Denise Aregba-Driollet

CONVERGENCE OF THE LATTICE BOLTZMANN METHOD WITH OVERRELAXATION FOR A NONLINEAR CONSERVATION LAW

Keywords:

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Consider a nonlinear conservation law [START_REF] Aregba | Kinetic approximation of a boundary value problem for conservation laws[END_REF] ∂ t u + divA(u) = 0 where u(x, t) ∈ U a real interval, (x, t) ∈ R D × R and A ∈ C 1 (U; R). In what follows we assume that A is not constant and, without loss of generality, that U = R and A(0) = 0. Numerical approximations of this equation can be designed by using a discrete kinetic system of BGK type, as introduced in [START_REF] Natalini | A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws[END_REF], [START_REF] Aregba | Discrete kinetic schemes for systems of conservation laws[END_REF]:

(2) ∀l ∈ {1, . . . , L}, ∂

t f ε l + D d=1 v ld ∂ x d f ε l = - 1 ε (f ε l -M l (u ε )) , u ε = L l=1 f ε l .
Here, the v ld are fixed real coefficients, the "maxwellian functions" M l are defined on R and ε is a positive relaxation parameter. Moreover the following compatibility conditions are fulfilled:

(3) ∀u ∈ R,

L l=1 M l (u) = u, L l=1 v ld M l (u) = A d (u), d = 1, . . . , D.
If f ε is a solution of (2) then, denoting w ε j = L l=1 v lj f ε l , we have

             ∂ t u ε + D d=1 ∂ x d w ε d = 0, ∂ t w ε j + D d=1 L l=1 (v ld v lj ∂ x d f ε l ) = - 1 ε w ε j -A j (u ε ) , j = 1, . . . , D.
Hence formally if f ε → f when ε tends to 0, then in the limit w = A(u) where u is a solution of equation [START_REF] Aregba | Kinetic approximation of a boundary value problem for conservation laws[END_REF]. Actually, let u 0 ∈ L 1 (R D ) ∩ L ∞ (R D ) ∩ BV(R D ) be an initial data for (1):

(4) u(x, 0) = u 0 (x), x ∈ R.

The convergence towards the unique weak entropy solution of the Cauchy problem (1)(4) has been rigorously proved in [START_REF] Natalini | A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws[END_REF]. Convergence also holds in the presence of boundary conditions, see [START_REF] Milišić | Stability and convergence of discrete kinetic approximations to an initial-boundary value problem for conservation laws[END_REF]. An essential assumption for convergence is that the maxwellian functions M l are monotone nondecreasing:

(5) ∀u ∈ [-µ ∞ , µ ∞ ], M l (u) ≥ 0, l = 1, . . . , L where (6) µ ∞ = u 0 ∞ .

In [START_REF] Aregba | Discrete kinetic schemes for systems of conservation laws[END_REF], [START_REF] Aregba | Discrete kinetic schemes for multidimensional systems of conservation laws[END_REF], [START_REF] Aregba | Kinetic approximation of a boundary value problem for conservation laws[END_REF] we use system [START_REF] Aregba | Convergence of relaxation schemes for conservation laws[END_REF] to construct finite volume schemes for [START_REF] Aregba | Kinetic approximation of a boundary value problem for conservation laws[END_REF]. The procedure is as follows:

(1) For fixed ε > 0, construct a numerical approximation of the BGK system (2) by splitting it into a set of linear transport equations which are solved by a monotone upwind scheme, and a system of ordinary differential equations which can be solved analytically. (2) Make ε → 0 in this approximation to obtain a finite volume "relaxed" scheme for [START_REF] Aregba | Kinetic approximation of a boundary value problem for conservation laws[END_REF]. Under condition [START_REF] Baty | A robust and efficient solver based on kinetic schemes for magnetohydrodynamics (MHD) equations[END_REF], we established L ∞ , L 1 and BV estimates which are uniform with respect to ε. We proved convergence for f ε for all ε > 0 and also convergence of the relaxed scheme to the unique weak entropy solution of (1)(4). Systems of form [START_REF] Aregba | Convergence of relaxation schemes for conservation laws[END_REF] can also be used to construct lattice Boltzmann schemes for [START_REF] Aregba | Kinetic approximation of a boundary value problem for conservation laws[END_REF], see [START_REF] Graille | Approximation of mono-dimensional hyperbolic systems: a lattice Boltzmann scheme as a relaxation method[END_REF], [START_REF] Coulette | High-order implicit palindromic discontinuous Galerkin method for kinetic-relaxation approximation[END_REF], [START_REF] Coulette | Vectorial kinetic relaxation model with central velocity. Application to implicit relaxations schemes[END_REF], [START_REF] Hélie | Schéma de relaxation pour la simulation de plasmas dans les tokamaks[END_REF], [START_REF] Baty | A robust and efficient solver based on kinetic schemes for magnetohydrodynamics (MHD) equations[END_REF], [START_REF] Bellotti | Numerical analysis of lattice Boltzmann schemes: from jundamental issues to efficient and accurate adaptive methods[END_REF], [START_REF] Bellotti | Monotonicity for genuinely multi-step methods: Results and issues from a simple lattice boltzmann scheme[END_REF]. The velocities v (l) = (v l1 , . . . , v lD ) must be such that if x α is a node of the lattice then x α -∆tv (l) is also a node of the lattice, which we denote x α l . Splitting again system (2) into the linear transport part and an ordinary differential system, we study here a cartesian lattice Boltzmann scheme with space step ∆x = (∆x d ) 1≤d≤D :

x α = (x d,α d ) 1≤d≤D = (α d ∆x d ) 1≤d≤D , C α = D d=1 x d,α d - ∆x d 2 , x d,α d + ∆x d 2 , α ∈ Z D .
We We denote V = 1≤d≤D ∆x d the volume of C α .

Initialization:

(8) u 0 α = 1 V Cα u 0 (x)dx, α ∈ Z D , (9)
f 0 l,α = M l (u 0 α ), α ∈ Z D , l = 1, . . . , L. Then for n ≥ 0:
Step 1: stream phase. Each transport equation is solved exactly: [START_REF] Caetano | A result of convergence for a mono-dimensional two-velocities lattice Boltzmann scheme[END_REF] f

n+ 1 2 l,α = f n l,α l with α l = α -j l , l = 1, . . . , L, and (11)
u n+ 1 2 α = L l=1 f n+ 1 2 l,α .
Step 2: collision phase. For each node x α we solve numerically on [t n , t n+1 ] the system

(f ε l ) = - 1 ε (f ε l -M l (u ε )) , l = 1, . . . , L, u ε = L l=1 f ε l with f ε l (t n ) = f n+ 1 2
l,α . We drop the superscript ε for the sake of simplicity. By the compatibility conditions (3), we have

u(t) = u(t n ) so that we set (12) u n+1 α = u n+ 1 2 α .
Note that the exact solution of the system is

(13) f l,α (t n+1 ) = 1 -exp(- ∆t ) M l (u n+ 1 2 α ) + exp(- ∆t )f n+ 1 2 l,α , l = 1, . . . , L
while the explicit Euler scheme gives

f n+1 l,α = 1 - ∆t f n+ 1 2 l,α + ∆t M l (u n+ 1 2 α
), l = 1, . . . , L.

Here we choose a positive parameter ω and we set ( 14)

f n+1 l,α = (1 -ω)f n+ 1 2 l,α + ωM l (u n+ 1 2 α
), l = 1, . . . , L.

Note that we have then [START_REF] Coulette | High-order implicit palindromic discontinuous Galerkin method for kinetic-relaxation approximation[END_REF].

If ω ∈]0, 1] the Lattice Boltzmann Method (abbreviated LBM) ( 10)-( 11)-( 14) can be interpreted as ( 10)-( 11)-( 13) by setting ω = 1 -exp(-∆t ), the asymtotic limit ε = 0 being the case ω = 1. Therefore all the estimates of [START_REF] Aregba | Discrete kinetic schemes for multidimensional systems of conservation laws[END_REF] apply. Giving a fixed value to the ratio ∆t/ε means that ∆t (and therefore ∆x) and ε tend simultaneously to 0. This case has been treated in [START_REF] Caetano | A result of convergence for a mono-dimensional two-velocities lattice Boltzmann scheme[END_REF] for the D1Q2 approximation.

The overrelaxation case ω > 1 cannot be treated by the same method. Nevertheless this case is meaningful by considering the interpretation as an Euler explicit scheme with ω = ∆t . Several numerical experiments show that the LBM is more accurate for ω > 1, see [START_REF] Caetano | A result of convergence for a mono-dimensional two-velocities lattice Boltzmann scheme[END_REF], [START_REF] Hélie | Schéma de relaxation pour la simulation de plasmas dans les tokamaks[END_REF] for example.

In the general nonlinear scalar case, as far as we know, the only result of convergence with ω > 1 is due to T. Bellotti ([7], [START_REF] Bellotti | Monotonicity for genuinely multi-step methods: Results and issues from a simple lattice boltzmann scheme[END_REF]) in the case of the D1Q2 model: the LBM is interpreted as a multistep finite difference method, and by monotony convergence is proved for ω ∈]1, 2[.

In the present paper, we highlight the fact that mononicity properties can be retrieved in the case ω > 1, for a large class of multidimensional models, thus allowing to prove convergence of the LBM. We do not follow the multistep interpretation of [START_REF] Bellotti | Numerical analysis of lattice Boltzmann schemes: from jundamental issues to efficient and accurate adaptive methods[END_REF] but rather keep the scheme under the form ( 10)-( 11)- [START_REF] Dubois | Equivalent partial differential equations of a lattice Boltzmann scheme[END_REF]. We show that the choice of the velocities v ld dictates the range of values of ω for which monotonicity holds.

The plan of the paper is the following. In section 2 we present the LBM in detail and study monotonicity conditions. In section 3, some 1D and 2D examples are given. Section 4 is devoted to the proof of convergence. Some numerical tests illustrate our result in section 5.

Detailed presentation of the LBM and monotonicity conditions

We remark that except in the case ω = 1 the scheme ( 10)-( 11)-( 14) cannot be explicited as a one-step finite volume approximation of [START_REF] Aregba | Kinetic approximation of a boundary value problem for conservation laws[END_REF]. We have to deal with the f l .

For a given initial data u 0 ∈ L ∞ (R D ) we define the f 0 l,α by ( 8)- [START_REF] Bouchut | Entropy satisfying flux vector splittings and kinetic BGK models[END_REF]. For n ≥ 0 the scheme can be written as:

f n+1 l,α = (1 -ω)f n l,α l + ωM l L k=1 f n k,α k , l = 1, . . . , L and
u n+1 α = L k=1 f n k,α k .
We consider here the most usual case where

(15) M l (u) = a l u + D d=1 b ld A d (u), l = 1, . . . , L,
where the a l and b ld are real coefficients. In order to satisfy the compatibility conditions (3), we have to impose ( 16)

L l=1 a l = 1, L l=1 b ld = 0, L l=1 v ld a l = 0, L l=1 v ld b lj = δ dj .
Note that M(0) = 0.

We can write the scheme as follows:

(17) f n+1 l,α = S l (f n 1,α 1 , . . . , f n L,α L ) (l = 1, . . . , L), and u n+1 α = L k=1 f n+1 k,α with S l (f ) = (1 -ω)f l + ωM l L k=1 f k , l = 1, . . . , L, or (18) S l (f ) = (1 -ω(1 -a l ))f l + ωa l k =l f k + ω D d=1 b ld A d L k=1 f k , l = 1, . . . , L.
It is useful to remark that by the compatibility conditions (3)

(19) ∀u ∈ R, S l (M(u)) = M l (u), l = 1, . . . , L.
Using notation [START_REF] Bellotti | Monotonicity for genuinely multi-step methods: Results and issues from a simple lattice boltzmann scheme[END_REF], we define (20)

m l = M l (-µ ∞ ), M l = M l (µ ∞ ), V = L l=1 [m l , M l ].
We denote f = (f 1 , . . . , f L ). We define the following monotonicity condition:

(21) ∀k ∈ {1, . . . , L}, ∀l ∈ {1, . . . , L}, ∀f ∈ V, ∂ k S l (f ) ≥ 0. Proposition 2.1. Suppose that (5) is satisfied. For f ∈ V , we denote u = L l=1 f l . Then u ∈ [-µ ∞ , µ ∞ ], ( 22
) 0 ≤ M l (u) = a l + D d=1 b ld A d (u) ≤ 1, l = 1, . . . , L,
and condition (21) is satisfied if and only if the following condition is satisfied:

(23) ∀u ∈ [-µ ∞ , µ ∞ ], ∀l ∈ {1, . . . , L}, ω ≤ 1 1 -a l - D d=1 b ld A d (u)
.

