ 5 -6300 σ 2 4 σ 5 -σ 5 α 2 - 4 + 798 σ 4 -3192 α σ 3 5 -95760 σ 3 5 -

 555524455 234 α σ 5 + 361 β σ 4 + 171 α σ 361 β σ 5 .

(57) σ 4 = σ 5 

 5 and this relation induces some a priori relationship between the shear viscosity µ of relation (50) and the bulk viscosity ζ b presented in (54). Moreover, the relations (51) (52) must be satisfied and Hénon's parameters of the multiple relaxation times have to follow, with an ordering proposed in (44), the complementary conditions (58) σ 10 = σ 11 = σ 12 = 1 6 σ 5 , σ 13 = σ 14 = σ 15 = σ 5 , σ 16 = σ 17 = σ 18 = 1 6 σ 5.

5 σ 5 Figure 1 .

 551 Figure 1. Fourth order isotropy parameters for the D3Q27 lattice Boltzmann scheme.

2 = E α 2 1 W 2 ∂ 2

 2122 + O(∆t) .We re-write the second line of the expansion of the equation (84) at second order accuracy:Y ∂t 2 + O(∆t 3 ) = = S E W + (I -S) Y + ∆t C 1 W + D 1 Y ) + ∆t 2 C 2 W + D 2 Y ) + O(∆t 3 )and we get

= α 1 +

 1 ∆t α 2 + . . . ∆t k-1 α k W + O(∆t k ) Y = E + ∆t β 1 + ∆t 2 β 2 + . . . ∆t k β k W + O(∆t k+1 ) . A. Augier, F. Dubois, B. Graille and P. LallemandWe expand the relation (84) at order k + 2, we eliminate the zeroth order term and divide by ∆t. 1 A j W + B j Y + O(∆t k+1 ) .

  Y + O(∆t k+2 ) = = S E W + (I -S) Y + k+1 j=1 ∆t j C j W + D j Y + O(∆t k+2 ) .

∆t β 1

 1 + ∆t 2 β 2 + . . . + ∆t k β k + . . . W = E + ∆t β 1 + ∆t 2 β 2 + . . . + ∆t k β k + . . . ∂ j t W = E + ∆t β 1 + ∆t 2 β 2 + . . . + ∆t k β k + . . . α 1 + ∆t α 2 + • • • + ∆t ℓ α ℓ + . . . j Wrelatively to the space derivatives. Then with the help of the induction hypothesis j j+ℓ , j ≥ 0 , we identify the two expressions of the coefficient of ∆t k+1 issued from the equation (101):(103) S β k+1 = C k+1 + k+1 j=1 D j β k+1-j -

  , σ 10 and σ 11 are strictly positive if it is the case for σ 3 and σ 4 . In this case, the stability conditions[START_REF] Landau | Fluid Mechanics[END_REF] are satisfied for the coefficients σ 3 , σ 4 , σ 8 , σ 10 and σ 11 . With the choice (40) the nontrivial algebraic expressions of the previous conditions

	the coefficients σ 8 (36), (37) and (38) can be written as
	(42)	r eq =	29 30	λ 4 J , ε eq 2 = -	1189 130	α +	7546 65	λ 4 ρ , ε eq 3 =	547 39	+	145 156	α λ 4 ρ .
	With the above conditions (40) and (42) the equivalent equations of the D2Q13 lattice
	Boltzmann scheme at fourth order are made explicit in (28), with the associated coeffi-
	cients, except ζ 4 , given according to:
	(43)
		σ 10 = σ 11 =	3973 45 a 155 σ 4 . 43 a -16610 89 a -20680 +	1 24 5 c 2 s -4 7 a -1391 a -308 1189 c 2 σ 3 + s -828 154 1395 7 a -1391 89 a -20680 1189 c 2 1 , σ 3 725 c 2 s -418 s -828	a σ 4 ,
	(41)	c 2 s <	418 25	, -28 < α ≤ -	9432 725	≃ -13 , 155 < a <	1391 7	≃ 198 ,
										

If the parameters c s and α relative to the non-dimensionalized sound velocity are linked together thanks to (33) and if the new parameter a is chosen such that

Acknowledgments

This work has been financially supported by the French Ministry of Industry (DGCIS) and the "Région Ile-de-France" in the framework of the LaBS Project. The authors thank Yves Benoist (Centre National de la Recherche Scientifique and department of Mathematics in Orsay) for an enlighting discussion about the representation of groups. The authors thank also the referees for helpful comments and suggestions.

in the matricial formal series (99). For j = 0, the power in relation (99) is the identity. Then (104) Γ 0 0 = I , Γ 0 ℓ = 0 , ℓ ≥ 1 . When j = 1, the initial series is not changed. Then (105)

For j = 2, we have to compute the square of the initial series, paying attention that the matrix operators α ℓ do not commute. Observe that with the formal Chapman-Enskog method used e.g. in [START_REF] Humières | Generalized Lattice-Boltzmann Equations[END_REF], non-commutation relations have also to be taken into consideration for higher order terms in the case of several conserved moments. We have

and we have in particular

and in consequence (107)

We have in particular for j = 3 and j = 4:

(108)

For the explicitation of the coefficients K j k+1 , we can replace the power of the formal series of the relation (99) in the relation (102). We obtain, with the notation

For j = 0, we deduce (110) K 0 0 = E , K 0 p = 0 , p ≥ 1 and for j = 1, we have a simple product of two formal series:

We specify some particular values of the coefficients K j j+p when j = 2, j = 3 and for j = 4:



A. Augier, F. Dubois, B. Graille and P. Lallemand

• It is now possible to make explicit up to fourth order to fix the ideas the matricial coefficients of the expansion (86) of the nonconserved moments and of the associated partial differential equation (87). We have, following the natural order of the algorithm:

(113)

Observe that with the explicit relations (113), the computer time for deriving formally the equivalent partial equation like (97) at fourth order of accuracy has been reduced by three orders of magnitude (!) in comparison with the algorithm presented in the contribution [START_REF] Dubois | Towards higher order lattice Boltzmann schemes[END_REF].

