Andreas Günther
email: guenther@zib.de

Hans Lamecker

Martin Weiser

Direct LDDMM of Discrete Currents with Adaptive Finite Elements

We consider Large Deformation Diffeomorphic Metric Mapping of general m-currents. After stating an optimization algorithm in the function space of admissable morph generating velocity fields, two innovative aspects in this framework are presented and numerically investigated: First, we spatially discretize the velocity field with conforming adaptive finite elements and discuss advantages of this new approach. Second, we directly compute the temporal evolution of discrete m-current attributes.

Introduction

The Large Deformation Diffeomorphic Metric Mapping (LDDMM) approach initiated in [START_REF] Dupuis | Variational problems on flows of diffeomorphisms for image matching[END_REF][START_REF] Trouvé | An infinite dimensional group approach for physics based models in pattern recognition[END_REF] has attracted considerable attention over the last few years in medical imaging. It allows to match highly deformed objects and as such is capable of performing inter-individual registration. LDDMM constructs a space mapping by evolving a displacement field along a velocity field, we call wind. Depending on the regularity of the wind, either diffeomorphisms [START_REF] Beg | Computing large deformation metric mappings via geodesic flows of diffeomorphisms[END_REF][START_REF] Marsland | Constructing diffeomorphic representations for the groupwise analysis of non-rigid registrations of medical images[END_REF] or homeomorphisms [START_REF] Younes | Shapes and diffeomorphisms[END_REF] of the embedded space can be obtained. Thus, it provides a basis for many applications of anatomical shape analysis, where a one-to-one correspondence between different spatial objects is required.

The LDDMM technique is commonly applied for matching currents [START_REF] Durrleman | Statistical models of currents for measuring the variability of anatomical curves, surfaces and their evolution[END_REF]. Currents provide a unified mathematical description of geometrical objects of dimension 0 (points), 1 (curves), 2 (surfaces) or 3 (volumes) [START_REF] Federer | Geometric measure theory[END_REF][START_REF] Morgan | Geometric measure theory[END_REF] which are embedded in R 3 . The spaces of m-currents are linear and equipped with an inner product and hence are a suitable tool for statistical shape analysis [START_REF] Durrleman | Statistical models of currents for measuring the variability of anatomical curves, surfaces and their evolution[END_REF]. The induced norm provides a similarity measure for matching of source and target objects.

In [START_REF] Cotter | The variational particle-mesh method for matching curves[END_REF] a particle-mesh method has been applied to 1-currents in 2D. Therein curves of same topology represented by a parameterization were approximated by a finite point set without any tangential information regardless of some geometrical level of detail for matching. Although attaching a momentum vector at each vertex of the source shape can be proven to be the optimal wind parameterization, Cotter proposes to use cubic B-splines on a fixed grid. In an enclosed efficiency discussion this sub-optimal parameterization is still competitive due to the simpler wind structure. However the numerical scheme there relies on equidistant cartesian grids with constant diffusity in order to apply FFT techniques.

In contrast to [START_REF] Cotter | The variational particle-mesh method for matching curves[END_REF] we apply in 3D the Orthogonal Matching Pursuit (OMP) proposed in [START_REF] Durrleman | Statistical models of sets of curves and surfaces based on currents[END_REF] to obtain a sparse representation for general m-currents at a given geometric resolution (spectral length) in terms of a sum of discrete Dirac delta m-currents. Currently, the LDDMM evolution of this representation is only done indirectly via an approximative scheme [START_REF] Durrleman | Statistical models of currents for measuring the variability of anatomical curves, surfaces and their evolution[END_REF]Rem. 4.13] as depicted in Figure 1 (left) for m = 2, because it would otherwise require the computation of the Jacobian of the diffeomorphism, which is a challenge when discretizing the wind using Gaussian kernels [START_REF] Glaunès | Large deformation diffeomorphic metric curve mapping[END_REF][START_REF] Glaunès | Diffeomorphic matching of distributions: a new approach for unlabelled point-sets and sub-manifolds matching[END_REF][START_REF] Vaillant | Surface matching via currents[END_REF]. Furthermore, one looses the connectivity of the input mesh structure in this case. However, this is not a significant problem, since the connectivity can be recovered by applying the final displacement field to the input meshes afterwards. In this paper, we study the direct evolution of Dirac delta m-currents (right of Figure 1). We show that the direct approach allows to uniformly treat mcurrents for m = 0, . . . , 3 (Sec. 2), which to the best of our knowledge has not been shown before. We show also how to compute the Jacobian in this setting by using finite elements (FEM) to discretize the wind in the LDDMM framework (Sec. 3). Since the compactly supported basis functions are fixed in space the computation is significantly simplified. Although equidistant grids can of course be incorporated, we also in contrast to [START_REF] Cotter | The variational particle-mesh method for matching curves[END_REF] consider locally refined meshes and exploit the decoupling of the wind and current discretization for an adaptive current representation, giving a significant reduction of degrees of freedom (Sec. 4). Based on our results, the increased spatial flexibility of adaptive FEM may be exploited in the future by implementing hierarchical schemes, error estimators and non-constant spatial anisotropic diffusity.