Moreover if (5) and (21) are satisfied, then ω ∈]0, 2[.

Proof. Suppose that (5) is satisfied. If f ∈ V we set u = L k=1 f k . The compatibility conditions (3) insure that u ∈ [-µ ∞ , µ ∞ ], so for k = l (24) ∂ k S l (f ) = ωM l (u) ≥ 0.
Consequently condition [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF] is satisfied if and only if ∂ l S l ≥ 0 on V for all l.

By (3), L l=1
M l (u) = 1 which with ( 5) proves ( 22). Now we have (25)

∂ l S l (f ) = 1 -ω (1 -M l (u)) = 1 -ω 1 -a l - D d=1 b ld A d (u)
which gives condition [START_REF] Milišić | Stability and convergence of discrete kinetic approximations to an initial-boundary value problem for conservation laws[END_REF].

We prove now that ω ∈]0, 2[. As

L l=1 M l (u) = 1, if L > 2 then
there exists l and u such that

M l (u) < 1 2 , hence ω < 2. If L = 2
we are in one space dimension. If M l (u) ≥ 1 2 for all l and u, then M l (u) = 1 2 for all l and u. This happens only in the trivial case of a constant flux A. Therefore ω < 2.

In view of these properties, we can proceed as follows:

• u 0 being given, we choose ∆x.

• We fix the velocity scale λ (and hence ∆t) and ω such that conditions ( 5) and ( 23) are satisfied. We now illustrate the method by some 1D and 2D examples.

Examples in 1D and 2D

3.1. The D1Q2 model. In 1D, the minimal number of velocities is 2, and in that case system (2) is the wellknown Jin and Xin's model [START_REF] Shi | The relaxation schemes for systems of conservation laws in arbitrary space dimensions[END_REF]:

v 2 = -v 1 = λ > 0,
and

M 1 (u) = 1 2 u - A(u) λ , M 2 (u) = 1 2 u + A(u) λ . Condition (5) is satisfied if and only if ∀u ∈ [-µ ∞ , µ ∞ ], |A (u)| ≤ λ. Condition (23) is satisfied if and only if (26) ω ≤ 2 1 + max u∈[-µ∞,µ∞] |A (u)| λ .
Note that this condition appears in T. Bellotti's proof of convergence ( [START_REF] Bellotti | Numerical analysis of lattice Boltzmann schemes: from jundamental issues to efficient and accurate adaptive methods[END_REF]).

As a consequence, ω can take all values in ]0, 2[, provided that λ is large enough. The numerical scheme can be written as

             f n+1 1,α = (1 -ω)f n 1,α+1 + ω 2 u n+1 α - A(u n+1 α ) λ f n+1 2,α = (1 -ω)f n 1,α-1 + ω 2 u n+1 α + A(u n+1 α ) λ u n+1 α = f n 1,α+1 + f n 2,α-1 . Remark 3.1. We can express the variables u = L l=1 f l and w = L l=1 v l f l ,
showing that u is constant in the collision phase while w is not at its asymptotic equilibrium A(u) except for ω = 1.

Using the notation (7) we have

             u n+ 1 2 α = L l=1 f n l,α-j l w n+ 1 2 α = L l=1 v l f n l,α-j l and    u n+1 α = u n+ 1 2 α w n+1 α = w n+ 1 2 α -ω(w n+ 1 2 α -A(u n+ 1 2 α
)).

A D1Q4 model.

We consider the following model:

(27) -v 1 = v 4 = 2λ, -v 2 = v 3 = λ
where λ > 0, and

                       M 1 (u) = u 4 - A(u) 6λ , M 2 (u) = u 4 - A(u) 6λ , M 3 (u) = u 4 + A(u) 6λ , M 4 (u) = u 4 + A(u) 6λ . Condition (5) is satisfied if and only if ∀u ∈ [-µ ∞ , µ ∞ ], |A (u)| ≤ 3 2 λ. (28)
Condition ( 23) is here

∀u ∈ [-µ ∞ , µ ∞ ], ω ≤ 1 3 4 + |A (u)| 6λ .
The maximal value for ω is then 4 3 . The numerical scheme can be written as

                                 f n+1 1,α = (1 -ω)f n 1,α+2 + ω u n+1 α 4 - A(u n+1 α ) 6λ f n+1 2,α = (1 -ω)f n 2,α+1 + ω u n+1 α 4 - A(u n+1 α ) 6λ f n+1 3,α = (1 -ω)f n 3,α-1 + ω u n+1 α 4 + A(u n+1 α ) 6λ f n+1 4,α = (1 -ω)f n 4,α-2 + ω u n+1 α 4 + A(u n+1 α ) 6λ u n+1 α = f n 1,α+2 + f n 2,α+1 + f n 3,α-1 + f n 4,α-2 .

A D2Q4 model. This model is the direct extension of the D1Q2 model in 2D:

(29)

v (1) = λ 1 (-1, 0), v (2) = λ 2 (0, -1), v (3) = λ 1 (1, 0), v (4) = λ 2 (0, 1),
where λ 1 > 0, λ 2 > 0, and

                         M 1 (u) = u 4 - A 1 (u) 2λ 1 , M 2 (u) = u 4 - A 2 (u) 2λ 2 , M 3 (u) = u 4 + A 1 (u) 2λ 1 , M 4 (u) = u 4 + A 2 (u) 2λ 2 Condition (5) is satisfied if and only if ∀u ∈ [-µ ∞ , µ ∞ ], 2|A d (u)| ≤ λ d , d = 1, 2. (30)
Condition [START_REF] Milišić | Stability and convergence of discrete kinetic approximations to an initial-boundary value problem for conservation laws[END_REF] reads as

ω ≤ 4 3 + 2 |A d (u)| λ d , d = 1, 2.
The maximal value of ω is ω = 4 3 . 3.4. A D2Q8 model. One can consider more velocities, : for (8) .

λ 1 > 0, λ 2 > 0 we define (31) v (1) = (-λ 1 , 0) = -v (5) , v (2) = (-λ 1 , -λ 2 ) = -v (6) v (3) = (0, -λ 2 ) = -v (7) , v (4) = (λ 1 , -λ 2 ) = -v

and

(32)

                         M 1 (u) = u 8 - A 1 (u) 6λ 1 M 2 (u) = u 8 - A 1 (u) 6λ 1 - A 2 (u) 6λ 2 , M 3 (u) = u 8 - A 2 (u) 6λ 2 M 4 (u) = u 8 + A 1 (u) 6λ 1 - A 2 (u) 6λ 2 , M 5 (u) = u 8 + A 1 (u) 6λ 1 M 6 (u) = u 8 + A 1 (u) 6λ 1 + A 2 (u) 6λ 2 , M 7 (u) = u 8 + A 2 (u) 6λ 2 M 8 (u) = u 8 - A 1 (u) 6λ 1 + A 2 (u) 6λ 2 . Proposition 3.2. We denote c d = max u∈[-µ∞,µ∞] 8|A d (u)| 3λ d . A sufficient condition for (5) to be satisfied is that c d ≤ 1 for d = 1, 2. In this case condition (23) is verified if ω ≤ 8 7 + 1 2 (c 1 + c 2 )
.

Proof. For all l we have

a l + 2 d=1 b ld A d (u) = 1 8 1 + 1 2 2 d=1 d 8A d (u) 3λ d , d ∈ {-1, 0, 1}. Hence 0 ≤ a l + 2 d=1 b ld A d (u) ≤ 1 8 1 + 1 2 (c 1 + c 2 ) ≤ 1 4
so condition ( 5) is satisfied. Moreover

a l + 2 d=1 b ld A d (u) ≥ 1 8 1 - 1 2 (c 1 + c 2 ) so that condition (23) is satisfied if ω ≤ 8 7 + 1 2 (c 1 + c 2 )
.

The above examples show that monotonicity can hold for ω > 1, and that the maximal value of this parameter depends on λ, which value determines the velocity scale. The spacestep being fixed, when |λ| increases, ∆t decreases.

Convergence of the LBM

In the remaining of the paper, we consider the cartesian LBM with equilibrium functions under the form [START_REF] Godlewski | Hyperbolic systems of conservation laws, volume 3/4 of Mathématiques & Applications[END_REF]. We take an initial function u 0 ∈ L 1 (R) ∩ L ∞ (R) ∩ BV(R) and initialize the scheme by ( 8) and [START_REF] Bouchut | Entropy satisfying flux vector splittings and kinetic BGK models[END_REF]. Then the scheme is given by ( 17) with [START_REF] Guillon | Stability analysis of the vectorial lattice-Boltzmann method[END_REF].

Denoting χ n α the characteristic function of C α × [t n , t n+1 [ and χ α the characteristic function of C α we define f ∆ (x, t) = ∞ n=0 α∈Z D f n α χ n α (x, t), u ∆ (x, t) = L l=1 f ∆,l (x, t) f n ∆ (x) = α∈Z D f n α χ α (x), u n ∆ (x) = L l=1 f n ∆,l (x)
f n = (f n α ) α∈Z D .
Our goal is to prove the convergence of u ∆ to the unique weak entropy solution of (1)(4) when ∆x (and hence ∆t) tends to zero.

The initial value u 0 being given, we fix the v ld and ω such that conditions ( 5) and ( 23) are satisfied. Therefore the ratios ∆t/∆x d are constant and when we make ∆t vary, then ∆x varies in the same proportion.

We establish L ∞ , L 1 , BV and entropy estimates in the spirit of Crandall and Majda ( [START_REF] Michael | Monotone difference approximations for scalar conservation laws[END_REF]). In this reference, the authors consider monotone schemes in conservative form for [START_REF] Aregba | Kinetic approximation of a boundary value problem for conservation laws[END_REF]. They prove convergence to the unique weak entropy solution of the Cauchy problem. Our framework is different, as we cannot express u n+1 α as a function of some u n β where the C β are some neighbour cells of C α . Nevertheless, we prove some properties on the f n α and u n α which allow us to use the same mathematical tools.

We recall that

u n ∆ ∞ = sup α∈Z D |u n α |, u n ∆ 1 = V α∈Z D |u n α |, TV(u n ∆ ) = α∈Z D D d=1 V ∆x d |u n α+e d -u n α |
where we denoted (e 1 , . . . , e D ) the canonical basis of R D . Also:

f n ∆ ∞ = sup α∈Z D L l=1 |f n l,α |, f n ∆ 1 = V α∈Z D L l=1 |f n l,α |, TV(f n ∆ ) = α∈Z D D d=1 L l=1 V ∆x d |f n l,α+e d -f n l,α | . Lemma 4.1.
The discretization of the initial data satisfies the following properties:

(33) u 0 ∆ ∞ ≤ µ ∞ , u 0 ∆ 1 ≤ u 0 1 , TV(u 0 ∆ ) ≤ TV(u 0 ) and (34) f 0 ∆ ∞ = u 0 ∆ ∞ , f 0 ∆ 1 = u 0 ∆ 1 , TV(f 0 ∆ ) = TV(u 0 ∆ ). Proof.
The two first inequalities in (33) are clear, the last one is proved in [START_REF] Michael | Monotone difference approximations for scalar conservation laws[END_REF]. To prove (34) we use (3), the monotony property [START_REF] Baty | A robust and efficient solver based on kinetic schemes for magnetohydrodynamics (MHD) equations[END_REF] and the fact that M(0) = 0.

f 0 ∆ ∞ = sup α∈Z D L l=1 |M l (u 0 α ) -M l (0)| = sup α∈Z D | L l=1 (M l (u 0 α ) -M l (0))| so f 0 ∆ ∞ = u 0 ∆ ∞ .
The proof is the same for the L 1 norm.