Appendix 2. A specific algebraic coefficient

• With Hénon's coefficients σ j defined according to [START_REF] Lallemand | Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability[END_REF], a numbering of the D3Q27 moments proposed in (60), the equilibrium of the energy [START_REF] Humières | Generalized Lattice-Boltzmann Equations[END_REF] parametrized by α, and the parameter ψ introduced in (70), the coefficient ζ 4 for the fourth order term in (28) can be evaluated according to:  On rotational invariance of lattice Boltzmann schemes


Review of the Rise of Discourse Markers (Heine et al. 2021) By Eric Mélac for Journal of Linguistics

The Rise of Discourse Markers, by Bernd Heine, Gunther Kaltenböck, Tania Kuteva, and Haiping Long is an outstanding reference work for any linguist interested in discourse and language change. It provides a review of the scattered literature describing the role and evolution of discourse markers (from now 'DMs') in several languages, as well as a clear synthesis of the most recent developments in the theories of language change. Even though DMs are indispensable, pervasive tools in everyday conversations, they had been neglected by several generations of grammarians, before becoming a vibrant field of study in the past three decades. The functions of DMs are notoriously difficult to investigate, which has led linguists to develop new analytical frameworks. The various, overlapping terminologies and concepts that have emerged to account for the behaviour of DMs make the literature particularly challenging to unravel. This challenge was successfully taken up by the four authors, and the Rise of Discourse Markers will answer many questions of any researcher who has been waiting for a work making sense of the heterogeneous contributions to the field.

The book contains the following sections: (1) an introduction on the development of discourse markers, which discusses the properties that the previous literature has identified to define DMs (chapter 1). It also presents the several hypotheses that have been investigated to account for their rise, and finally examines the problematic specificities regarding the emergence and evolution of discourse markers which are hard to reconcile with an allgrammaticalization hypothesis;

(2) a presentation of the concepts that are used in this book (ch. 2). It indicates the main criteria defining grammaticalization and cooptation, which are both involved in the rise of DMs. It presents an innovative, comprehensive description of cooptation, which is argued to be a key stage in the process;

(3) a selection of case studies on the development of DMs in four languages (ch. 3, 4, 5, and 6):

twelve English DMs (after all, like, well…), six French DMs (à la rigueur, en fait…), eight Japanese DMs (demo, ga…), and four Korean DMs (icey, maliya…). This section provides a detailed step-by-step account of the rise of those DMs in order to test the grammaticalization-cooptation hypothesis;

(4) an analysis of the behaviour of discourse markers in language contact (ch. 7), illustrating the processes of borrowing in a variety of languages. This chapter also offers theoretical analyses on the functions of discourse as they are revealed by the transfer of DMs from one language to another. A case study of Arabic Yaa'ni, as an example of the areal development of DMs, concludes the section;

(5) a final discussion and a conclusion (ch. 8 and 9), which summarize the authors' views on the role of grammaticalization and cooptation in the rise of DMs, and give an overview of the origins and classifications of DMs.

The introduction provides examples that explain clearly how the same form can function as a simple lexical item or a DM. It offers a definition of DMs, and a useful list of their properties, thus arguing why they should be considered a separate, coherent category [START_REF] Fraser | Types of English discourse markers[END_REF][START_REF] Brinton | Pragmatic Markers in English: Grammaticalization and Discourse Functions[END_REF][START_REF] Hansen | Alors and donc in spoken French: A reanalysis[END_REF][START_REF] Aijmer | Pragmatic markers in translation: a methodological proposal[END_REF]. Always adopting a very intelligible style, it also synthesizes the literature on the different hypotheses regarding their development. Providing a concise, structured review of such a disputed research field was a necessary, but far from easy task, and the authors have managed to complete it successfully. Even though one has to accept that, like for other linguistic categories, the boundaries can be fuzzy, additional syntactic and semantic tests, as well as more explicit contrastive examples, could have been useful. It is sometimes difficult to distinguish DMs from neighbouring items such as interjections or comment clauses, and one may also wonder whether certain sentence adverbs (apparently, clearly…), or exclamatives (wow, sure…) should be classified as DMs.

The authors then present how DMs have been dealt with in the different approaches of language change. They provide a very valuable, concise account of the development of DMs from the perspective of grammaticalization, pragmaticalization, lexicalization, cooptation, constructionalization, as well as other lines of research [START_REF] Traugott | The role of the development of discourse markers in a theory of grammaticalization[END_REF][START_REF] Hayashi | A cross-linguistic exploration of demonstratives in interaction: With particular reference to the context of word-formulation trouble[END_REF][START_REF] Wischer | Grammaticalization versus lexicalization: 'Methinks' there is some confusion[END_REF][START_REF] Heine | On discourse markers: Grammaticalization, pragmaticalization, or something else?[END_REF][START_REF] Traugott | Constructionalization and constructional changes[END_REF]. This section brings to light the salient points and difficulties that each approach has revealed, and demonstrates the necessity of the authors' work since none of these theories suffices to explain the phenomenon in its entirety. The main issue seems to be that previous approaches have failed to account for all the stages that markers have to go through before reaching the status of DMs. It is also disputed whether DMs belong to grammar, since they do not behave like lexical items, but are distinct from typical grammatical items on various aspects: their discursive function, textual scope, syntactic detachment, prosodic independence, and flexible position [START_REF] Fraser | Types of English discourse markers[END_REF][START_REF] Traugott | The role of the development of discourse markers in a theory of grammaticalization[END_REF][START_REF] Waltereit | Imperatives, interruption in conversation, and the rise of discourse markers: A study of Italian guarda[END_REF][START_REF] Brinton | Lexicalization and language change[END_REF]. Each property that makes DMs stand apart from other linguistic forms correspond to diachronic mechanisms that need to be examined, and documented chronologically in order to explain how they relate to one another when DMs arise.

The authors have thus efficiently covered a very large area in the introduction, and have remarkably presented most of the literature, giving us all the background knowledge necessary to fully understand the following content of the book.

Chapter 2, entitled 'Concepts of Analysis', expounds with great clarity the framework and the methods adopted in the book, as well as the limits of the study. The authors are well aware that diachronic investigations present numerous difficulties such as piecemeal data, and non-verifiable ambiguities.

This is all the more true when dealing with items that flourish in spoken language, such as DMs. This chapter presents a very convincing demonstration of the necessity to integrate cooptation as a major process of language change. DMs belong to the larger category of theticals, which are defined as items encoding discourse grammar instead of sentence grammar [START_REF] Kaltenböck | On thetical grammar[END_REF]). Cooptation is one step in the development of theticals, and this instantaneous process can occur before and/or after the grammaticalization of DMs. Cooptation is presented both as a cognitive and linguistic phenomenon.