t = 0 t = T t = 0 x0 n0 xt t = T xT nT

Continuous Matching Problem

For given shapes S, T ⊂ R 3 we aim at constructing a sufficiently smooth bijection φ of R 3 such that the distance between φ(S) and T is minimal. Here we fix the formalisms to describe the matching problem as an optimization task.

Currents

Currents are mathematical tools suited for describing geometrical objects such as points, space curves, surfaces and volumes embedded in R 3 . Their precise definition from [START_REF] Federer | Geometric measure theory[END_REF][START_REF] Morgan | Geometric measure theory[END_REF] requires notation for differential forms taken from [START_REF] Morita | Geometry of differential forms[END_REF]. Let for m = 0, 1, 2, 3 the set

D m := C ∞ c (R 3 , Λ m R 3 ) denote the vector space of all C ∞ differential m-forms on R 3 with compact support. A m-current is an element of D m , the dual space of D m . The elementary Dirac delta m-currents δ u1∧...∧um x ∈ D m act on ω ∈ D m as δ u1∧...∧um x (ω) = ω(x)(u 1 ∧ ... ∧ u m ) .
Following the discussion in [5, Sect. 1.5.1] it turns out that for the purpose of matching currents the testspace of all C ∞ differential m-forms is not suited due to a missing bound in variation. Moreover the space D 0 can be identified with scalar C ∞ c functions on R 3 . For m = 1 and m = 2 the space D m is isomorph to the space of vector-valued C ∞ c functions from R 3 to R 3 . An element of D 3 can be written as a scalar C ∞ c function times the determinant form on R 3 . Both aspects motivate the use of Reproducible Kernel Hilbert Spaces (RKHS) W m as testspaces.

Definition 1. Let d m = 1 for m ∈ {0, 3} and d m = 3 for m ∈ {1, 2}. For m = 0, 1, 2, 3 let W m denote the dense span of d m -vectorfields of the form ω(x) = k m (x, y)a, where x, y ∈ R 3 , a ∈ R dm and k m (x, y) = exp(-x -y 2 /σ 2 m ). The space W m can be equipped with the inner product k m (•, x)a, k m (•, y)b W m = a * k m (x, y)b.
Here the symbol * denotes the transpose operation.

An m-current in R 3 is a continuous linear functional on W m . W m denotes the vector space of all m-currents in R 3 .

For x ∈ R 3 and attribute a ∈ R dm we define the elementary Dirac delta m-currents δ a

x ∈ W m acting on ω ∈ W m as δ a x (ω) = a * ω(x).

The above inner product induces a norm on W m , which can be computed efficiently via FGT even for a large number of linear combinations of the above basis functions. The chosen Gaussian kernel k m can be considered as Green's function for some differential operator L W (see [START_REF] Beg | Computing large deformation metric mappings via geodesic flows of diffeomorphisms[END_REF][START_REF] Durrleman | Statistical models of sets of curves and surfaces based on currents[END_REF][START_REF] Glaunès | Large deformation diffeomorphic metric curve mapping[END_REF]). With the above objects at hand the Riesz representation theorem provides a unique operator K m W : W m → W m reflecting the canonical isometry between W m and W m defined via

K m W f, g W m = f, g Wm,W m = f (g)
for all f ∈ W m and g ∈ W m . It provides for the m-current S m the Riesz representant K m W S m as unique d m -vectorfield on R 3 .