TV(f 0 ∆ ) = α∈Z D D d=1 L l=1 V ∆x d |M l (u 0 α+e d ) -M l (u 0 α )|
and we conclude with the same arguments.

4.1. Convergence to a weak solution.

Here we obtain the same kind of estimates as in [START_REF] Aregba | Discrete kinetic schemes for multidimensional systems of conservation laws[END_REF]: supremum norm bound, L 1 contraction, TVD property and convergence to equilibrium, in order to obtain convergence of u n ∆ to a weak solution of the problem (1)(4). Proposition 4.2. Suppose that conditions (5) and (23) are satisfied. Then for all n ≥ 0,

(35) ∀α ∈ Z D , u n α ∈ [-µ ∞ , µ ∞ ] and f n α ∈ V where V is defined in (20). Proof. By lemma 4.1 u 0 α ∈ [-µ ∞ , µ ∞ ],
which is the domain where the M l are monotone. Therefore f 0 α ∈ V . By recurrence, suppose that (35) is true for a given n ≥ 0. We remark that by [START_REF] Hélie | Schéma de relaxation pour la simulation de plasmas dans les tokamaks[END_REF] S l (m 1 , . . . ,

m L ) = m l , S l (M 1 , . . . , M L ) = M l . Denoting f n α = (f n 1,α 1 , . . . , f n L,α L
), m = (m 1 , . . . , m L ):

f n+1 l,α -m l = S l (f n α ) -S l (m) = 1 0 L k=1 ∂ k S l (m + θ(f n α -m))(f n k,α k -m k )dθ ≥ 0
by [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF]. In the same way, M l -f n+1 l,α ≥ 0 so that

f n+1 α ∈ V and hence u n+1 α ∈ [-µ ∞ , µ ∞ ].
Corollary 4.3. Suppose that conditions (5) and (23) are satisfied.

(36) u ∆ ∞ ≤ µ ∞ and f ∆ ∞ ≤ L l=1 max(|m l |, |M l |).
Lemma 4.4. Suppose that conditions (5) and (23) are satisfied. For all f ∈ V , g ∈ V :

(37)

L l=1 |S l (g) -S l (f )| ≤ L l=1 |g l -f l |. Proof. We denote u = L k=1 f k , v = L k=1 g k . S l (g) -S l (f ) = (1 -ω) (g l -f l ) + ω (M l (v) -M l (u)) = (1 -ω) (g l -f l ) + ω 1 0 M l (u + θ(v -u)) L k=1 (g k -f k )dθ. Denote w θ = u + θ(v -u). w θ ∈ [-µ ∞ , µ ∞ ] and S l (g) -S l (f ) =(g l -f l ) 1 0 (1 -ω(1 -M l (w θ )))dθ + ω k =l (g k -f k ) 1 0 M l (w θ )dθ.
By proposition 2.1 the terms in the integrals are non negative (see (25)) so

L l=1 |S l (g) -S l (f )| ≤ L l=1 |g l -f l | 1 0 (1 -ω(1 -M l (w θ )))dθ + ω L l=1   k =l |g k -f k | 1 0 M l (w θ )dθ  
which can be written as

L l=1 |S l (g) -S l (f )| ≤ L l=1 |g l -f l |(1 -ω) + ω L l=1 L k=1 |g k -f k | 1 0 M l (w θ )dθ . As L l=1
M l (w θ ) = 1 we obtain the desired inequality.

Proposition 4.5. Suppose that conditions (5) and (23) are satisfied. Let f n α and g n α be two numerical solutions, with initial data u 0 and v 0 respectively,

u 0 , v 0 ∈ L 1 (R) ∩ L ∞ (R) ∩ BV(R) satisfying u 0 ∞ ≤ µ ∞ , v 0 ∞ ≤ µ ∞ . Then for all n ≥ 0 (38) g n+1 ∆ -f n+1 ∆ 1 ≤ g n ∆ -f n ∆ 1 ≤ v 0 -u 0 1 and there exists C > 0 such that (39) f n+1 ∆ -f n ∆ 1 ≤ C ∆t TV(u 0 ). Proof. g n+1 ∆ -f n+1 ∆ 1 = V α∈Z D L l=1 |g n+1 l,α -f n+1 l,α |.
We apply inequality (37):

α∈Z D L l=1 |g n+1 l,α -f n+1 l,α | = α∈Z D L l=1 |S l (g n 1,α-j1 , . . . , g n L,α-j L ) -S l (f n 1,α-j1 , . . . , f n L,α-j L )| ≤ α∈Z D L l=1 |g n l,α-j l -f n l,α-j l | = α∈Z D L l=1 |g n l,α -f n l,α |. Moreover α∈Z D L l=1 |g 0 l,α -f 0 l,α | = α∈Z D L l=1 |M l (v 0 α ) -M l (u 0 α )| = α∈Z D | L l=1 (M l (v 0 α ) -M l (u 0 α ))|
by condition [START_REF] Baty | A robust and efficient solver based on kinetic schemes for magnetohydrodynamics (MHD) equations[END_REF]. Then by ( 3) and ( 8)

α∈Z D L l=1 |g 0 l,α -f 0 l,α | = α∈Z D |v 0 α -u 0 α | ≤ 1 V v 0 -u 0 1
which proves (38). Let us prove the second inequality.

α∈Z D L l=1 |f n+1 l,α -f n l,α | = α∈Z D L l=1 |S l (f n 1,α-j1 , . . . , f n L,α-j L ) -S l (f n-1 1,α-j1 , . . . , f n-1 L,α-j L )| ≤ α∈Z D L l=1 |f n l,α-j l -f n-1 l,α-j l | ≤ α∈Z D L l=1 |f n l,α -f n-1 l,α | ≤ α∈Z D L l=1 |f 1 l,α -M l (u 0 α )|.
Moreover

α∈Z D L l=1 |f 1 l,α -M l (u 0 α )| = α∈Z D L l=1 |S l (f 0 1,α-j1 , . . . , f 0 L,α-j L ) -S l (M 1 (u 0 α ), . . . , M L (u 0 α ))| ≤ α∈Z D L l=1 |M l (u 0 α-j l ) -M l (u 0 α )| ≤ α∈Z D L l=1 |u 0 α-j l -u 0 α |
by [START_REF] Lax | Systems of conservation laws[END_REF]. Consequently

f n+1 ∆ -f n ∆ 1 ≤ C max d ∆x d TV(u 0 )
which gives (39).

Corollary 4.6. Suppose that conditions (5) and (23) are satisfied.

(40) ∀t > t > 0, f ∆ (•, t ) -f ∆ (•, t) 1 ≤ C(t -t + ∆t)TV(u 0 ).
The inequality (37) gives a BV estimate. Proceeding as in the proof of proposition 4.5 we obtain Proposition 4.7. Suppose that conditions (5) and (23) are satisfied. Then for all n ≥ 0 (41)

TV(f n+1 ∆ ) ≤ TV(f n ∆ ) ≤ TV(u 0 )
and

(42) TV(u n ∆ ) ≤ TV(f n ∆ ) ≤ TV(u 0 ).
We now estimate the distance between f n and the equilibrium.

Proposition 4.8. Suppose that conditions (5) and (23) are satisfied. There exists a constant C > 0 such that

(43) ∀n ≥ 0, M(u n ∆ ) -f n ∆ 1 ≤ C TV(u 0 ) |1 -ω| 1 -|1 -ω| ∆t.
Remark that if ω = 1 then the collision part of the scheme is just the projection on equilibrium: M l (u n α ) = f n l,α . In this case the inequality (43) is trivially satisfied.

Proof. We denote

E n = M(u n ∆ ) -f n ∆ 1 . E n+1 = V α∈Z D L l=1 |M l (u n+1 α ) -f n+1 l,α | = V|1 -ω| α∈Z D L l=1 |M l (u n+1 α ) -f n l,α-j l | ≤ V|1 -ω| α∈Z D L l=1 |M l (u n α-j l ) -f n l,α-j l | + |M l (u n+1 α ) -M l (u n α-j l )| ≤ |1 -ω| E n + V α∈Z D L l=1 |u n+1 α -u n α-j l | ≤ |1 -ω| (E n + C TV(u 0 )∆t) .
By recurrence, as E 0 = 0, we obtain the result.

The propositions 4.2, 4.5, 4.7, 4.8 allow us to prove the following theorem.

Theorem 4.9. Suppose that conditions (5) and ( 23) are satisfied. Let f n α be the numerical solution given by ( 17) with [START_REF] Guillon | Stability analysis of the vectorial lattice-Boltzmann method[END_REF], with initial data

u 0 ∈ L 1 (R)∩L ∞ (R)∩BV(R) satisfying u 0 ∞ ≤ µ ∞ .
Let (∆t i ) a sequence of time steps tending to 0. There exists a subsequence (∆t i k ) and a function f

: [0, +∞[→ L 1 (R D ) such that (44) lim k→∞ max 0≤t≤T f ∆ k (•, t) -f (•, t) 1 = 0 for T > 0 and, denoting u = l f l , lim k→∞ max 0≤t≤T u ∆ k (•, t) -u(•, t) 1 = 0 for T > 0.
Moreover, f = M(u) and u is a weak solution of the problem (1)(4).

Proof. The first step is to prove that f ∆ converges to a function f . We proceed as in [START_REF] Michael | Monotone difference approximations for scalar conservation laws[END_REF]: using the fact that bounded subsets of L 1 (R D ) ∩ BV(R D ) are precompact in L 1 loc and the time equicontinuity (40), we have the convergence of a subsequence (f ∆ k ) (44). The convergence of (u ∆ k ) is immediate. The fact that f = M(u) is the consequence of the estimate (43).

In a second step, in order to prove that u is a weak solution of the problem, we rewrite the scheme in the framework of conservative difference schemes.

We remark that the scheme can be written as (45)

                   f n+ 1 2 l,α = f n l,α - D d=1 ∆t ∆x d F n l,α+ e d 2 -F n l,α- e d 2 , u n+1 α = L l=1 f n+ 1 2 l,α , f n+1 l,α = (1 -ω)f n+ 1 2 l,α
+ ωM l (u n+1 α ). The numerical flux functions are given by ( 46)

F l,α+ e d 2 = -λ d -1 i=j ld f l,α-ie d if v ld < 0, 0 if v ld = 0, λ d j ld -1 i=0 f l,α-ie d if v ld > 0.
We recall that v ld = λ d j ld and λ d

∆t ∆x d = 1. Therefore F l,α+ e d 2 = F ld (f n l,α-j ld e d , . . . , f n l,α-e d ) if v ld < 0, 0 if v ld = 0, F ld (f n l,α , . . . , f n l,α-(j ld -1)e d ) if v ld > 0, with (47) ∀g l ∈ R, F ld (g l , . . . , g l ) = v ld g l .
As a consequence, if g l = M l (u) then by (3)

(48) ∀u ∈ [-µ ∞ , µ ∞ ], L l=1 F ld (M l (u), . . . , M l (u)) = A d (u).
For the unknown u n α we have

u n+1 α = u n α - D d=1 ∆t ∆x d A n α+ e d 2 -A n α- e d 2 with A n α+ e d 2 = L l=1 F n l,α+ e d 2 . A n α+ e d 2 = l/v ld <0 F ld (f n l,α-j ld e d , . . . , f n l,α-e d ) + l/v ld >0 F ld (f n l,α , . . . , f n l,α-(j ld -1)e d ).
So

A n α+ e d 2 = A d f n l,α+|j ld |e d , . . . , f n l,α-|j ld |e d -e d 1≤l≤L
with

∀(g 1 , . . . , g l ) ∈ R L , A d ((g l , . . . , g l ) 1≤l≤L ) = L l=1 v ld g l . and (49) ∀u ∈ [-µ ∞ , µ ∞ ], A d (M l (u), . . . , M l (u)) 1≤l≤L = A d (u).
We then follow the same steps as the Lax-Wendroff theorem ( [START_REF] Lax | Systems of conservation laws[END_REF], [START_REF] Michael | Monotone difference approximations for scalar conservation laws[END_REF]) but here when ∆t tends to zero, ∆t/∆x d being kept constant we take also into account the fact that u ∆ tends to equilibrium, so that the difference of flux tends to ∂ x d A d (u).