The authors argue for a multidimensional characterization of this process that involves meaning, function, syntax, prosody, semantic-pragmatic scope, and placement. This approach rightfully draws our attention to the frequent co-occurrence of mechanisms impacting different linguistic levels, thus justifying that cooptation is observable through the presence of several symptoms. However, because cooptation is multidimensional, it can be difficult to pinpoint when it starts, and when it ends. The authors draw a sharp distinction between (a) 'You probably know already that our chairman will resign next month' and (b) 'Our chairman, you probably know already, will resign next month' (ex. 7, p. 65).

They argue that the clause in bold in (b) is the coopted version of the one in (a) because '(i) its meaning is not part of its host sentence, (ii) its function can be said to be metatextual In the following chapters (3, 4, 5, and 6), the authors have compiled in-depth case studies in English, French, Japanese and Korean that justify their framework, and document the cross-linguistic relevance of their analyses. The DMs in each language are investigated with the same depth, which is rare in typological work, given that the general descriptions of some languages are more advanced than others. This section brilliantly shows that the diachronic phenomena the authors have revealed are observable in unrelated languages. The authors humbly admit that they have failed in their investigation of other language families, and this is perfectly understandable since some data are far more available for some languages than others. Nevertheless, a more diversified language sample would have better supported the universal ambition of the book. French and English belong to the Indo-European family, and have influenced each other greatly, while Japanese and Korean are typologically comparable. It would have been useful to read a few examples of DMs in isolating or polysynthetic languages, just to see whether the category is also relevant for them. The criteria for constituting the sample of DMs in each language could also have been made more explicit. In French, for example, very frequent DMs, such as là, or du coup [START_REF] Forget | Là: un marqueur de pertinence discursive[END_REF][START_REF] Rossari | Du coup et les connecteurs de conséquence dans une perspective dynamique[END_REF]Chanet 2000 inter alia), are not mentioned whereas the rather infrequent à la rigueur or à propos are analyzed in detail. However, the authors have probably been wise to include only instances that were backed up by a large amount of verifiable historical data. We are thus presented an impressive investigation In conclusion, the Rise of Discourse Markers is a long-awaited, imposing reference work that must have required numerous years of data mining and scientific collaboration. This well-crafted book is an unprecedented synthesis of a fruitful research field that is of great interest for specialists and general linguists alike. It presents with a clear, structured style an impressive number of well-selected references as well as exclusive, cutting-edge investigations.

  […], (iii) it is syntactically unattached […), (iv) it is likely to be set off prosodically[…], and (v) it can be moved to other positions of the sentence' (p. 66). These observations confirm the existence of discourse grammar as distinct from sentence grammar, but I wonder whether all the listed mechanisms were actually involved in the development of (b) out of (a). As the authors themselves point out, (a) and (b) mean 'essentially the same' (p. 65), and I would argue that (a) also has a metatextual meaning, since it is a clause that participates in the management of discourse. Although this example is presented as a textbook case, certain characteristics of cooptation, such as semantic shift, do not seem to be relevant here. It seems that the most determining difference between (a) and (b) is the syntactic independence of the clause in bold in (b), which can be tested by its freedom of placement. It can thus be argued that a narrower definition of cooptation may be more operationalizable, e.g. 'the syntactic detachment of a metatextual item', while the other dimensions, such as semantic change, scope variability, or prosodic detachment are just frequent by-products, and not determining criteria for the diagnosis of cooptation per se.

  of the step-by-step development of a variety of DMs, and future work will likely confirm the observed patterns in other, less-investigated languages.Chapter 7, entitled 'Discourse Markers in Language Contact', provides a good overview of the borrowing or contact-induced replication of DMs. Despite the little amount of historical data available in the literature, the authors provide examples from several languages that clearly illustrate the borrowing of DMs. Most of the chapter is devoted to the communicative and cognitive functions of DMs that can explain their easy movement from one language to another. The authors offer a wellwritten, accessible synthesis of what DMs are used for. Given how complex it is to grasp the discursive functions of DMs in bilingual and monolingual speech, such a concise, well-structured description is a very valuable contribution for linguistics, but also for other fields, such as psychology, and sociology.Further discourse analysis and psycholinguistic work could also have been cited in order to substantiate and measure some of the claims[START_REF] Berman | The psycholinguistics of developing text construction[END_REF][START_REF] Al-Surmi | Discourse Markers and Reading Comprehension: Is there an effect? Theory & Practice in Language Studies[END_REF] Rasenberg et al. 2018; Bosker et al. 2021 inter alia), but the focused style of the chapter makes it very pleasant to read. As far as the sociological dimension is concerned, I wonder whether it is necessary to use the terms 'socioeconomic values' and 'emblematic status', instead of referring to the more established concepts of 'overt and covert prestige'[START_REF] Labov | The effect of social mobility on linguistic behavior[END_REF]. The paradoxical prestige of DMs could also have been confronted, since some DMs, such as like in English or genre in French are usually stigmatized, and are frequently used compulsively despite their functional redundancy(Schiffrin 1987: 310; Andersen 1998: 150). The discussion and conclusion of the book are engaging since they provide both a clear synthesis of what has been developed, and open new perspectives. The authors admit that even though the processes leading to the rise of DMs are now well documented, little is known about what actually motivates their development. They explore several hypotheses, thus paving the way to new, compelling areas of research. They also offer a clear overview of what we know about the origins of DMs, and the precise criteria that make certain linguistic items potential sources for them.


Eric Mélac

Nicolas Tournadre

The Polyfunctionality of 'give'

. Its polyfunctional quality seems to be universally attested, as the verb root is frequently redeployed into other syntactic categories, and commonly appears 1 We would like to thank Amanda Edmonds, Xénia De Heering, Françoise Robin, Camille Simon, and Debra Ziegeler for their valuable comments, which allowed us to deepen our understanding of this paper's topic.

as a light verb [START_REF] Jespersen | A Modern English Grammar on Historical Principles Part IV: Morphology[END_REF][START_REF] Montaut | Aspects, voix et diathèses en hindi moderne : syntaxe, sémantique, énonciation[END_REF]Newman, Ed., 1997;[START_REF] Mohanan | Grammatical verbs (with special reference to light verbs)[END_REF]Tournadre & Pezechki, forthcoming, inter alia). However, the cross-linguistic descriptions collected on this verb and its equivalents still have to be confronted with more data from typologically and genetically diverse languages, which is the aim of this volume. Newman (Ed., 1997) has explored the use of 'give' in many languages belonging to several language families, but no comprehensive study has been conducted on 'give' in a Tibetic language so far. Our description of 'give' in standard spoken Tibetan will lead us to investigate the link between the transfer construction, the development of light verbs and the honorific domain. It will also allow us to formulate hypotheses on the cognitive and social motivations behind the evolutionary patterns of 'giving verbs'.