Homeomorphisms and Diffeomorphisms

Let Ω be an open bounded subset of R 3 and consider functions v t : Ω → R 3 that vanish on ∂Ω. For given final time T > 0 and a time-dependent wind v = (v t ) t∈[0,T ] we consider the temporal evolution of the identity map

∂φ v t ∂t = v t (φ v t ) with φ v 0 (x) = x . ( 1
)
In what follows it will be useful to define the trajectory x t := φ v t (x) for some fixed space point x ∈ R 3 and the map

φ v st := φ v t • (φ v s ) -1
, describing the movement of a particle starting in x at time s towards φ v st (x) at time t. It is well known (see [START_REF] Younes | Shapes and diffeomorphisms[END_REF]Thm. C.3]), that (1) is uniquely solvable when for some x 0 ∈ Ω the integral

T 0 v t (x 0 ) R 3 +Lip(v t ) dt is bounded. Furthermore its solution φ v t : R 3 → R 3 is a homeomorphism of Ω for all times t ∈ [0, T ].
Under more restrictive assumptions onto the spatial smoothness of the wind, i.e. 1) is even a diffeomorphism of Ω for all times t ∈ [0, T ] (see [START_REF] Younes | Shapes and diffeomorphisms[END_REF]Thm. 8.7]). For convenience we look for the wind v t in some Hilbert space V . Such spaces can be constructed by defining inner products associated to differential operators. Let therefore L : V → L 2 (R 3 ) be a differential operator and equip the Hilbert space V with the inner product

v t ∈ C 1 0 (Ω, R 3 ) ∀t ∈ [0, T ] and T 0 v t 1,∞ dt < ∞ the unique solution of (
v t , g V = Lv t , Lg L 2 = L * Lv t , g V * ,V .
Here L * denotes the adjoint operator. For this work we use

S := L * L = (-div(σ 2 V ∇) + I) k = (-σ 2 V Δ + I) k (2)
and k = 1 or k = 2 giving the Sobolev spaces H k (see [START_REF] Glaunès | Large deformation diffeomorphic metric curve mapping[END_REF]). For given f ∈ V * we consider solutions v t ∈ V of Sv t = f with homogeneous Dirichlet boundary conditions for v t (and v t if k = 2). Here the real parameter σ V > 0 balances between smoothing and data fitting of the right hand side f . For other choices of L * L and boundary conditions see [START_REF] Marsland | Constructing diffeomorphic representations for the groupwise analysis of non-rigid registrations of medical images[END_REF]. Dealing with natural boundary conditions is also possible, but requires a sufficiently large domain to keep all trajectories therein. Analogous to K m W we introduce the isometry operator [START_REF] Glaunès | Large deformation diffeomorphic metric curve mapping[END_REF][START_REF] Glaunès | Diffeomorphic matching of distributions: a new approach for unlabelled point-sets and sub-manifolds matching[END_REF][START_REF] Vaillant | Surface matching via currents[END_REF][START_REF] Vaillant | Statistics on diffeomorphisms via tangent space representations[END_REF].

K V : V * → V . A mathematically equivalent approach of constructing V consists in defining K V via the Green's function k V (x, y) of L * L, see for instance

Diffeomorphic Deformation of Currents

For m = 0, 1, 2, 3 let currents S m ∈ W m be given. Let φ denote a diffeomorphism on R 3 and d x φ the Jacobian of φ at x. The pushforward φ (S m ) ∈ W m of S m under φ is rigorously defined in [START_REF] Morita | Geometry of differential forms[END_REF] via the pullback of differential forms. For our purpose it is sufficient to mention that if S m is associated to a submanifold in R 3 its pushforward φ (S m ) under φ corresponds to the deformed submanifold φ(S m ). This important property justifies to write also φ(S m ) ∈ W m . The explicitly calculated pushforwards for elementary Dirac delta m-currents taken from [5, Table 1.2] are given in Table 1.