4.2.

Convergence to the entropy solution. We recall that the Cauchy problem (1)(4) admits a unique weak entropy solution which is characterized by ( [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF]):

(50) |u -c|∂ t ϕ + sgn(u -c) D d=1 (A d (u) -A d (c))∂ x d ϕ dxdt ≥ 0 for any c ∈ R and ϕ ∈ C ∞ 0 (R D × (0, T ))
, ϕ ≥ 0, and, for any interval I of R D :

(51) lim

T →0 + 1 T T 0 I |u(x, t) -u 0 (x)|dx dt = 0.
As in [START_REF] Natalini | A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws[END_REF], we associate to η c (u) = |u -c| the kinetic entropy-entropy flux pair:

(52) H l,c (g l ) = |g l -M l (c)|, G l,c (g l ) = v ld |g l -M l (c)|, d = 1, . . . , D.
As the transport part of the scheme is monotone, the following discrete entropy inequality holds for every l ∈ {1, . . . , L}:

(53) H l,c (f

n+ 1 2 l,α ) -H l,c (f n l,α ) ∆t + D d=1 Q n l,c,α+ e d 2 -Q n l,c,α- e d 2 ∆x d ≤ 0 with (54) Q l,c,α+ e d 2 = -λ d -1 i=j ld |f l,α-ie d -M l (c)| if v ld < 0, 0 if v ld = 0, λ d j ld -1 i=0 |f l,α-ie d -M l (c)| if v ld > 0. Q l,c,α+ e d 2 = Q l,c,d (f n l,α-j ld e d , . . . , f n l,α-e d ) if v ld < 0, 0 if v ld = 0, Q l,c,d (f n l,α , . . . , f n l,α-(j ld -1)e d ) if v ld > 0, with (55) ∀g l ∈ R, Q l,c,d (g l , . . . , g l ) = v ld |g l -M l (c))|.
As a consequence, if g l = M l (u), as M l is non decreasing and by (3)

(56) ∀u ∈ [-µ ∞ , µ ∞ ], L l=1 Q l,c,d (M l (u), . . . , M l (u)) = sgn(u -c)(A d (u) -A d (c)). Lemma 4.10. Define (57) ∀f ∈ R L , H c (f ) = L l=1 H l,c (f l ), Q n c,α+ e d 2 = L l=1 Q n l,c,α+ e d 2 , (1 ≤ d ≤ D).
The following inequality holds:

(58) H c (f n+1 α ) -H c (f n α ) ∆t + D d=1 Q n c,α+ e d 2 -Q n c,α- e d 2 ∆x d ≤ 0.
Proof. We use lemma (4.4). Noting that M l (c) = S l (M(c)) we can write

L l=1 |f n+1 l,α -M l (c)| = L l=1 |S l (f n+ 1 2 α ) -S l (M(c))| ≤ L l=1 |f n+ 1 2 l,α -M l (c)| ≤ L l=1 H l,c (f n+ 1 2 l,α ).
Then by (53) we obtain the result.

Proceeding again as in the proof of Lax-Wendroff theorem we can state our result.

Theorem 4.11. Suppose that conditions (5) and (23) are satisfied. Let f n α be the numerical solution given by [START_REF] Graille | Approximation of mono-dimensional hyperbolic systems: a lattice Boltzmann scheme as a relaxation method[END_REF] with [START_REF] Guillon | Stability analysis of the vectorial lattice-Boltzmann method[END_REF], with initial data u 0 ∈ L 1 (R)∩L ∞ (R)∩BV(R) satisfying u 0 ∞ ≤ µ ∞ . Let (∆t i ) a sequence of time steps tending to 0. (u ∆ti (•, t)) converge in L 1 (R D ), uniformly on all [0, T ], T > 0, to u(•, t), where u is the unique weak entropy solution of the Cauchy problem (1)(4).

Numerical experiments

5.1. One-dimensional computations. We approximate the entropy solution of the Burgers equation

∂ t u + ∂ x u 2 2 = 0 with initial condition u 0 (x) = 1 if x < 0, u 0 (x) = 0 else. Then ∀t > 0, u(x, t) = 1 if x < t 2 , u(x, t) = 0 else.
The computational domain is [-1, 1]. We do not add any order increasing procedure because our goal is to study the influence of the conditions of monotony ( 5) and ( 23) on the behaviour of the solution. As the choice of λ determines the number of time steps, we fix λ and to make ω vary.

For the D1Q2 model presented in section 3, condition [START_REF] Baty | A robust and efficient solver based on kinetic schemes for magnetohydrodynamics (MHD) equations[END_REF] gives

λ ≥ 1 = A (u 0 ) ∞ .
We set λ = 5, so that condition ( 23) is satisfied if

ω ≤ ω 0 = 5 3 .
For the D1Q4 model presented in section 3, condition [START_REF] Baty | A robust and efficient solver based on kinetic schemes for magnetohydrodynamics (MHD) equations[END_REF] gives

λ ≥ 2 3 .
We set λ = 5, so that condition ( 23) is satisfied if

ω ≤ ω 1 = 60 47 .
We have 1.276 < ω 1 < 1.277. With ∆x = 1 50 and a final time of computation T max = 0.8, the supremum bound of the solution takes the values indicated in Table 1. We can see that for the D1Q2 model, if ω > ω 0 then u(•, T max ) ∞ > 1: the scheme does not preserve the extrema. The D1Q4 model is very diffusive so that the value 1 is not reached for ω ≤ 1.70. 1. The values of u ∞ when ω varies, at time T max = 0.8 with 100 points on [-1, 1], λ = 5.

ω D1Q2 u(•, T max ) ∞ D1Q4 u(•, T max ) ∞ 1.
In Figure 1, we show the solution as a function of x at the final time T max = 0.8, for some values of ω. The shock is more and more accurately approximated when ω increases but the maximum of the solution is not preserved. This is classical in the context of hyperbolic conservation laws: accuracy and stability are often contradictory. This indicates that convergence in a weaker sense than the one proved here could hold. On the left picture we show the D1Q2 results, while the D1Q4 results are depicted on the right. The D1Q4 model computes a more diffusive solution than the D1Q2 one. It is also the case with a 1000 points mesh, as shown in Figure 2 left if we take the same value of ω for both models. In Figure 2 right, we take ω = 1.67 for the D1Q2 model in order to keep the monotony property but for D1Q4 we can take ω = 1.8 and we obtain the same accuracy, without loss of stability. The extended stencil seems to bring more stability than expected by the monotonicity conditions. For ν = (cos θ, sin θ) fixed, we define for (x, t) ∈ R 2 × R:

u(x, t) = u(x • ν, t).
This defines a solution of the two-dimensional equation

∂ t u + ∂ x1 (A(u) cos θ) + ∂ x2 (A(u) sin θ) = 0.
We take the same initial value as for the one-dimensional tests:

∀x ∈ R 2 , u 0 (x) = 1 if x • ν < 0, u 0 (x) = 0 else.
The rotation angle is θ = π/12. The final simulation time is equal to T max = 0.8. The test is performed on a 100 × 100 uniform mesh of [-1, 1] × [-1, 1]:

∆x 1 = ∆x 2 = 1 50 .
As a consequence λ 1 = λ 2 = λ > 0. We test the D2Q4 and D2Q8 models presented in section 3.

For the D2Q4 model the condition (5) reads as 2 max(cos θ, sin θ) ≤ λ. The supremum bound of the solution takes the values indicated in Table 2. We observe that the L ∞ bound of the solution is preserved for ω ≤ 1.30. Table 2. The values of u ∞ when ω varies, at time T max = 0.8 with 100x100 points on [-1, 1] × [-1, 1], λ = 10.

In Figure 3 we represent the solution along the axis which contains (0, 0) and is orthogonal to the direction of propagation of the shock for several values of ω, beginning with ω 2 for D2Q4 and ω 3 for D2Q8. We also represent the two-dimensional isovalues for ω = 1.60 in Figure 4. We remark that the D2Q8 model is more diffusive than the D2Q4 one.

Conclusion

In this article the monotony properties of the considered lattice Boltzmann Method allowed us to obtain its convergence for ω > 1, for a scalar multidimensional conservation law. In particular, for the D1Q2 model, the maximal value is ω = 2. The one and two dimensional tests show that the L ∞ norm of the solution can exceed the L ∞ norm of the initial data for values of ω greater than the theoritical value given by [START_REF] Milišić | Stability and convergence of discrete kinetic approximations to an initial-boundary value problem for conservation laws[END_REF]. Except for the D1Q2 model, it seems that some diffusion process preserves this norm on a larger interval.

Hyperbolic systems of conservation laws can be approximated by the same scheme, but monotony does not hold anymore for systems, so that the method presented here cannot work. Some investigations were already done for systems, with enlightening results in the linear case, see [START_REF] Guillon | Stability analysis of the vectorial lattice-Boltzmann method[END_REF] and references in this paper.

  denote λ = (λ 1 , . . . , λ D ) the velocity scale: λ d ∆t ∆x d = 1 for d = 1, . . . , D. The set of characteristic velocities is such that (7) v ld = j ld λ d , j ld ∈ Z, d = 1, . . . , D, l = 1, . . . , L and there exist l and d such that |j ld | = 1. Therefore ∀d ∈ {1, . . . , D}, ∀l ∈ {1, . . . , L}, α l,d = α d -j ld and, denoting j l = (j ld ) 1≤d≤D , ∀l ∈ {1, . . . , L}, α l = α -j l .

Figure 2 .

 2 Figure 2. Shock solution of Burgers equation at time T max = 0.8 with 1000 points on [-1, 1], λ = 5. Left : ω = 1.3 for both D1Q2 and D1Q4 models. Right: D1Q2 model with ω = 1.67 and D1Q4 model with ω = 1.80.

Figure 3 .. 3 (

 33 Figure 3. Shock solution of Burgers equation with rotated data in 2D, at time T max = 0.8 with 100 × 100 points on [-1, 1] × [-1, 1], λ = 10. Solution along the axis containing (0, 0) and orthogonal to the direction of propagation of the shock. Left : D2Q4 model. Right: D2Q8 model.

ω

  D2Q4 u(•, T max ) ∞ D2Q8 u(•, T max ) ∞ 1

Figure 4 .

 4 Figure 4. Shock solution of Burgers equation with rotated data in 2D, at time T max = 0.8 with 100 × 100 points on [-1, 1] × [-1, 1], λ = 10, ω = 1.60. Top : isovalues for the D2Q4 model. Bottom: isovalues for the D2Q8 model.


Dr Riccardo De Marco

Dr Valerio Giuso

Dr Thierry Achard

Dr Marzio Rancan

Prof. Dr Marco Baron

Prof. Dr Lidia Armelao

Dr Matteo Mauro

Stéphane Bellemin-Laponnaz

Cristina Tubaro
email: cristina.tubaro@unipd.it

Exploring the Coordination Properties of Phosphonium-Functionalized N-Heterocyclic Carbenes Towards Gold

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Introduction

The interest towards N-heterocyclic carbenes (NHCs) is continuously growing driven by their application either as organocatalysts [1][START_REF] Biju | Carbenes Organocatalysis[END_REF][START_REF] Flanigan | [END_REF] or as ligands in functional organometallics. [4- 7] NHC-metal complexes have shown indeed excellent performances in homogeneous catalysis, medicinal chemistry and material science. [START_REF] Mercs | [END_REF][9][10][11] Consequently, there is a very intense research effort toward the design of novel NHC architectures with different moieties. In this view, a very common approach is the introduction of a second functionality at nitrogen wingtip group(s) or as a dangling substituent in the backbone of the heterocyclic ring. To date, several examples of backbone functionalized NHCs that feature neutral (like for example -OR, -NR2, -PR2, -SR, -SeR groups), anionic (like -O -, -NR -, -S -, -SO3 -, -COO -groups) or even cationic (like -NR3 + , -PR3 + groups) moieties have been described. [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] The exploitation of the second functionality allows for example to obtain polynuclear complexes, [28,29] the anchoring of the complex to polymers, [30] surfaces or nanoparticles. [31] This also increases the robustness of the whole molecular system in the case of catalytic activity. It can also provide additional functionality, especially in medicinal chemistry, by seeking specific targeting. [32] In the series of functionalized NHCs, those bearing a phosphonium group have recently been extensively investigated by Canac and co-workers with particular interest towards palladium(II) and rhodium(I) centres. [33][34][35][36][37][38][39][40] Importantly, this type of compounds can act as bidentate chelating ligands through the formation of a phosphonium ylide in addition to the NHC donor. Although phosphonium ylides are known since the early 20 th century, there is now a renaissance in the study of their coordination properties. [41,42] They formally have a negatively charged carbon atom directly attached to a positively charged phosphorus atom and behave as σ-donor ligands, stronger than NHCs. [43,44] A second benefit of the positive charge present on the phosphonium group is related to the possibility of using the obtained metal complexes in medicinal chemistry. Indeed since late 60s, triphenylphosphonium moieties have been shown to act perfectly as a delocalized lipophilic cation (DLC), allowing the functionalized compound to efficiently penetrate and accumulate in the mitochondria. [45,46] In addition, the higher membrane potentials of cancer cells compared to normal cells, selectively promote the accumulation of DLCs in carcinoma cells. Considering both these aspects related to the use of phosphonium-functionalized NHC ligands, we report here a study on their coordinating properties towards gold. In particular, we show that the simple addition of LiBr salt in the reaction media allows the formation of a NHC gold(I) mononuclear complex, instead of a dinuclear complex, the two compounds having been characterized by X-ray diffraction studies. Gold(III) species were also isolated and fully characterized. Preliminary results of the biological effects on MCF7 cancer cells are also reported.