The various lexical items corresponding to 'give' in Tibetan

In order to answer the question of how one says 'give' in Tibetan, it is first essential to identify the semantic features that are generally included in the verb 'give', as well as what we mean by the 'Tibetan language'.

The polysemy of 'give'

'Give' is a highly polysemous verb in English, and its most literal meaning is to 'hand an object over to someone' [START_REF] Newman | Give: A cognitive linguistic study[END_REF]. However, it is distinguished from the verbs 'hand' or 'pass' by its abstractness. It is perfectly acceptable to use 'give' when the giver, the transferred object and the recipient are not physically in the same place, or even when the three elements of the giving process are abstract entities.

(1) The station has been giving time to stories on education, government, the arts and community issues. (COCA, SPOK;npr_TalkNation, 2000) In this example, the giver 'station', the transferred entity 'time' and the recipient 'stories' are all immaterial. The meaning that an English speaker can naturally extract from this sentence is more or less that the program director of the radio station has decided that stories on education, government, the arts and community issues should be allowed a certain amount of time. This example shows that in English the verb 'give' has become a quick, spontaneous and convenient tool to express complex, abstract ideas.

According to the Oxford English Dictionary, the general sense of 'give' is 'make another the recipient of (something that is in the possession, or at the disposal, of the subject)'.2 However, as the entry contains 112 senses and 17 subentries, it is quite obvious that the definition of 'give' is highly complex, and its diverse categories of meaning are not easy to map. In English as well as in probably many of the world's languages, 'give' corresponds to a semantic area made up of multiple protrusions with indefinite, ever-evolving boundaries.

'Tibetan' and the Tibetic language family

Until the end of the 20 th century, most scholars spoke of 'Tibetan dialects' to refer to a number of languages derived from Old Tibetan and spoken mainly in the Tibetan cultural area, currently covering six countries (China, India, Bhutan, Nepal, Pakistan and Myanmar). This label is problematic because the so-called 'Tibetan dialects' do not allow mutual intelligibility. In the beginning of the 21 st century, the term 'Tibetan dialects' used to designate the languages derived from Old Tibetan has been progressively replaced by 'Tibetan languages' [START_REF] Zeisler | Relative tense and aspectual values in Tibetan languages: a comparative study[END_REF]Gawne & Hill, 2017). [START_REF] Tournadre | The Tibetic languages and their classification[END_REF] proposed to adopt the term 'Tibetic' to refer to this well-defined language family, and the term is now widely used (Sun, 2014;Gawne & Hill, 2017;[START_REF] Yliniemi | Copulas in Denjonke or Sikkimese Bhutia[END_REF][START_REF] Hyslop | An overview of some epistemic categories in Dzongkha[END_REF][START_REF] Chirkova | Evidentials in Pingwu Baima[END_REF][START_REF] Suzuki | The evidential system of Zhollam Tibetan[END_REF][START_REF] Delancey | Evidentiality in Tibetic[END_REF]. 3The Tibetic family includes at least 50 languages. However, the total number of dialects and varieties certainly amounts to more than 200. The term 'Tibetic languages' is preferable to 'Tibetan languages' because these languages are spoken not only by Tibetans per se, but also by other ethnic groups such as Ladakhi, Balti, Lahuli, Sherpa, Bhutanese, Sikkimese, etc. who do not actually consider themselves to be Tibetan. 4 They also do not call their languages 'Tibetan' (BOD.SKAD in Tibetan). Similarly, we do not talk of Latin or Italian languages, but of Romance languages, and do not think of French, Portuguese, Italian, Catalan or Romanian as various 'dialects' of Latin or Italian [START_REF] Tournadre | The Tibetic languages and their classification[END_REF]. Finally, let us note that Literary Tibetan is vastly different from the modern spoken languages.

In order to circumscribe a manageable dataset and conduct an in-depth analysis, we will focus here on one Ü-Tsang spoken dialect that belongs to the Central Tibetic language group, and which is usually referred to as Lhasa spoken Tibetan or standard spoken Tibetan. We will also consider Old and Classical Tibetan in order to understand the evolutionary paths that 'giving verbs' have followed leading to the current linguistic situation in standard spoken Tibetan.

2.3.'Give' in Old and Classical Tibetan

The basic verb expressing 'give' in Tibetan has evolved from SBYIN/BYIN in Old Tibetan to SPRAD 5 in contemporary standard spoken Tibetan. 6 In Old Tibetan, the verb SPROD/SPRAD barely exists whereas SBYIN/BYIN is commonly used:

(2) 'di btsun.mo gnang.cen-la byin dang sras yong-pa-r In the modern Tibetic languages, the most frequently attested verb is SBYIN, but the other verbs are also found in some languages. For example, BTANG is the main verb for 'give' in Ladakh and GTOD is used in Spiti (Tournadre & Suzuki, forthcoming). In some languages such as Amdo and Sherpa, SBYIN and STER are used as suppletive verbs to express the completed past versus the imperfective/future of the basic verb 'give' (Robin, forthcoming;Tournadre et al., 2009).

'Give' in standard spoken Tibetan

The basic way of saying 'give' in standard spoken Tibetan is SPRAD. In this dialect, STER 'give, offer, treat' is a synonym of SPRAD, with a more specific meaning. The verb GNANG is the honorific form of SPRAD, and is also a frequent light verb. PHUL is the humilific form of SPRAD, and can also be used as a light verb. SBYIN is not used as a verb in standard spoken Tibetan, but the morpheme is found in the compound form BYIN.RLABS 'blessing' or SBYIN.BDAG 'a benefactor'.