Table 1. Pushforwards of Dirac delta m-currents under φ m = 0 d0 = 1 c ∈ R φ (δ c x ) = δ c φ(x) m = 1 d1 = 3 τ ∈ R 3 φ (δ τ x ) = δ dxφ(τ ) φ(x) m = 2 d2 = 3 n ∈ R 3 φ (δ n x ) = δ |dxφ|dxφ - * (n) φ(x) m = 3 d3 = 1 ρ ∈ R φ (δ ρ x ) = δ |dxφ|ρ φ(x)
Let some wind v be given and consider the family (φ v t ) t of diffeomorphisms generated via [START_REF] Beg | Computing large deformation metric mappings via geodesic flows of diffeomorphisms[END_REF]. The following theorem describes the direct evolution of mcurrent attributes a m ∈ R dm under (φ v t ) t , where denotes the time derivative.

x t = v t (x t ) with x(0) = x 0 τ t = (d xt v t )τ t with τ (0) = τ 0 n t = n t tr(d xt v t ) -(d xt v t ) * n t with n(0) = n 0 ρ t = ρ t tr(d xt v t ) with ρ(0) = ρ 0 .
Proof.

Abbreviating J t = d x0 φ v t and A t = d xt v t there holds (see [1]) J t = A t J t with J(0) = I 3 .
Observing the evolution of the Wronskian [13, Thm. 2.14] or via Jacobi's formula one obtains

|J t | = |J t |tr J -1 t J t = |J t |tr J -1 t A t J t = |J t |tr(A t ),
where tr(A) denotes the trace of a matrix A and A - * = (A -1 ) * . Now from Table 1 we read out

x t = φ v t (x 0 ) , τ t = J t τ 0 , n t = |J t |J - * t n 0 , ρ t = |J t |ρ 0 .
Differentiation of the above equations with respect to t yields

x t = φ v t (x 0 ) = v t (φ v t (x 0 )) = v t (x t ) τ t = J t τ 0 = A t J t τ 0 = A t τ t n t = |J t | J - * t n 0 + |J t |(J - * t ) n 0 = |J t |tr(A t )J - * t n 0 -|J t |A * t J - * t n 0 = n t tr(A t ) -A * t n t ρ t = |J t | ρ 0 = |J t |tr(A t )ρ 0 = ρ t tr(A t ),
which proves the assertion.

Remark 1. The authors emphasize the striking advantage that Theorem 1 enables to find the final position and attribute of a Dirac delta m-current without computing the Jacobian of the deformation. The appearing ODEs only involve the Jacobian of the velocity fields, which will be given in a closed form in any case.

Optimization Problem in Function Space

Let source S m ∈ W m and target current T m ∈ W m be given for m = 0, . . . , 3. For given wind v we define the deformed current S m t := φ v t (S m ) at time t. Matching means the minimization of the distance of the deformed source current at final time S m T with its target current T m , i.e. minimizing the dual norm

φ v T (S m ) -T m Wm = S m T -T m
Wm in the space of m-currents.

Given a regularization parameter γ > 0 and matching weights ω m ≥ 0 we consider for v ∈ L 2 ([0, T ], V ) the following optimization problem:

J(v) := γ T 0 v t 2 V dt + 3 m=0 ω m φ v T (S m ) -T m 2 Wm → min . ( 3
)
Here the first summand involves the kinetic energy of the wind. The existence of a solution for (3) is proven in [START_REF] Glaunès | Transport par difféomorphismes de points, de mesures et de courants pour la comparaison de formes et l'anatomie numérique[END_REF], however it is generally not unique [START_REF] Cao | Large deformation diffeomorphic metric mapping of vector fields[END_REF]. Following [START_REF] Glaunès | Diffeomorphic matching of distributions: a new approach for unlabelled point-sets and sub-manifolds matching[END_REF] the gradient of J in L 2 ([0, T ], V ) at fixed v is given by (∇J) t = 2γv t +2K V (f t ), where f t ∈ V * is defined by

f t (u) = 3 m=0 ω m S m t , ∇(K m W (S m T -T m ) • φ v tT ) * u Wm,W m ∀u ∈ V .
For further discussion concerning the choice of the gradients metric we refer the reader to [START_REF] Beg | Computing large deformation metric mappings via geodesic flows of diffeomorphisms[END_REF]. With the above quantities at hand one is able to state a steepest descent optimization algorithm in the function space of velocity fields v.