Results and Discussion

Synthesis of phosphonium functionalized NHC proligands

A convenient access to the phosphonium functionalized NHC proligands 1 and 2 is the quaternization of the proper Nsubstituted imidazole via nucleophilic substitution using (3bromopropyl)triphenylphosphonium bromide (Scheme 1). [35] Scheme 1. Synthesis of the phosphonium functionalized NHC proligands 1 (89% yield) and 2 (91% yield).

The 31 P NMR spectra of the phosphonium salts in deuterated acetonitrile show a singlet at d = 24.0 and 24.1 ppm for 1 and 2, respectively. The successful synthesis of these compounds was supported in the 1 H NMR spectra by the presence of a singlet at d around 9.3-9.6 ppm associable to the acidic C2-H. Moreover, the molecular structure of compound 2 was determined by X-ray diffraction (Figure 1).

Gold(I) complexes bearing phosphonium functionalized NHC ligands

Starting from proligands 1 and 2, two mononuclear gold(I) complexes were synthesized without by-products formation via the so-called weak base route (Scheme 2). Complexes 3 and 4 were isolated by using the proper proligand, 5 equiv. of K2CO3 and 5 equiv. of LiBr to avoid the formation of the bis(NHC) complex and heating the mixture in acetonitrile at 40 °C for ca. 5 hours. [19] The benzyl-substituted gold(I) complex 4 is stable and was isolated in excellent yield (86%). The successful coordination of the carbene to the gold(I) centre was confirmed by means of 1 H NMR with the disappearance of the singlet assigned to the acidic C2-H of the imidazolium as well as by the presence of a signal at 173.1 ppm in the 13 C NMR spectrum, which is the signature of the C2 carbene coordinated to the gold(I) centre. [47] The chemical shift of the other functional groups both in the 1 H and 13 C NMR spectra are similar to those observed in the starting imidazolium salt. The 31 P NMR spectrum presents a signal at d 23.5 ppm slightly shifted (Dd +0.6 ppm) in comparison with the analogous signal in the ligand precursor 2. The ESI-MS spectrum shows a fragment at 737.20 m/z associable to the loss of the bromide, counter ion of the phosphonium moiety. The analogous methyl-substituted gold(I) complex 3 was more difficult to isolate. The complex appeared stable in the reaction mixture (i.e. in the presence of LiBr and K2CO3), so that for example, colourless crystals of 3 suitable for single-crystal X-ray diffractometric (SC-XRD) analysis formed directly in the NMR tube containing a sample of the reaction mixture. The molecular structure of complex 3 together with selected bond distances and angles is shown in Figure 2 (see below for discussion). However, the complex undergoes decomposition after filtration of the inorganic salts and evaporation of the solvent under vacuum. In fact, during evaporation of the solvent, the colour of the solution turned yellow and a brown residue, attributed to the formation of metallic gold, formed at the bottom of the reaction flask. This residue was treated with deuterated acetonitrile, transferred into an NMR tube and the 1 H NMR spectra of this sample mixture were recorded after 1, 3, 5 and 8 days (Figure S7). A second set of signals, slightly downfield shifted with respect to the analogous set of the gold(I) complex, increases with time. The ESI-MS spectrum of the sample after 8 days displays two main fragments, one associable as the cation of the pristine gold(I) complex (fragment at 661.14 m/z, corresponding to loss of one bromide ion) and one to the species [AuBr3(L)] + (822.83 m/z, attributed to the corresponding gold(III) complex with L=NHC ligand) (Figure S8). The partial formation of the gold(III) complex 7 and its characterisation were also confirmed by direct oxidative bromine addition experiments to the mixture, with the aim of isolating only the gold(III) species (see later in the text the section on gold(III) complexes). The partial oxidation of gold(I) to gold(III) could be the result of a partial disproportionation of the gold(I) complex. This type of reactivity has already been reported in the literature. [48][49][50][51][52] Scheme 2. Synthesis of the mononuclear gold(I) complexes 3 and 4 (86% yield), and of the dinuclear ones 5 (73% yield) and 6 (82% yield).

Complex 3 presents a linear coordination geometry, as expected for a metal center in a d 10 electronic configuration, with one NHC and a bromide ligand coordinated to the gold(I) (C-Au-Br angle of 177.54°, see Figure 2). The Au-C and Au-Br bond distances are in agreement with literature data (1.984 Å and 2.3976 Å, respectively). [19,53] Interestingly, we found that the result of the complexation on gold(I) strongly depends on the presence of LiBr. In fact, when the synthesis is carried out without an excess of bromide ions, two novel gold(I) complexes, 5 and 6, could be isolated rather than the mononuclear ones 3 and 4. Complexes 5 and 6 are dinuclear species, characterized by the coordination of two C,C-bridging NHC-phosphonium ylide ligands. The deprotonation of the methylene group in alpha position to the phosphonium group has been already reported in the literature for palladium(II) complexes with similar ligands. [33][34][35]39] In our case, the formation of an ylidemetal bond was indeed suggested on the basis of the 31 P NMR spectrum, which exhibited two resonances around d 29 ppm (29.6 (minor) and 29.5 (major) ppm for complex 5; 29.6 (major) and 28.8 (minor) ppm for complex 6) shifted of ca. 5 ppm to lower fields relative to phosphonium precursors or the gold(I) complexes 3 and 4. The presence of two signals in the 31 P NMR spectrum can be justified by the occurrence of two conformational isomers in solution, [54] characterised by different orientation of the bridge between the two donors or of two diasteroisomersd due to the presence of the stereogenic ylidic carbon atoms. [35,36,[38][39][40] Finally, the molecular structure of dinuclear complex 6 was definitively established by X-ray diffraction analysis on colourless crystals isolated by slow diffusion of diethyl ether into an acetonitrile solution of 6 (Figure 3). Complex 6 is a dinuclear species with a head-to-tail bridging coordination of the two NHC-phosphonium ylide ligands. The stereochemical structure of complex 6 found in the crystal is the meso isomer. The gold(I) centres present an almost linear coordination geometry (Ccarbene-Au-Cylide of 176.78 °). The Au-Ccarbene and Au-Cylide bond distances (2.017 and 2.085 Å, respectively) are in agreement with literature data, with the Au-Cylide bond slightly longer than the Au-Ccarbene one. [34,55] The Au•••Au distance (3.2265 Å) suggests the presence of aurophilic interactions, since the distance is shorter than the sum of van der Waals of two gold atoms (i.e. 3.32 Å from the 1.66 Å radii). [56] When crystals of complex 6 are dissolved in deuterated DMSO a singlet is observed (Figure S15) in the 31 P NMR spectrum at d 28.8 ppm. The spectrum changes with time and after 72 h, it becomes superimposable to the one registered for the isolated solid at the end of the synthesis reaction (two signals at d 29.6 and 28.8 ppm). This confirms the possibility of two isomers in equilibrium in solution. A change is also observed in the 1 H NMR spectrum (Figure S14), in which the most striking difference is the benzylic CH2 group: in the just-dissolved sample an AB system is observed, whereas it becomes a singlet with time, suggesting an increase in mobility of the benzyl substituent.

Au I -NHC complexes and their multi-metallic assemblies are known to display interesting and sometimes intense luminescence properties, [57][58][59][60][61] in particular when short Au•••Au are present. Regrettably, complexes 5 and 6 display virtually absent photoluminescence very close to the detection limit of the in the solid state and no mechanochromic behavior.

Gold(III) complexes bearing phosphonium functionalized NHC ligands

Mononuclear gold(III) complexes 7 and 8 were isolated by oxidative addition of molecular bromine to gold(I) complexes (Scheme 3). We used complex 4 in its isolated pure form and complex 3 in solution and not purified because of its instability (vide supra). The 1 H NMR spectra of the two synthesized gold(III) complexes are similar to those observed for the pristine gold(I) complexes, except for a slight shift to higher ppm for all signals.

In contrast, the 13 C NMR spectra show a signal for the carbene carbon around 136 ppm, in agreement with values reported in the literature, but high-field shifted by about 40 ppm from the carbene carbon coordinated to a gold(I) centre. [53] The ESI-MS spectra shows a fragment at 822.80 m/z and 896.82 m/z for complex 7 and 8, respectively, associable to the loss of the bromide counterion of the phosphonium moiety. This behaviour is similar to that observed for gold(I) complexes and could be easily justified given the cationic nature of the ligand.

Antiproliferative activity of the mononuclear gold(I) and gold(III) complexes

As mentioned in the introduction, one of the possible applications of phosphonium functionalized complexes is their use as antitumoral agents. The synthesised mononuclear gold(I) and gold(III) complexes (except for complex 3, which demonstrated to be unstable in solution), were tested on breast cancer cells MCF7 and their activity compared with that of the well-known anti-cancer drug cisplatin. The stability of complexes 4 and 8 was assessed by UV-vis spectroscopy in a mixture of water/dmso 95/5 v/v (for 8) or NaCl (0.9%)/dmso 95/5 v/v solution (for 4, concentration of about 0.15 mM), observing a decrease in absorbance, but no change in band position for 24 h. The determined half-inhibitory concentrations, IC50 values, are reported in Table 1 and they are in the same order of magnitude of the values measured for cisplatin and for the analogous Au(I) complexes bearing unfunctionalized NHC ligands. [19,62] The antiproliferative activity does not strongly depend on the oxidation state of the metal, but this is not surprising considering that gold(III) complexes of the type [AuBr3(NHC)] are well known to be reduced to the corresponding gold(I) complex under physiological conditions.

Conclusion

The coordinating properties of two phosphonium-functionalized NHC ligands towards gold(I) and gold(III) metal centres have been explored. The mononuclear gold(I) complexes of general formula [AuBr(NHC)] were isolated by base-assisted metalation only if a large excess of LiBr is added in the reaction mixture. In the absence of added bromides, dinuclear gold(I) complexes with bridging NHC-phosphonium ylide ligands are isolated. This possibility to obtain NHC-phosphonium ylide bidentate ligand was already reported for palladium(II) complexes; in that case the chelating coordination of the ligand was observed while we reported here for the first time a bridging coordination for the NHC-phosphonium ylide ligand. Finally, the mononuclear gold(III) complexes [AuBr3(NHC)] were isolated by oxidative addition of bromine to the gold(I) centres.

The mononuclear complexes displayed moderate antiproliferative activity, within the usual range observed for this type of compounds. Although these studies show activity in the same range as non-functionalized gold complexes, it would be interesting to conduct mechanistic studies to see if mitochondrial function is altered. Future studies will aim to extend this study to the preparation of complexes featuring phosphonium moieties with different substituents from phenyl groups. The reactivity of the phosphonium ylide group coordinated to the gold(I) centres will be investigated as well.