In English, a verb like 'give' has many non-literal senses and frequently appears as a light verb in (semi-)fossilized constructions. However, in standard spoken Tibetan, SPRAD is very rarely used figuratively and does not appear as a light verb. It might be surprising to note that SPRAD, as well as the archaic form SBYIN/BYIN, have not really developed figurative meanings, and do not serve as light verbs. Old Tibetan already had a set of light verbs and many of them are still used today. Among them, BTANG can be seen as a verb expressing 'give, send, let go' that was already widely used as a light verb in Old Tibetan, and it seems that the other basic verbs expressing 'give', such as SBYIN/BYIN and SPROD/SPRAD, have never competed for this status.

'Give' as a light verb

Jespersen (1965) coined the term 'light verb' to refer to verbs appearing in an English 'verb + noun phrase construction' as in 'have dinner' or 'take a shower'. Light verb constructions are a type of complex verbal lexeme which usually result from the collocational association of a nominal element and a verbal element, the latter being called a 'light verb'. 8 One of the salient properties of light verbs is their limited semantic weight and their involvement in a linguistic sequence that functions as a complex predicate. It is therefore the verb complement that mostly carries the semantic weight of the construction (Tournadre & Lessan-Pezechki, forthcoming).

Lexical verbs that are good candidates to function as light verbs belong to various semantic classes. There are mainly 'action verbs' ('do', 'make', 'hit', 'eat'), 'transfer verbs' ('give', 'put', 'carry', 'set', 'send', 'take'), 'motion verbs' ('go', 'come', 'run') and 'state verbs' ('be', 'have', 'become', 'stay') (Tournadre & Lessan-Pezechki, forthcoming).

Light verbs are not as 'light' as they seem, as they may convey significant lexical information, and they may also encode some grammatical meaning, particularly in the fields of aspect, modality, voice and diathesis [START_REF] Simon | Dérivation causative en tibétain (Lhasa)[END_REF]Tournadre & Lessan-Pezechki, forthcoming).

'Give' appears in the list of the verbs that are commonly selected to be used as light verbs cross-linguistically [START_REF] Mohanan | Grammatical verbs (with special reference to light verbs)[END_REF] Tournadre & Lessan-Pezechki, forthcoming). In English, 'give' is a light verb in many constructions. As expected with the light verb constructions, their fossilization is quite idiosyncratic and will differ greatly from one language to another, and even from one dialect to another. Light verb constructions raise a problem for translation, since it is not possible to document all of them in a monolingual or in a bilingual dictionary. Therefore, when translating [START_REF] Tournadre | Manuel de tibétain standard[END_REF][START_REF] Tournadre | Manual of Standard Tibetan: language and civilization[END_REF][START_REF] Delancey | Ergativity and the cognitive model of event structure in Lhasa Tibetan[END_REF][START_REF] Kopp | Verbalizers in Lhasa Tibetan[END_REF][START_REF] Tournadre | Final auxiliary verbs in literary Tibetan and in the dialects[END_REF][START_REF] Bartee | The role of animacy in the verbal morphology of Dongwang Tibetan[END_REF][START_REF] Randall | The Properties of Lhasa Tibetan Verbalizers[END_REF]Mélac, Robin & Simon, 2014, inter alia).

'Give an explanation' 'GREL.BSHAD BRGYAB (lit., explain hit); 'give an answer' This analysis and listing of constructions that would be appropriate translations of the English verb 'give' reveal several points that are essential to further our analysis of 'give' in Tibetan. Firstly, 'give' is not only highly abstract and polysemous in English, it also appears as a light verb in a great variety of entrenched constructions, making the predictability of the use of 'give' instead of another light verb for a given idiomatic construction quite low. 10 Secondly, although the translation of 'give' when it has a basic meaning of 'handing something over to someone' is quite straightforward in standard spoken Tibetan (SPRAD), the translation of figurative 'give' is in no way easy to systematize. Finally, this contrastive analysis illustrates that the Tibetan language has also developed very common light verbs, but SPRAD is not one of them.

The distribution of SPRAD, GNANG and PHUL in standard spoken

Tibetan

Now that we know that the answer to the question 'How do you say "give" in Tibetan?' is not as simple as it seems, we can adopt a form-to-function approach starting from Tibetan. We will focus on the most common words that can express the notion of 'giving' in standard spoken Tibetan and briefly explore their evolution from Old Tibetan.

The Tibetan verb SPRAD

The verb SPRAD is the most common way of saying 'give' when it means 'hand something over'. However, SPRAD is quite monosemous, literal and unproductive. In Old Tibetan, and partly in Classical Tibetan, the meaning of SPROD/SPRAD was different from its contemporary use in standard spoken Tibetan, since it usually meant 'meet' or 'bring together'.

( and 'make an announcement' are perfect collocations while ' ?? make a talk' and ' ?? give an announcement' are not.

In standard spoken Tibetan, the verb SPRAD has lost that meaning and only signifies 'give'. It appears 11 times in the TSC, where it is always associated with a concrete object (money, food, things, socks, gloves…; see example 4).

This further confirms that the use of SPRAD is quite different from 'give' in English, the latter having most often a figurative meaning as shown by large corpus statistics [START_REF] Gilquin | What you think ain't what you get: Highly polysemous verbs in mind and language. Du fait grammatical au fait cognitif[END_REF].

The Tibetan verb GNANG

The verb GNANG is used as a plain lexical verb in Old Tibetan to mean 'give' in the honorific. It can have both a concrete or an abstract meaning. Since the agent of the transfer has to be a high-profile person, it can quite often be translated as 'grant'.

(6) bdag-gyi pha ltar dgum-ba-r ci gnang zhes gsol=lo In standard spoken Tibetan, it is used as a lexical verb to convey the honorific meaning of both SPRAD 'give, hand over' and BYED 'do' [START_REF] Tournadre | Manuel de tibétain standard[END_REF][START_REF] Tournadre | Manual of Standard Tibetan: language and civilization[END_REF]. It also occurs as an honorific morpheme after an honorific lexical verb such as GSUNG ('say') or after the light verb BTANG (see below).

In the TSC, it appears 24 times. In no occurrence, is it the honorific form of (lit., turn hit). In order to examine why the humilific and honorific forms PHUL and GNANG are actually more productive than the ordinary verb SPRAD, it is first necessary to understand the mechanics of the honorific domain in Tibetan.