Discrete Matching Problem

Discretization of the Wind by Finite Elements (FE)

In the field of optimal current matching mainly wind discretizations of the form

v t (x) = j k V (x j,t , x)α j,t (4)
have been considered. Here α j,t ∈ R 3 are the time-dependent momentum vectors and k V denotes a Gaussian kernel with some global kernel parameter σ V > 0, describing the coherent movement of neighboring particles. In order to apply Fast Gauss Transform (FGT) for efficient evaluation, σ V is necessarily a constant. Although (4) can be proven to be the optimal wind parameterization, the spatial movement of non-compactly supported basis functions along trajectories x j,t may cause numerical difficulties. Too small distances between them cause a redundant or badly conditioned description of the velocity field while the absence of trajectories in a part of the domain produces almost no wind there for small kernel sizes. The trajectory density varies during optimization and hence is difficult to control. Because the trajectories' starting points are the spatial components of the Dirac delta source currents the number of trajectories is fixed and hence a notion of adaptivity for the velocity field can hardly be introduced. Finally, as mentioned in Sect. 2.2, C ∞ smoothness is not required to solve the evolution equation.

In [START_REF] Risser | Simultaneous fine and coarse diffeomorphic registration: Application to atrophy measurement in alzheimer's disease[END_REF] and [START_REF] Sommer | A multi-scale kernel bundle for LDDMM: Towards sparse deformation description across space and scales[END_REF], some of the above mentioned drawbacks are overcome by incorporating multiple kernel shapes at different scales σ V .

Similar to the particle-mesh method proposed in [START_REF] Cotter | The variational particle-mesh method for matching curves[END_REF], we follow another suboptimal approach completely decoupling the discretization of the space of mcurrents W m from the spatial velocity space V . Keeping in mind that fast point evaluation of the wind is essential for performance, we consider adaptive hexahedral grids for Ω with hanging nodes saved as an octree. Over such hexahedral grids we construct either C 1 conforming Hermite finite elements of third order or simpler C 0 conforming Lagrange finite elements of first order. The wind for fixed time t ∈ [0, T ] in the FE basis {ϕ j } j takes the form

v t (x) = n j=1 ϕ j (x)α j,t .
(

In contrast to radial basis functions, locally constant functions are contained in the ansatz space and allow to represent local or even global translations with few degrees of freedom (DOF). Due to the compactly supported basis functions there is no need for an approximate evaluation like FGT with further unknown tolerance parameters. Since the basis functions are fixed in space, the underlying mesh provides a natural clustering which can be exploited via a smart parallel octree search algorithm for point evaluation. Furthermore this approach provides a multilevel wind hierarchy with a fraction of DOFs on the coarsest mesh level completely decoupled from the m-current discretization. These advantages also appear in the particle-mesh method with tensor-products of cubic B-splines for instance. But since we do not apply FFT for wind evaluation, we are more flexible with adaptive meshes and do not require a box domain. Moreover non-constant anisotropic diffusity σ V (x) ∈ R 3×3 may be incorporated in future.

A difficulty arises in the computation of the L 2 ([0, T ], V )-gradient. It permanently involves the solution of a second (k = 1) or fourth (k = 2) order elliptic PDE in every time-step and every iteration. It is clear that one should employ suited preconditioners and / or multigrid solvers. Using existing FE libraries limits implementation overhead. We chose libMesh [START_REF] Kirk | libMesh: A C++ Library for Parallel Adaptive Mesh Refinement/Coarsening Simulations[END_REF], which provides conforming C 1 finite elements on adaptive hexahedral meshes.

The development of adaptive mesh refinement is beyond the scope of this work. Here, we provide a proof of concept that adaptive grids can easily be incorporated. Therefore we simply geometrically refine near S m ∪T m considered as subsets of R 3 equally for all times. More sophisticated error indicators suggesting refinements could be the scalar fields

|K m W (S m T -T m ) • φ v tT |, |v t | or |L * Lv t |.
The latter one measures the smoothness of v t . Moreover thinking of hierarchical error estimators one could compute

v 1 t -I h v 1 t L 2 (Q) or even v 1 t -v 0 t L 2 (Q) on hexahedrons Q,
where v i denote the numerical solutions for the C i conforming FE discretization and I h is the usual Lagrange interpolation operator.