Experimental Section

Materials and methods

Reagents were purchased from Sigma-Aldrich as high-purity products and used as received. MCF7 cells were provided by Imagif, Gif sur Yvette (France). All manipulations were carried out using standard Schlenk techniques under an atmosphere of argon. NMR spectra were recorded on a Bruker Avance 300 MHz operating at 300.

Synthesis of the imidazolium salts 1 and 2

In a flame dried Schlenk, the (3-bromopropyl) triphenylphosphonium bromide (2.00 g, 4.31 mmol, 1 equiv.) and the proper 1-alkylimidazole (1.5 equiv.) were heated at 120 ˚C in dry chlorobenzene (ca. 15 mL) under argon for 15 h. After this time, the solvent was evaporated via a «trap-totrap» technique. The crude residue was suspended in Et2O (20 mL) then sonicated for 10 min; this action was repeated three times, to remove the excess of 1-alkylimidazole, to afford after filtration a white powder.

Synthesis of the mononuclear gold(I) complexes 3 and 4

The proper imidazolium salt 1 or 2 (1 equiv.), the Au(I) precursor AuCl(SMe2) (1.05 equiv.), LiBr (5 equiv.) and an excess of K2CO3 (5 equiv.) were introduced into a two-or three-neck round bottom flask. Acetonitrile (15 mL) was then added via a syringe under inert atmosphere.

The mixture was stirred at 40 °C for 5 hours, then filtered. Finally, the solvent was removed under vacuum giving a white solid. For complex 3, the reported 1 H and 31 P NMR, as well as the ESI-MS, characterizations are based on a withdrawal from the reaction mixture, before the work-up for isolation of the solid, which causes decomposition of the compound.

The most important 13 C signals were extrapolated from the 2D NMR spectrum of the mixture isolated after work-up.

Synthesis of complexes 5 and 6

The proper imidazolium salt 1 or 2 (1 equiv.), the Au(I) precursor AuCl(SMe2) (1.05 equiv.) and an excess of K2CO3 (5 equiv.) were introduced into a two-or three-neck round bottom flask. Acetonitrile (15 mL) was then added via a syringe under inert atmosphere. The mixture was stirred at 40 °C for 5 hours, then filtered. Finally, the solvent was removed under vacuum giving a white solid.

Synthesis of complexes 7 and 8

The proper gold(I) complex was dissolved in acetonitrile (ca. 10 mL) under Argon atmosphere. A solution of bromine in acetonitrile was added to the solution of the gold(I) complex with a syringe (Au:Br2 ratio 1:1.2). The solution was left under stirring at room temperature for 2 hours. Finally, the solvent and the excess of bromine were removed at reduced pressure. The methyl-substituted gold(III) complex 7 was synthesised starting from the mixture of gold(I) and gold(III) complexes, isolated in the work-up reaction of 3. 7. 1

X-ray crystal structure determination

Data were collected using an Oxford Diffraction Gemini E diffractometer, equipped with a 2K × 2K EOS CCD area detector and sealed-tube Enhance (Mo) and (Cu) X-ray sources. A suitable single crystal was fastened on a nylon loop and measured at room temperature. Detector distance has been set at 45 mm. The diffraction intensities have been corrected for Lorentz/polarization effects as well as with respect to absorption. Empirical multi-scan absorption corrections using equivalent reflections have been performed with the scaling algorithm SCALE3 ABSPACK. Data reduction, finalization and cell refinement were carried out through the CrysAlisPro software. Accurate unit cell parameters were obtained by least squares refinement of the angular settings of strongest reflections, chosen from the whole experiment. The structures were solved with Olex2 [63] by using ShelXT [64] structure solution program by Intrinsic Phasing and refined with the ShelXL [65] refinement package using leastsquares minimization. In the last cycles of refinement, non-hydrogen atoms were refined anisotropically. Hydrogen atoms were included in calculated positions, and a riding model was used for their refinement. The structure of complex 6 contains a mixture of chloride and bromide anions occupying the same positions due to substitutional disorder. These atoms were split and modelled through EXYZ and EADP constrains. The unit cell contains circa 3 chloride and 1 bromide (Cl 2.9; Br 1.1). Details for each structure refinement are reported in the crystallographic table (Table S1). Deposition Numbers 2221949 (for 3), 2221950 (for 6) and 2221951 (for 2) contain the supplementary crystallographic data for this paper. These data are provided free of charge by the joint Cambridge Crystallographic Data Centre and Fachinformationszentrum Karlsruhe.

Cytotoxicity tests of NHC-gold(I/III) complexes

Cytotoxicity tests have been carried out at the "Institut de Chimie des Substances Naturelles CNRS -UPR2301, Gif-Sur-Yvette, France". Samples were prepared by dissolution of the compounds in DMSO (except for cisplatin that was dissolved in water since it rapidly decomposes in DMSO) at stock concentrations of 10 mM. MCF7 (breast cancer cells) cell lines were maintained as monolayers in RPMI 1640 medium supplemented with 10% fetal calf serum, in the presence of penicilline, streptomycine and fungizone in 75 cm 2 flask under 5% CO2. Cells were plated in 96-well tissue culture plates in 200μl complete medium at a density of 1000-2500 cells per well and treated 24h later with 2μl of compounds using a Biomek 3000 automation workstation (Beckman-Coulter). Controls received the same volume of the appropriate vehicle (DMSO, EtOH or water, 1% final volume). After 72 h exposure, MTS reagent (CellTiter 96 ® Aqueous One, Promega) was added and incubated for 3 h at 37 °C: the absorbance was monitored at 490nm and results expressed as the inhibition of cell proliferation calculated as the ratio [(1-(OD490 treated/OD490 control))×100] in triplicate experiments after subtraction of the blank without cells. Positive controls (cells incubated with a reference drug at its IC50 concentration) were routinely added to check the responsiveness of cells. For IC50 determination [50% inhibition of cell proliferation], cells were incubated for 72 h following the same protocol with compound concentrations ranged 5 nM to 100 μM in separate duplicate experiments. Values are given as means ± SD from at least three independent experiments performed in triplicate.

Figure 1 .

 1 Figure 1. Crystal structure of 2. Colour code: C, grey; P, purple; N, blue. H atoms, anions and solvent molecules omitted for clarity. Two carbon atoms in the imidazole ring are affected by positional disorder. Disordered moiety with minor occupation is displayed as translucent atoms.

Figure 2 .

 2 Figure 2. Crystal structure of complex 3. Colour code: C, grey; Au, dark blue; P, purple; N, blue. H atoms and uncoordinated anions omitted for clarity. Selected bond lengths (Å) and angles (°): Au1-Br1 2.3976(5), Au1-C1 1.984(4); C1-Au1-Br1 177.54(9).

Figure 3 .

 3 Figure 3. Crystal structure of complex 6. Colour code: C, grey; Au, dark blue; P, purple; N, blue. H atoms and anions omitted for clarity. Selected bond lengths (Å) and angles (°): Au1-Au1' 3.2265(4), Au1-C1 2.017(5), Au1-C13' 2.085(4); C1-Au1-C13' 176.78(16). Symmetry operation: ' 1-x,1-y,1-z.

Scheme 3 .

 3 Scheme 3. Synthesis of the mononuclear gold(III) complexes 7 and 8 (67% yield), by Br2 oxidative addition to the corresponding gold(I) complexes 3 and 4.

Table 1 .

 1 Half-inhibitory concentrations IC50 (in µM) values against MCF7 cancer cells for selected gold(I) and gold(III) complexes.[a]

	Complex	IC50 (in µM)
	4	12.3 ± 2.1
	7	20.0 ± 5.4
	8	18.3 ± 1.7
	cisplatin	11.9 ± 0.7

[a] After 72h of incubation; stock solutions in DMSO for all complexes; stock solutions in H2O for cisplatin.

  2.17 ppm.31 P{ 1 H} NMR (121 MHz, CD3CN, 298 K, ppm) d: 24.1 ppm.

	1. Yield 89%. mp: 269 °C. 1 H NMR (300 MHz, CD3CN, 298 K, ppm) d: 3.66
	(m, 2H, CH2P), 3.86 (s, 3H, CH3), 4.56 (t, J = 6.8 Hz, 2H, CH2N), 7.35 (m,

1H, CH), 7.58 (m, 1H, CH), 7.65 -7.92 (m, 15H, Ph), 9.36 (s, 1H, NCHN), the signal of one CH2 group is superimposed with the water peak at 2.17 ppm.

31

P{ 1 H} NMR (121 MHz, CD3CN, 298 K, ppm) d: 24.0 ppm. 2. Yield 91%. mp: 174 °C. 1 H NMR (300 MHz, CD3CN, 298 K, ppm) d: 3.63 (m, 2H, CH2P), 4.53 (t, J = 6.9 Hz, 2H, CH2N), 5.40 (s, 2H, CH2Ph), 7.33 -7.50 (m, 6H, Ph + CH), 7.58 (m, 1H, CH), 7.66 -7.91 (m, 15H, Ph), 9.59 (s, 1H, NCHN), the signal of one CH2 group is superimposed with the water peak at

Acknowledgements

Financial support by Dipartimento di Scienze Chimiche (P-DiSC#08BIRD2021-UNIPD, PI: C. T.) is gratefully acknowledged. This work was supported by La Ligue contre le Cancer CCIR Est.

Keywords: cytotoxicity • gold complexes • NHC ligands • phosphonium group • phosphorous ylides


V Saey
email: verosaey@hotmail.com

H Van Loo

A Gryspeerdt

S Pronost

F Gasthuys

K Rosiers

D Cassart

J Govaere

studie om de prevalentie van acht infectieuze oorzaken van abortus en perinatale sterfte bij het paard in België te bepalen en ten tweede om een onder veldomstandigheden bruikbaar "polymerase-chainreaction"protocol op te stellen. Een "real-time polymerase chain reaction" voor acht pathogenen geassocieerd met abortus/perinatale sterfte bij het paard werd uitgevoerd bij 105 gevallen. In 37% van de gevallen werd tenminste één infectieus agens gedetecteerd, met als koploper equine herpesvirus-1 (49%) gevolgd door Streptococcus equi subspecies zooepidemicus (28%), Coxiella burnetii (18%), Leptospira interrogans (3%) en Neospora caninum (3%). Equine viral arteritis, Equine herpesvirus-4 and Chlamydophila spp. werden niet gedetecteerd.

In deze studie wordt aangetoond dat PCR nuttig kan zijn om een co(-infectieuze) oorzaak van abortus, doodgeboorte of neonatale sterfte bij het paard te detecteren, vooral in praktijkomstandigheden waarbij autolyse en contaminatie een uitgebreid post-mortemprotocol met klassieke microbiologische analyses kunnen verhinderen.

BSTRACT

Equine abortion, stillbirth and neonatal death cause major economic losses to the equine industry worldwide. Both non-infectious and a wide range of infectious causes have been described. However, the relative contribution of pathogens to equine abortion, stillbirth and neonatal death is poorly documented, since available studies involve only a limited number of pathogens. Therefore, the objectives of the present retrospective monitoring study were to determine the prevalence of infectious agents associated with equine abortion and perinatal mortality in Belgium, and to set up a protocol usable under field conditions using polymerase chain reaction targeting. A real-time simple polymerase chain reaction for eight different abortifacient pathogens was conducted leading to the detection of at least one infectious agent in 37% of 105 analyzed cases. In the diagnosed cases, equine herpesvirus-1 was the most detected pathogen (49%), followed by Streptococcus equi subspecies zooepidemicus (28%), Coxiella burnetii (18%), Leptospira interrogans (3%) and Neospora caninum (3%). None of the analyzed cases was positive for equine viral arteritis, equine herpesvirus-4 and Chlamydophila spp.

In this study, PCR targeting is shown to have its value to detect a (co)-infectious cause in equine abortion, stillbirth and neonatal death, especially in field conditions where autolysis and contamination might preclude a full post-mortem examination protocol based on classical microbiological examination.