'Give' and the honorific domain

The honorific domain in Tibetan

We are using the term 'domain' here, as it is slightly misleading to call the honorific system of Tibetan a register, since 'a register' refers to a certain linguistic variety that has a diffuse impact on a speaker's speech and is particularly dependent on social settings. The use of Tibetan honorifics is more specific and systematized than what is usually meant by the notion of register, as it reflects the speaker's attitude toward the referents of the words he uses by positioning them on the social scale.

In all the world's languages, there are words, constructions, terms of address and/or grammatical paradigms that encode respect towards the addressee, as well as the things and people that the speaker refers to [START_REF] Ide | Formal forms and discernment: Two neglected aspects of universals of linguistic politeness[END_REF]Agha, 1994, inter alia). However, several Asian languages possess an honorific domain that is more pervasive and systematized. For example, there have been numerous studies on the honorific systems of Japanese and Korean [START_REF] Okamoto | Situated politeness: manipulating honorific and nonhonorific expressions in Japanese conversations[END_REF]Strauss & Eun, 2005, inter alia). Several scholars have also investigated the honorific domain of Tibetan [START_REF] Hajime | The honorifics in Tibetan[END_REF]Rdorje et al., 1993;[START_REF] Delancey | Semantic categorization in Tibetan honorific nouns[END_REF][START_REF] Tournadre | Manuel de tibétain standard[END_REF][START_REF] Tournadre | Manual of Standard Tibetan: language and civilization[END_REF][START_REF] Denwood | Tibetan[END_REF][START_REF] Dorje | Tibetan Honorific Speeches[END_REF].

Regarding the general conditions of the use of the honorific domain in Tibetan, it can first be noticed that it is quite limited both in regional and social dialects. The Lhasa aristocracy is reputed to be the only Tibetan speakers that use it 'perfectly'. The vast majority of speakers of standard Tibetan employ an honorific system that is less pervasive and simpler than in the Lhasa upper-class sociolect [START_REF] Tournadre | Manuel de tibétain standard[END_REF][START_REF] Tournadre | Manual of Standard Tibetan: language and civilization[END_REF]. Secondly, it is true that the general register of the sentence will influence the use of the honorific domain, and Tibetan speakers tend to use more honorifics in formal situations. However, in order to really understand how the honorific system of Tibetan functions, it is necessary to take into account several other linguistic parameters.

First of all, the honorific domain affects many linguistic forms, but is systematized differently according to the syntactic category of a given word.

In Tibetan, the honorific domain may concern nouns, verbs, pronouns, suffixes and terms of address. [START_REF] Tournadre | Manuel de tibétain standard[END_REF][START_REF] Tournadre | Manual of Standard Tibetan: language and civilization[END_REF] distinguish four planes for the honorific domain in addition to the ordinary plane: the honorific, the humilific, the high honorific and the double honorific. The plane for a given linguistic item is chosen according to the social status of the participants mentioned in the sentence (explicit or implicit) with respect to the speaker. The honorific domain not only refers to people, but also to their spheres, that is, the objects and other entities that are related to them. What is special about verbs is that they generally connect several participants in a sentence, which makes all of these planes relevant for some verbs, whereas nouns generally distinguish between only the ordinary and honorific forms at most. The high honorific plane concerns very few verbs and is used to convey a highly reverential attitude. In order to illustrate the other planes of the honorific domain (honorific, humilific and double honorific), we will take SPRAD ('give, hand over') as an example.

The functioning of 'give' in the Tibetan honorific system

The special relation that the verb 'give' bears with the honorific domain has already been investigated by [START_REF] Loveday | Explorations in Japanese sociolinguistics[END_REF] in his study of Japanese from a sociolinguistic perspective. In Japanese, 'give' possesses several translations depending on the social relationship between the two participants [START_REF] Loveday | Explorations in Japanese sociolinguistics[END_REF], cited in Newman, 1996).

In Tibetan, in order to understand the honorific system of transitive and ditransitive verbs, the social relationship between the speaker and the addressee is relevant, as well as the social relationship between the participants in the sentence. The basic rules for the use of SPRAD (honorific form GNANG, humilific form PHUL and double honorific form PHUL.GNANG)

are the following:

a. The honorific form GNANG encodes that the agent has a higher status than the speaker b. The humilific form PHUL encodes that the agent has a lower status than the recipient c. The double honorific form PHUL.GNANG encodes that the agent has a higher status than the speaker and that the agent has a lower status than the recipient

In order to illustrate these diverse possibilities, let us consider a few protagonists: some ordinary people (the speaker, Tenzin and Tsering), two monks (high on the social scale) and a Rinpoche (a reincarnated lama, considered very high on the social scale). What can be noticed from this schematic presentation is first that the humilific plane is not the symmetrical opposite of the honorific plane. The honorific form GNANG can encode a downward transfer, and also a horizontal transfer if the agent is higher than the speaker. The humilific form can only be used for an upward transfer with an agent that is not higher than the speaker, and the double honorific form will be appropriate for an upward transfer when the agent is superior to the speaker.

Nevertheless, in addition to the high variation that exists in the use of the honorific system, what makes it even more complex is the strict convention in Tibetan culture for a speaker to pretend that he is not higher on the social scale than anyone else. Therefore, even when a monk or a Rinpoche is the speaker, they will never use honorific words in reference to themselves or express that it is an upward transfer when they are recipients.

The emergence of GNANG and PHUL as light verbs of the honorific domain

Is it now possible to offer some explanations as to why SPRAD has not redeployed as a light verb, whereas the honorific form GNANG and the humilific form PHUL have been quite productive? 11It is undeniable that in Tibetan there is a special relation between the honorific domain and the verb 'give'. We would argue that this phenomenon is not just a random idiosyncrasy of Tibetan, but may rely on the special cognitive status of the notion of transfer, as capturing social hierarchy quite accurately.

What the cross-linguistic research on light verbs has demonstrated is that light verbs generally come from frequent verbs with a basic meaning (Butt & Lahiri, 2003;[START_REF] Mohanan | Grammatical verbs (with special reference to light verbs)[END_REF]. It is the case in Tibetan since the three most frequent light verbs BYED, BTANG, and BRGYAB respectively mean 'do', 'give, send', and 'hit, throw'. What is special about the verb 'give' however may be that it refers to one of the most basic interpersonal actions of the human behavioural repertoire. Therefore, 'give' is particularly relevant for the honorific domain, and it is indeed one of the rare verbs in standard spoken Tibetan that possesses a humilific form. 12 The verb SPRAD is also one of the only two verbs that possess a double honorific form PHUL.GNANG (together with BSHAD 'say', whose double honorific form is ZHU.GNANG). The verb 'say' is similar to 'give' in the sense that they are both basic verbs prototypically connecting two human beings.