All appearing ODEs are numerically integrated via the explicit method of Heun on an equidistant decomposition of the time interval [0, T ].

Current Compression and Direct Evolution

For approximating a m-current S m ∈ W m as Ŝm = sm i=1 δ ai xi ∈ W m we use the Orthogonal Matching Pursuit (OMP) proposed in [START_REF] Durrleman | Statistical models of sets of curves and surfaces based on currents[END_REF]. This method iteratively selects the most important points x i and computes corresponding attributes a i (i.e. c i , τ i , n i , ρ i ) of a general m-current via a greedy algorithm. It has the advantage of compressing the current information for a characteristical spectral length σ m > 0 towards a fraction. This enables the design of highly efficient numerical solution algorithms. The approximation error in OMP is controlled by a threshold parameter and the grid size of a uniform testgrid.

The obvious drawback of loosing the connectivity between vertices (for m ≥ 1) can be compensated by applying the obtained optimal diffeomorphism to all connected vertices whenever it is required. This only requires one additional forward flow computation at the end.

In [START_REF] Vaillant | Surface matching via currents[END_REF]Sec. 3.2] two methods to deform a 2-current Ŝ2 under a family of diffeomorphisms (φ v t ) t are described. In contrast to all previous work, we will pursue the direct approach motivated by Theorem 1. For 2-currents, only 1 instead of 3 trajectories is needed to evolve the normal n 0 (Fig. 1). In general, the direct approach requires only one trajectory per attribute, hence decreasing the number of variables in the computation, whereas in the indirect case an artificial m-simplex with m + 1 vertices is attached.

Remark 2. To quote Rem. 4.13 in [START_REF] Durrleman | Statistical models of currents for measuring the variability of anatomical curves, surfaces and their evolution[END_REF] the direct evolution of current attributes is closer to the analytical concept of currents and is particularly suited for OMP, where no connectivity between the points is provided. But [START_REF] Durrleman | Statistical models of currents for measuring the variability of anatomical curves, surfaces and their evolution[END_REF] indicates the need of Jacobi matrices (as they arise in Theorem 1) as a disadvantage for numerical implementation. At least for the gradient computation in the next section we benefit from the simpler structure of v t in [START_REF] Durrleman | Statistical models of currents for measuring the variability of anatomical curves, surfaces and their evolution[END_REF], which in Lemma 2 enables easy evaluations of d xt v t = n j=1 α j,t ∇ϕ j (x t ) * and hence tr(d xt v t ), (d xt v t )w and (d xt v t ) * w for a vector w ∈ R 3 . Note that all sums over j are local sums due to the compact support of the basis functions ϕ j .

Discrete Optimization Problem and its Gradient

Let a(•, •) denote the bilinear form corresponding to the elliptic differential operator S from (2). We define the sparse symmetric, positive definite matrix S := [a(ϕ i , ϕ j )] n i;j=1 using the FE basis {ϕ j } j from Sect. 3.1. Moreover we introduce the block vectors α t := [α i,t ] n i=1 , x t := x m t := [x i,t ] sm i=1 and a t := a m t := [a i,t ] sm i=1 . This notation allows to write the matching terms as

E m = E m (x T , a T ) = φ v T ( Ŝm ) -T m 2 Wm = sm i=1 δ ai,T xi,T - rm j=1 δ bj yj 2 Wm .
Finally the discrete form of the current matching problem (3) is

Ĵ(α t ) := γ T 0 v t (α t ) 2 V dt + 3 m=0 ω m φ v T ( Ŝm ) -T m 2 Wm → min or even shorter via (5) and v t (α t ) 2 V = a(v t (α t ), v t (α t )) = α * t Sα t Ĵ (α t ) = γ T 0 α * t Sα t dt + 3 m=0 ω m E m (x T , a T ) → min . ( 6
)
The analytical computation of the gradient at given α t becomes manageable though the simpler wind representation [START_REF] Durrleman | Statistical models of currents for measuring the variability of anatomical curves, surfaces and their evolution[END_REF]. Numerically the computation is more involved due to presence of Hessians of basis functions. But these are easily provided via the already mentioned libMesh library.