SAMENVATTING

Abortus, doodgeboorte en neonatale sterfte bij de merrie veroorzaken jaarlijks wereldwijde economische verliezen. Zowel infectieuze als niet-infectieuze oorzaken worden beschreven in de literatuur. Het relatieve belang van de verschillende pathogenen bij abortus, doodgeboorte en neonatale sterfte bij het paard is echter onduidelijk, aangezien in de meeste studies slechts een beperkt aantal pathogenen onderzocht werd. Daarom waren de doelstellingen van deze retrospectieve monitorings-

A

Detection of infectious agents in equine pregnancy loss, stillbirth and neonatal death

Detectie van infectieuze agentia bij abortus, doodgeboorte en neonatale sterfte bij het paard

INTRODUCTION

Equine abortion, stillbirth and neonatal death (ASND) are major problems in the horse industry as they cause great economical and emotional losses worldwide [START_REF] Bosh | Impact of reproductive efficieny over time and mare financial value on economic returns among Thoroughbred mares in central Kentucky[END_REF][START_REF] Parameswaran | Analysis of economic loss due to Equine Herpes Viral infection[END_REF]. Spontaneous abortion in horses is defined as pregnancy loss prior to three hundred days of gestation. Fetuses dying after three hundred days of gestation are conventionally termed as stillbirths, since foals born from that time on may be capable to survive [START_REF] Acland | Abortion in mares[END_REF]. Cases of foals that die within two days after birth are classified as neonatal death.

Both infectious and non-infectious causes involved in equine ASND have been reported worldwide [START_REF] Bain | Foetal losses during pregnancy in the Thoroughbred mare: a record of 2562 pregnancies[END_REF][START_REF] Giles | Causes of abortion, stillbirth and perinatal deaths in horses: 3527 cases (1986-1991)[END_REF][START_REF] Smith | A survey of equine abortion, stillbirth and neonatal deaths in UK from 1988 to 1997[END_REF][START_REF] Léon | Identification of pathogenic Leptospira strains in tissues of a premature foal by use of polymerase chain reaction analysis[END_REF][START_REF] Léon | Molecular detection of Coxiella burnetii and Neospora caninum in equine aborted foetuses and neonates[END_REF]), but regional differences highlight the importance of local, systematic monitoring. In Belgium, limited information exists on the prevalence of equine causes of ASND.

Continuous surveillance of ASND in horses is important as some disorders, e.g. equine herpesvirus-1 (EHV-1) infections [START_REF] Damiani | A severe equine herpesvirus type 1 (EHV-1) abortion outbreak caused by a neuropathogenic strain at a breeding farm in northern Germany[END_REF] and the mare reproductive loss syndrome (MRLS) [START_REF] Sebastian | Review paper: mare reproductive loss syndrome[END_REF], require immediate preventive measures, encouraging laboratories to focus on specific infectious causes. However, examination is often challenging, especially under field conditions. Cases of ASND often show extensive post-mortal decay or autolysis, which hampers routine bacteriological and histological examination. Moreover, some pathogens are difficult to detect because of an intracellular location, e.g. N. caninum and C. burnetii, or a strenuous culture protocol (Leptospira). This underlines the importance of the use of PCR assays.

The aims of this study were (1) to determine the prevalence of the most important infectious causes of equine abortion, stillbirth and neonatal death in Belgium, and (2) to study the possible value of a panel of real-time polymerase chain reaction tests in the detection of infectious agents in equine abortion, stillbirth and neonatal death.

MATERIALS AND METHODS

Sample population

Between February 2015 and June 2017 (time period determined by availability of funding), a total of 133 cases (128 abortions/stillbirths, and 5 neonatal deaths) in different horse breeds (predominantly warmbloods) in Belgium (95 in Flanders, 38 in Wallonia) were declared by the owner. Within 24 hours of case declaration, the fetus and, in the majority of cases, corresponding fetal membranes and umbilical cord, or fetal organs were collected by the veterinarian and transported to the laboratory of Animal Health Services Flanders (DGZ Vlaanderen) or the Faculty of Veterinary Medicine of the University of Liège. The location of referral depended only on the location where the ASND case happened (DGZ Vlaanderen for Flanders; University of Liège for Wallonia).

From the initial 133 cases, 28 cases were excluded because of incompleteness of submitted material, resulting in a final study population of 105 cases (Figure 1).

Necropsy protocol

Processing of the fetus/foal was performed within 24 hours after arrival at the laboratory facilities. At necropsy, the fetus/foal was weighed, placed in rightsided recumbency and the crown-rump length (cm) was measured. In cases where no gestational length was known, the gestational length was calculated based on fetal crown-rump length, as described by [START_REF] Mcgeady | Age determination of the embryo and foetus[END_REF].

The fetus/foal, umbilical cord and placenta were checked for macroscopic abnormalities. The thoracic and abdominal cavities were cut-opened, followed by tissue sampling of the lung and liver. The skull was disarticulated at the atlantooccipital joint followed by sagittal dissection and tissue sampling of the brain. Finally, the placenta underwent tissue sampling.

All samples were separately packed and sent under cooled conditions (4°C) to LABÉO (Caen, France) for PCR examination.

Molecular analysis

The presence of EHV-1, equine herpesvirus-4 (EHV-4), equine viral arteritis (EVA), Lepstospira interrogans (L. interrogans), Streptococcus equi subspecies zooepidemicus (S. zooepidemicus), and Coxiella burnetii (C. burnetii) was systematically investigated by a simple real-time PCR on a pool sample of liver and lung tissue. The real-time PCR for the detection of Neospora caninum (N. caninum) and Chla-mydophila spp. (Chlamydophila abortus, psittaci and pecorum) was performed on brain and placenta, respectively. Nucleic acids of the collected organs were extracted using the Qiagen (QIAamp DNA/RNeasy mini kits) procedure according to the manufacturer's instructions. The real-time PCR tests were performed according to previously described procedures [START_REF] Léon | Identification of pathogenic Leptospira strains in tissues of a premature foal by use of polymerase chain reaction analysis[END_REF][START_REF] Diallo | Multiplex real-time PCR for the detection and differentiation of equid herpesvirus 1 (EHV-1) and equid herpesvirus 4 (EHV-4)[END_REF][START_REF] Pronost | Description of the first recorded major occurrence of equine viral arteritis in France[END_REF][START_REF] Léon | Molecular detection of Coxiella burnetii and Neospora caninum in equine aborted foetuses and neonates[END_REF][START_REF] Pronost | Outbreak of equine herpesvirus myeloencephalopathy in France: a clinical and molecular investigation[END_REF].

RESULTS

The majority of analyzed cases (66%) happened between the 7 th and 10 th month of gestation (range 3-11 months). Forty percent of the analyzed cases were abortions, 57% were stillbirth cases (>300 days of gestation), while 3% of the cases were neonatal deaths.

In 35% of the analyzed cases (37/105), an infectious agent could be detected. In 65% (68/105) of the cases, no infectious agent was detected, nor was there an apparent cause of abortion seen.

Viral infection

The main infectious agent that could be detected was EHV-1. This virus was present in 49% of the diagnosed cases (19/39). In 58% of the EHV-1 positive cases (11/19), post-mortem examination revealed lesions such as icterus, and an enlarged liver with or without pinpoint necrotic foci was noticed (Figure 2).

No cases were positive for EVA.

Bacterial infection

In 28% of the diagnosed cases (11/39), S. zooepidemicus was detected. Coxiella burnetii was detected in 18% of the diagnosed cases (7/39). In two of these C. burnetii positive cases, a co-infection was noticed (one with S. zooepidemicus; one with EHV-1). Only one case was positive for Leptospira interrogans (3%; 1/39), and no cases were positive for Chlamydophila spp.

Parasitic infection

Three percent of the diagnosed cases (1/39) was positive for N. caninum.

Non-infectious

In 4% of the analyzed cases (4/105), umbilical cord strangulation was present.

The results of the diagnostic analyses and the sampled tissues are summarized in Table 1.

In 5.7% of the analyzed cases (6/105), a presumptive diagnosis of placentitis was made based on gross abnormalities, such as thickening, edema and discoloration. However, no infectious agents could be detected in these cases.

DISCUSSION

This is the first overall survey in Belgium with the objective to establish an epidemiologic insight into equine ASND. In 37% of the analyzed cases (39/105), an infectious agent was detected.

In most large-case studies on aborted foals focussing on well-known pathogens causing abortion in horses (EHV-1 and S. zooepidemicus), standard microbiological, fungal examination, and viral isolation are used. In these surveys, prevalences of infectious causes of abortion ranging from 12% (Rickets et al., 2003) to 20% [START_REF] Smith | A survey of equine abortion, stillbirth and neonatal deaths in UK from 1988 to 1997[END_REF] have been reported. [START_REF] Léon | Molecular detection of Coxiella burnetii and Neospora caninum in equine aborted foetuses and neonates[END_REF] were the first to investigate an extensive panel of abortifacient pathogens (EHV, EVA, S. zooepidemicus, Leptospira, C. burnetii, N. caninum and C. abortus) in French horses using molecular analysis, resulting in a diagnostic rate of 27%. [START_REF] Szeredi | A survey of equine abortion and perinatal foal losses in Hungary during a three-year period (1998-2000)[END_REF] also used PCR to target EHV-1 and C. abortus in aborted foals. The higher diagnostic rate in the present study may be the result of the use of molecular analysis, or the higher prevalence of some of the analyzed pathogens. Additionally, the higher detection of infectious agents may also be caused by submission bias. Horse owners and their veterinarians may be more motivated to submit ASND cases for laboratory analysis when infectious causes are suspected.

In the present study, EHV-1 was the leading cause of equine ASND, similar to reports worldwide throughout all stages of pregnancy [START_REF] Macleay | A scoping review of the global distribution of cause and syndromes associated with mid-to late-term pregnancy loos in horses between 1960 and 2020[END_REF]. Both EHV-1 and EHV-4 are endemic in horses worldwide, but regional differences exist [START_REF] Allen | Advances in understanding of the pathogenesis, epidemiology and immunological control of equine herpesvirus abortion[END_REF][START_REF] Yildirim | Equine herpesvirus type 1 (EHV-1) and 4 (EHV-4) infections in horses and donkeys in northeastern Turkey[END_REF]. In the UK, EHV was detected in 7% of aborted foals [START_REF] Ricketts | A review of the causes of abortion in UK mares and means of diagnosis used in an equine studform practice in Newmarket[END_REF][START_REF] Smith | A survey of equine abortion, stillbirth and neonatal deaths in UK from 1988 to 1997[END_REF]. The large majority of these cases represented an EHV-1 infection (>90%) which corresponds with other studies [START_REF] Patel | Equine herpesviruses 1 (EHV-1) and 4 (EHV-4) -epidemiology, disease and immunoprophylaxis: a brief review[END_REF]. In France, [START_REF] Léon | Molecular detection of Coxiella burnetii and Neospora caninum in equine aborted foetuses and neonates[END_REF] identified EHV-1 infection in 8.8% of aborted foals. These percentages are low compared to the present study, which suggests a higher circulation of EHV-1 in Belgium than in some other countries. Belgium is known for its important role in equine breeding, and it is also a popular equestrian sports country. Consequently, there is a higher chance of direct contact between animals, which may lead to new outbreaks of EHV-1, as already shown in the past [START_REF] Carvelli | Clinical impact, diagnosis and control of Equine Herpesvirus-1 infection in Europe[END_REF]. In other important horse breeding areas, such as the USA and UK, EHV-1 positive cases have declined over time, which was attributed to a combination of vaccination and improved risk reduction [START_REF] Macleay | A scoping review of the global distribution of cause and syndromes associated with mid-to late-term pregnancy loos in horses between 1960 and 2020[END_REF]. The higher number of EHV-1 positive cases in the present study could be the result of the fact that the involved veterinarian might have been more inclined to refer In 7% of the analyzed cases, C. burnetii could be detected in liver and lung tissues of the fetus/foal. This prevalence suggests a high circulation of this bacterium in Belgium. Recently, 8.5% of Belgian bovine abortions originating from Wallonia, were tested positive for C. burnetii on the basis of a PCR test on a pool of fetal spleen and, if available, placental cotelydones [START_REF] Saegerman | Diagnosis of Coxiella burnetii cattle abortion: a one-year observational study[END_REF]. In Flanders, a percentage of only 1.5 % of bovine cases tested positive for C. burnetii. However, in these Flemish cases, only the fetal abomasal content was tested by PCR [START_REF] Loo | Retrospective study of factors associated with bovine infectious abortion and perinatal mortality[END_REF]. Not much literature is available regarding C. burnetii infections in horses. A pooled mean seroprevalence of 15.8% has been calculated in horses [START_REF] Marenzoni | Is the horse a reservoir or an indicator of Coxiella burnetii infection? Systematic review and biomolecular investigation[END_REF]. [START_REF] Léon | Molecular detection of Coxiella burnetii and Neospora caninum in equine aborted foetuses and neonates[END_REF] were the first to report the presence of C. burnetii in 1.5% of aborted foals. The placenta is the target organ of C. burnetii [START_REF] Clothier | Evaluation of bovine abortion cases and tissue stability for identification of infectious agents in California diagnostic laboratory cases from 2007 to 2012[END_REF]. In the study by [START_REF] Léon | Molecular detection of Coxiella burnetii and Neospora caninum in equine aborted foetuses and neonates[END_REF], both lung and placental tissues of equine fetuses were tested by PCR. However, it was not specified if there were significant differences in prevalence between both tested organs. In the present study, in 2/7 of the C. burnetii positive cases, co-infection was present (one co-infection with EHV-1 and one co-infection with S. zooepidemicus). Presuming that this EHV-1 infection was the cause of abortion, the percentage of C. burnetii positive ASND cases may be overestimated in the present study. Considering the zoonotic character of C. burnetii, extra attention should be paid to this pathogen [START_REF] Deyell | Q fever endocarditis: a case report and review of the literature[END_REF]. More studies, including histology and immunohistochemistry, are needed to further unravel the role of this pathogen in equine ASND.