In Tibetan, the honorific form GNANG has become a very frequent light verb (as confirmed by its frequent occurrences in the TSC), and its meaning has bleached so that it does not seem to encode much more than an honorific semantic feature in some of these constructions.

Although the verb PHUL is far less frequent and productive than GNANG, its meaning has also bleached when it is used as a light verb. Let us consider an example from the TSC:

(10) In this example, the speaker uses the verb PHUL, but it does not mean 'hand something over'. It expresses a communicative contact between two people, as well as a mark of respect that the agent shows to the recipient. As mentioned before, the ordinary form of ZHAL.PAR PHUL is KHA.PAR BRGYAB (or KHA.PAR BTANG), and not KHA.PAR SPRAD for this meaning, which again shows that the humilific feature of PHUL was more important in its selection as a light verb than its basic meaning of 'giving'.

In some constructions, the ditransitive feature of PHUL is not even retained.

As we saw in 3.4, PHUL is often used as a light verb for religious performances: MCHOD.PA PHUL 'make offerings', PHYAG PHUL 'prostrate', ZHABS.SKOR PHUL 'circumambulate', and DMAR.MCHOD PHUL (lit., sacrifice give) 'make a sacrifice'. In these cases, the meaning of PHUL has bleached further, as the constructions seem to refer to specific acts without any explicit transfer. However, we would argue that these acts imply an underlying beneficiary to which it is essential for Tibetans to show great respect. Tibetan people perform those rituals in a spirit of devotion to the Buddha or a deity.

When we know how essential religious devotion is in Tibetan culture, we can understand why the humilific PHUL has emerged as a light verb to refer to those spiritual performances.

Conclusion

We have collected data on the translation of 'give' into Tibetan in order to investigate both the functioning of light verbs and the system of honorific verbs in this language. We have shown that in standard spoken Tibetan, the most basic way of translating the verb 'give' when it has the literal meaning of 'handing over' is SPRAD. However, we noticed that this verb is not used as a light verb in Tibetan. The lack of productivity of SPRAD might result from the fact that it has only become the basic and frequent equivalent of 'give' in modern standard Tibetan while other 'giving verbs' had already been selected to participate in light verb constructions. The emergence of light verbs in a language can be a relatively slow process involving the semantic evolution of a lexical verb and the crystallization of specific word combinations. For example, the Tibetan verb BTANG went through that process, and it already held the position of a light verb involved in transfer constructions in Classical Tibetan, thus being a long-established competitor to SPRAD for that function.

What also drew our attention is that, although SPRAD has not become a light verb in standard spoken Tibetan, its honorific form GNANG, and to a lesser extent its humilific form PHUL, have undeniably gone through that process.

We argued that this is probably not a mere coincidence, since the honorific domain is particularly relevant when referring to a transfer between two people. The verb 'give' triggers a notional and linguistic representation that prototypically involves two human participants, and therefore captures a scene where social relationship is crucial. In order to encode the honorific domain for an abstract transfer, and later even for other types of actions, Lhasa Tibetan has promoted GNANG and PHUL to the status of light verbs, because they are emblematic of this social domain, which is particularly pervasive in the strictly hierarchical society of the Tibetan capital. As we suggested, it seems that both universal semantic associations and cultural sensitivities can shed light on the motivations behind the selection of those specific verbs as central pivots of the honorific system of Tibetan.

Figure 1 :Figure 2 :

 12 Figure 1: Schematic illustration of the Tibetan social scale

zhig sprad-byung one give-CMPL.PAST.REC

	The honorific and humilific forms to express 'give' are respectively GNANG
	and 'BUL/PHUL. They are very frequent verbs in Old and Classical Tibetan, as
	well as in standard spoken Tibetan.
	There are other lexical verbs that are attested in Old and Classical Tibetan to
	convey the concept of 'giving'. Among them are STER 'give, bestow, offer',
	GTONG/BTANG 'give, give away, send, let go', and GTAD/GTOD 'direct towards,
	hand over, give'. There are also the less frequent MCHOD 'offer, venerate, eat',
	STSAL 'bestow, give', and marginally, GSOL meaning 'ask, wear, dress, beg',
	DEM queen but also 'give' in some expressions such as MTSHAN GSOL 'give/award a name Nangchen-OBL give IMP son come-NMZ-OBL
	'gyur=ro title', and in the noun GSOL.RAS 'donation', and 'GYED/GYED 'offer, distribute'. zhes
	FUT= FCLT	QUOT
	slightly less frequent competitor. In the life of Milarepa (16 th c.), SPROD/SPRAD
	is used around one third of the times as a synonym of SBYIN/BYIN.
	(3)
	kho-'i	ska.rags-kyi mdud.pa bkrol nas	'jim.pa-'i	sho
	3SG-GEN belt-GEN	knot	loosen	CONN	clay-GEN	die
	'After loosening his belt, he handed a clay die to me.' (MLNT)
	5 Standard spoken Tibetan has replaced most of its inflectional verbal system by a system of
	verbal suffixes. Therefore, we use the allomorphs SPROD/SPRAD when referring to Old and
	Classical Tibetan, and only SPRAD for standard spoken Tibetan.
	6 It is possible to distinguish three main periods of written Tibetan or Literary Tibetan: Old
	Tibetan (8 th -12 th c.), Classical Tibetan (13 th -19 th c.) and Modern Literary Tibetan (20 th c.-
	now). However, some contemporary authors still write in a style close to Classical Tibetan
	(Tournadre & Suzuki, forthcoming). For Old Tibetan, our examples come from a text written
	in the 8 th century. It is one of the Tibetan version of Rāmāyaṇa stories found in Dunhuang
	caves (Pt 0981). The examples in Classical Tibetan come from the life of Milarepa, written
	by Tsangnyön Heruka in the 15 th century (mi la'i rnam thar, gtsang smyon he ru ka).