Theorem 2. The gradient of Ĵ in the L 2 -metric is

(∇ Ĵ ) t = 2γSα t + 3 m=0 ω m ((ϕ m t ) * η m t + (∂ α g m t ) * ζ m t ), (7)
with

ϕ m t = [ϕ j (x i,t )I dm ] i=1...sm;j=1...n ζ m t = ∇ aT E m + T t (∂ a g m s ) * ζ m s ds η m t = ∇ xT E m + T t (∂ x g m s ) * ζ m s ds .
The proof is postponed to the appendix. The remaining quantities

∇ xT E m , ∇ aT E m , ∂ α g m t , ∂ x g m t
and ∂ a g m t from Theorem 2 for each m are specified in the next two lemmas.

Lemma 1. Let f m (x) = sm i=1 k m (x i,T , x)a i,T - rm j=1 k m (y j , x)b j .
There hold

∇ xT E m = [2(d xi,T f m (x i,T )) * a i,T ] sm i=1 and ∇ aT E m = [2f m (x i,T )] sm i=1 .
Proof.

(∂ xT E m )η = 2 ∂ xT sm i=1 δ ai,T xi,T η (f m ) = 2a * T (d xT f m (x T )) η ∇ xT E m = 2(d xT f m (x T )) * a T = 2 sm i=1 (∇ 2 k m (x i,T , x T ))a * i,T - rm j=1 (∇ 2 k m (y j , x T ))b * j a T (∂ aT E m )η = 2 ∂ aT sm i=1 δ ai,T xi,T η (f m ) = 2η * f m (x T ) .
Lemma 2. For g m t in [START_REF] Kirk | libMesh: A C++ Library for Parallel Adaptive Mesh Refinement/Coarsening Simulations[END_REF] their sparse Jacobians are given via , where H ϕj (x i,t ) denote the Hessian of ϕ j at x i,t .

g 0 t = 0 ∂ α g 1 t = (τ * i,
Proof. The proof for all cases of m can easily be adapted from the case m = 2. For this choice the derivatives of g 2 t follow from direct calculations starting with

g 2 t = g 2 (α t , x t , n t ) = diag tr(d xi,t v t )I 3 -(d xi,t v t ) * s2 i=1 n t = diag n j=1 (α * j,t ∇ϕ j (x i,t ))I 3 -∇ϕ j (x i,t )α * j,t s2 i=1 n t = n j=1 n i,t (α * j,t ∇ϕ j (x i,t )) -∇ϕ j (x i,t )(α * j,t n i,t ) s2 i=1 , where diag[v] = [δ ij v i ] s i;j=1
for v ∈ R s and δ ij denotes the Kronecker delta.

Corollary 1.

If ω m = 0 for m > 0 Theorem 2 simply provides

(∇ Ĵ) t = 2γSα t + ω 0 (ϕ 0 t ) * ∇ xT E 0 .
Remark 3. The L 2 ([0, T ], V )-gradient of Ĵ is immediately obtained by applying S -1 from the left in equation ( 7).

Numerical Experiments

Since the numerical implementation is not yet fully tested, we postpone the investigation of the cases m > 0 and only consider the case m = 0, i.e. ω i = δ i0 . The surfaces S and T are depicted in Fig. 2. To both of them we apply the OMP with σ 0 = 8 towards Ŝ with s 0 = 1746 points and T with r 0 = 2141 points, which are sketched as set of spheres of diameter 8 in Fig. 3. A quantitative comparison between all different wind discretizations is issue of Table 2. Therein the column DOFs denote the number of freely choosable vectors α j,t for fixed t. All methods provide acceptable matches with respect to the fixed level of detail σ 0 = 8. Especially the surface S T corresponding to C 1 wind is also visually closest to T although it is obtained via less wind DOFs compared to the approach from Exoshape. This fact stresses the potential of suboptimal, albeit simpler wind parameterizations and decoupling the discretization of the spaces W m from V . Combining ( 9), [START_REF] Glaunès | Transport par difféomorphismes de points, de mesures et de courants pour la comparaison de formes et l'anatomie numérique[END_REF] Collecting ( 8), ( 13), ( 14) and ( 15) yields the assertion.