L. interrogans is considered to be one of the most widespread zoonosis worldwide [START_REF] Levett | Leptospirosis[END_REF]. Leptospirosis has also been described as a cause of placentitis in horses [START_REF] Ellis | Leptospirosis in horses[END_REF][START_REF] Donahue | Emergent causes of placentitis and abortion[END_REF]. Moreover, the disease has been documented as the second most common cause of equine abortion in North America [START_REF] Donahue | Diagnosis and prevalence of leptospira infection in aborted and stillborn horses[END_REF][START_REF] Macleay | A scoping review of the global distribution of cause and syndromes associated with mid-to late-term pregnancy loos in horses between 1960 and 2020[END_REF]. However, only 4% of equine cases of pregnancy loss positive for leptospirosis worldwide originated from Europe [START_REF] Macleay | A scoping review of the global distribution of cause and syndromes associated with mid-to late-term pregnancy loos in horses between 1960 and 2020[END_REF]. Up till now, no information is available concerning the role of leptospirosis in equine ASND in Belgium. In the present study, only 3% of the diagnosed cases were positive for L. interrogans. Differences in density of the horse population and in the presence of wild animal reservoirs may explain the lower prevalence of leptospirosis in our region compared to others.

Neospora caninum, an apicomplexa protozoan parasite, is recognized as a major cause of abortion in cattle [START_REF] Dubey | Neosporosis in cattle[END_REF][START_REF] Dubey | Diagnosis of bovine neosporosis[END_REF][START_REF] Loo | Retrospective study of factors associated with bovine infectious abortion and perinatal mortality[END_REF]. In horses, an association between equine reproductive disorders and N. caninum infec-tion has also been suggested [START_REF] Villalobos | Association between the presence of serum antibodies against Neospora spp. and foetal loss in equines[END_REF][START_REF] Veronesi | Neospora spp. Infection associated with equine abortion and/or stillbirth rate[END_REF]. The number of animals with elevated anti-Neospora specific antibodies is higher in aborting mares [START_REF] Pitel | Investigation of Neospora sp. antibodies in aborted mares from Normandy, France[END_REF][START_REF] Kligler | Seroprevalence of Neospora spp. among asymptomatic horses, aborted mares and horses demonstrating neurological signs in Israel[END_REF]. [START_REF] Léon | Molecular detection of Coxiella burnetii and Neospora caninum in equine aborted foetuses and neonates[END_REF] detected N. caninum by molecular analysis in 1% of aborted foals in France. PCR for N. caninum was positive in 23% of brain specimen from aborted foals in a study by [START_REF] Veronesi | Neospora spp. Infection associated with equine abortion and/or stillbirth rate[END_REF]. In the present study, N. caninum was detected in only one case in the brain. However, as this parasite is very easily transmitted from infected mother to fetus, it can also be found in cases where N. caninum is not the cause of abortion [START_REF] Dubey | Diagnosis of bovine neosporosis[END_REF]. Veronesi et al. ( 2008) also reported infections with N. caninum in aborted foals where the abortion event was assumed as primarily caused by EHV-1. Histology was not performed in this study, which could have been useful in order to correlate results of the PCR with the detection of N. caninum tachyzoites in fetal tissues [START_REF] Dubey | Diagnosis of bovine neosporosis[END_REF]. However, detecting these protozoal agents in brain specimens by histological examination may be challenging [START_REF] Veronesi | Neospora spp. Infection associated with equine abortion and/or stillbirth rate[END_REF].

In the present study, no EVA virus was detected. However, a serological study has shown that 10% of tested horses in Belgium presented specific antibodies to EVA (unpublished data, Lauwers, 1997(unpublished data, Lauwers, -1998)). In 2000, a severe outbreak of EVA infection happened in a Belgian Arabian stud farm [START_REF] Van Der Meulen | An outbreak of equine viral arteritis abortion in belgium[END_REF]. [START_REF] Gryspeerdt | Neonatal foal death due to infection with equine arteritis virus in Belgium[END_REF] reported also a small outbreak of neonatal death caused by EVA infection in Belgium.

Chlamydophila abortus, C. psittaci or C. pecorum were not detected in this study. There is only limited information available regarding the role of Chlamydophila spp in equine abortion. Abortions associated with C. psittaci have been reported in both Australia and Europe (France, Germany, Hungary and Switzerland) [START_REF] Macleay | A scoping review of the global distribution of cause and syndromes associated with mid-to late-term pregnancy loos in horses between 1960 and 2020[END_REF]. A high prevalence of C. psittaci in equine fetal membranes has been demonstrated in Hungary [START_REF] Szeredi | High prevalence of chlamydial (Chlamydophila psittaci) infection in foe-tal membranes of aborted equine foetuses[END_REF]. Reports based on molecular tests also associated C. psittaci with equine abortion [START_REF] Henning | Demonstration of Chlamydia from an equine abortion[END_REF]Telocnik et al., 2018).

Placentitis is known as one of the main causes of infectious ASND in horses [START_REF] Brinsko | Pregnancy loss[END_REF]. According to a study by [START_REF] Macleay | A scoping review of the global distribution of cause and syndromes associated with mid-to late-term pregnancy loos in horses between 1960 and 2020[END_REF], placentitis accounted for 16% of reported pregnancy loss in mares worldwide. However, this percentage may be an underestimation as the diagnosis of placentitis in most studies is only based on gross placental examination without further cause-specific laboratory tests or histology [START_REF] Macleay | A scoping review of the global distribution of cause and syndromes associated with mid-to late-term pregnancy loos in horses between 1960 and 2020[END_REF]. Histopathology has been described as the standard tool for definitive diagnosis of placentitis [START_REF] Cummins | Ascending placentitis in the mare: a review[END_REF]. Unfortunately, in the present study, no microbiological or histological examination on placental tissue was performed because of autolysis, contamination or scavenger damage in multiple cases. Studies in the US [START_REF] Tengelsen | A 12-year retrospective study of equine abortion in Michigan[END_REF] and France (Bain et al., 1996) using microbiologic examination of the placenta, showed an infection by S. zooepidemicus in 7% and 24% of equine abortions, respectively.

No analyses were performed to detect nocardioform placentitis. This specific type of placentitis is associated with gram-positive branching actinomycetes. The disease causes episodic abortions and weak neonates in horses, primarily in the USA [START_REF] Fedorka | Clinical, pathologic, and epidemiologic features of nocardioform placentitis in the mare[END_REF]. However, nocardioform placentitis is rather uncommon in Europe [START_REF] Cattoli | First case of equine nocardioform placentitis caused by Crossiella equi in Europe[END_REF], and therefore, this pathogen was not included in the current study.

Non-infectious conditions such as umbilical cord abnormalities were rarely detected in the present study. In the literature, umbilical cord disorders in horses have been reported with an incidence ranging from 0.8 to 46% [START_REF] Smith | A survey of equine abortion, stillbirth and neonatal deaths in UK from 1988 to 1997[END_REF][START_REF] Tengelsen | A 12-year retrospective study of equine abortion in Michigan[END_REF][START_REF] Ricketts | Equine abortion[END_REF][START_REF] Weber | Causes of abortion in horses -overview of the literature and own evaluations[END_REF]. Macleay (2022) reported umbilical cord torsion mainly in Europe and North America, in both mid and late gestation. In the current study, which included predominantly warmblood foals, 4% of the cases showed umbilical strangulations. Important to mention is that in half of these cases, no infectious pathogen was detected. In the other half of the cases, an infection with S. zooepidemicus was diagnosed. During agony, umbilical strangulation can occur, secondary to the primary cause of compromise. Twisted umbilical cords without vascular compromise can often be seen in aborted foals [START_REF] Vandenplassche | The twisted umbilical cord: an expression of kinesis of the equine foetus?[END_REF]. Umbilical strangulation in this study was only mentioned if edema, congestion or hemorrhage of the cord was present. Umbilical or placental disorders were possibly underestimated in this study due to the submission of incomplete placental tissue specimens and/ or the absence of the umbilical cord in multiple cases.

Miscellaneous conditions such as dystocia or twinning could be missed due to the limited availability of history in several cases, and the less likelihood of such cases to be submitted as the main aim of submission to a diagnostic lab is the exclusion of infectious causes. Twin pregnancy was the main cause of noninfectious abortion (21.1%) in the study by [START_REF] Weber | Causes of abortion in horses -overview of the literature and own evaluations[END_REF]. Late-term twin pregnancies are however quite rare in Belgian horses because of routine ultrasonographic examination in early pregnancy.

CONCLUSION

In this study, the importance of PCR tests is confirmed, as advanced autolysis or strenuous culture protocols are often limiting factors in standard investigation protocols of equine ASND.

EHV-1 was the predominant infectious cause of ASND in Belgian horses, which is comparable to results worldwide. The high prevalence of S. equi subspecies zooepidemicus in this study highlights the importance of molecular analysis for this pathogen in future studies. Given the detection of C. burnetii and Leptospira spp., these zoonotic pathogens should be taken into account as possible causes of equine ASND.

As an infectious agent was detected in only 37% of the cases, additional analyses to detect both infectious and non-infectious causes of ASND in horses are required to increase the diagnostic rate.

Figure 1 .

 1 Figure 1. Flow chart showing the initial and final study population and the reasons of exclusion.

Figure 2 .

 2 Figure 2. Enlarged liver with white necrotic foci on the capsular surface (rectangular) of an aborted foal diagnosed with EHV-1. These typical lesions were only seen in half of EHV-1 positive cases.

Table 1 . Results of diagnostic analyses in 105 cases of equine abortion, stillbirth and neonatal death. In 37% of the cases (39/105), at least one infectious abortifacient agent could be detected (= diagnosed cases).

 1

	Pathogen	% positive	% positive of	Sample type
		(n=105)	diagnosed cases (n=39)
	EHV-1	18.1	48.7	Fetal/foal liver and lung
	S. zooepidemicus	10.5	28.2	Fetal/foal liver and lung
	C. burnetii	6.7	17.9	Fetal/foal liver and lung
	L. interrogans	1.0	2.6	Fetal/foal liver and lung
	N. caninum	1.0	2.6	Fetal/foal brain
	Chlamydophila spp.	0	0	Placenta
	EHV-4	0	0	Fetal/foal liver and lung
	EVA	0	0	Fetal/foal liver and lung

EHV-1 = equine herpesvirus-1; S. zooepidemicus = Streptococcus equi subspecies zooepidemicus; C. burnetii = Coxiella burnetii; L. interrogans = Leptospira interrogans; N. caninum = Neospora caninum; EHV-4 = equine herpesvirus-4; EVA = equine viral arteritis only EHV-suspected cases for further investigation.

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