" 'Give it to the queen Nangchen, and you will have a son.', he said…" (Pt 0981, r099)

In Classical Tibetan, SBYIN/BYIN remained the most common verb meaning 'give' in a neutral register. However, SPROD/SPRAD had already emerged as a

  'give' from English into Tibetan, it is first necessary to analyse in what specific construction 'give' appears in English before looking for a Tibetan equivalent. We are now going to illustrate this difficulty with a few Tibetan translations of the light verb 'give' in English. Starting from English does not

	mean that English is considered a reference language, or even a language with
	a typical distribution of the verb 'give'. It simply aims to show that the precise
	mapping of the semantic territory of 'give' in two unrelated languages can be
	highly complex.
	a.	The English light verb 'give' can quite often be translated using
	another light verb in standard spoken Tibetan, especially BTANG, which is the
	most frequent light verb in this dialect:

'Give a hint' GO.BRDA BTANG (lit., notification send/give); 'give a pat' CAG.CAG BTANG (lit., smack send); 'give an order' BKOD.PA BTANG (lit., order send); 'give a punishment' NYES.CHAD BTANG (lit., punishment send); 'give a hand' ROGS.PA BTANG (lit., friend send), etc. 9 b.

Standard spoken Tibetan possesses two other very frequent light verbs that will be favoured when translating some instances of the English light verb 'give': BRGYAB and BYED

  In the TSC, the verb PHUL only appears once in the idiomatic expression What is interesting about the verb PHUL is that when it is used as a humilific light verb, it does not correspond to the verb SPRAD in the ordinary domain, but to other verbs instead. For example, the ordinary form of the humilific MTSHAN PHUL 'give a name' is MING BTAG, literally meaning 'attach a name'.The ordinary form of ZHAL.PAR PHUL 'give a phone call' is KHA.PAR BRGYAB (or KHA.PAR BTANG). In these two examples, the original meaning of the verb PHUL is not much altered, as the act of 'giving a name' or 'giving a phone call' can both be seen as a 'transfer of data' between two people.However, the meaning of the verb PHUL seems to have bleached further in constructions such as PHYAG PHUL, meaning 'prostrate' (lit., hand [hon.] give

	3.3. The Tibetan verb PHUL
	In Old Tibetan, 'BUL/PHUL is a lexical verb that is the humilific form of SPRAD
	'give, hand over' when the agent of the transfer is presented as inferior to the
	recipient.
	(8)
	de.nas phyag	byas 'phrin.yig dang	rtags so.rdub
	then	arm (H.)	LV	letter		CONN token	ring
	phul	te
	give (h)	CONN
	'After paying respect, he gave her the letter and the ring as a token…' (Pt
	0981, r219)
	The same function can be observed in Classical Tibetan:
	(9)
	nga-s	gser g.yu	kun phul	te
	1SG-ERG gold	turquoise	all	give (h) CONN
	ZHAL.PAR PHUL ('give a phone call', hum.), in which it is undeniably a light [hon.].give [hon.]) 'say' is an alternative to the simple form GSUNG, the honorific of BSHAD, etc. verb.

SPRAD ('give, hand over'), but it is the honorific of BYED ('do') 7 times in the corpus. It appears twice as part of the honorific imperative suffix -GNANG.ROGS. In the remaining 13 occurrences, it is an honorific light verb or is part of an honorific compound verb. It is compatible with different kinds of actions. Here are a few examples from the TSC: 'TSHAL.GNANG (lit., seek.give [hon.]) 'look for' is the honorific of 'TSHAL; DRAN.GNANG (lit., remember.give [hon.]) 'remember' is the honorific of DRAN; MOS.MTHUN GNANG 'agree' is the honorific of MOS.MTHUN BYED; BZHUGS.GNANG (lit., stay [hon.].give [hon.]) 'stay' is an alternative to the simple form BZHUGS, the honorific of BSDAD; GSUNG.GNANG (lit., say 'I gave you all the gold and Turquoise…' (MLNT) Just like the honorific verb GNANG, the humilific verb 'BUL/PHUL was already a well-established light verb in Classical Tibetan. The following constructions appear in the life of Milarepa:

PHYAG PHUL

(lit., arm [hon.]

give [hum.]) 'prostrate'; GZIGS PHUL (lit., eye [hon.] give [hum.]) 'show'; RGYU.MTSHAN PHUL (lit. reason give [hum.]) 'give some reasons'; GLU PHUL (lit. song give [hum.]) 'offer a song', etc. [hum.]) or ZHABS.SKOR PHUL, meaning 'circumambulate', i.e. 'walk around a religious emblem' (lit., foot [hon.].turn give [hum.]). The ordinary form of PHYAG PHUL is PHYAG 'TSHAL (lit., hand [hon.] beg), and the ordinary form of ZHABS.SKOR PHUL is SKOR.BA BRGYAB

Online version (consulted 10/05/2017).

The term 'Tibetic' has been used differently by other authors to refer to 'Bodish', 'Bodic', 'Himalayish' languages, or other intermediate groupings within the Tibeto-Burman family. For details, see[START_REF] Tournadre | The Tibetic languages and their classification[END_REF].

Conversely, some ethnic groups in Tibet are officially classified as Tibetans (zangzu in Chinese) by the People's Republic of China but their native languages are not Tibetic.

One can note however that the complement of a light verb is not always nominal, as it might look like a noun, but have no exact lexical equivalent in this syntactic category, such as in 'have a go'. In addition, many idiomatic constructions with a generic verb and an adjectival, prepositional or even onomatopoeic complement can be classified as 'light verb constructions', such as 'take into account', 'do wrong' or 'go boom'.

These word-to-word translations are just given as indications. Most words are polysemous, and many polymorphemic units can also be found outside of the light verb construction with an autonomous meaning.

The use of light verbs is in essence idiosyncratic, as the construction has to be memorised in its entirety. It is for example difficult to explain systematically why in English 'give a talk'

It has to be noticed however that there are a few figurative expressions involving SPRAD, and that GNANG has been far more productive than PHUL.

The other humilific verbs of Tibetan are ZHU 'eat, drink, say', the humilific form of ZA 'eat', BTHUNG 'drink' and BSHAD 'say', BCAR 'come close to sb., call on sb., meet sb.', the humilific form of 'GRO 'go' or THUG 'meet', and MJAL 'visit, see', the humilific form of THUG 'meet' and MTHONG 'see'.