Fig. 1 .

 1 Fig. 1. Usual discrete 2-current deformation (left) versus the direct approach (right)

Fig. 2 .Fig. 3 .

 23 Fig. 2. S (dark green), T (light red) Fig. 3. Ŝ (green), T (red), grid

Fig. 4 . 5 . 6 .

 456 Fig. 4. ST for C 0 wind Fig. 5. ST for C 1 wind Fig. 6. ST for C ∞ wind

t 0 F 0 Fu 0 F 0 FF 0 FF

 00000 ut ((∂ α g u ) αu + (∂ x g u )x u ) du = t ut (∂ α g u ) αu du + t 0 ut (∂ x g u )ϕ s αs ds du = t ut (∂ α g u ) + t u st (∂ x g s ) ds ϕ u αu du .In particular there holds ãT = T tT (∂ α g t ) + T t sT (∂ x g s ) ds ϕ t αt dt .[START_REF] Marsland | Constructing diffeomorphic representations for the groupwise analysis of non-rigid registrations of medical images[END_REF]

  t ∇ϕ j (x i,t ))I 3 i=1...s1;j=1...n ∂ α g 2 t = n i,t ∇ϕ j (x i,t ) * -∇ϕ j (x i,t )n * ∇ϕ j (x i,t ))I 3 -∇ϕ j (x i,t )α *

	∂ n g 2 t = diag	n j=1 (α * j,t j,t	s2 i=1
	∂ ρ g 3 t = diag	n j=1 α * j,t ∇ϕ j (x i,t )	s3 i=1
				i,t i=1...s2;j=1...n
	∂ α g 3 t = [ρ i,t ∇ϕ j (x i,t ) * ] i=1...s3;j=1...n
	∂ x g 1 t = diag	n j=1 α j,t τ * i,t H ϕj (x i,t )	s1 i=1
	∂ x g 2 t = diag	n j=1 n i,t (α * j,t H ϕj (x i,t )) -(α * j,t n i,t )H ϕj (x i,t )	s2 i=1
	∂ x g 3 t = diag	n j=1 ρ i,t α * j,t H ϕj (x i,t )	s3 i=1
	∂ τ g 1 t = diag	n j=1 α j,t ∇ϕ j (x i,t ) *	s1 i=1

Table 2 .

 2 One-sided surface distances between ST and T for m = 0Proof. First we consider the variation of the kinetic energy, i.e. ω m = 0 for all m. One directly calculates(∇ Ĵ) t = 2γSα t . (8)Let us now consider the contrary case, i.e. γ = 0. We aim to compute ∇ α E m for some fixed m.Variation of E = E m w.r.t. α t in direction αt gives Ẽm = (∂ xT E)x T + (∂ aT E)ã T .It remains to express ãt . We therefore introduce the flow dFst dt = (∂ a g t )F st with F tt = I and get

	vt	DOFs mean stddev rms max
	C 0 1•(568-133) = 435	1.18	1.14 1.64 11.33
	C 1	8•120 = 960	0.95	0.91 1.32	9.16
	C ∞	1746	0.97	0.93 1.34	9.70
	no wind, distx∈S(x, T )	5.08	3.87 6.39 21.05

t = g(α t , x t , a t ) = g t with a(0) = a 0 . (

11

)

Its variation in direction αt satisfies

ã t = (∂ α g t ) αt + (∂ x g t )x t + (∂ a g t )ã t with ã(0) = 0 .

  Since F st F ts = I and dF * st ds = -(∂ a g s ) * F * st we have in particular the integral form F * ts = I + s t (∂ a g u ) * F * us du. This helps to simplify ζ t = F * tT (∇ aT E) = I + F * sT (∇ aT E) ds = ∇ xT E +

	=	0	T	η * t ϕ t + ζ * t (∂ α g t ) αt dt .	(13)

and

[START_REF] Marsland | Constructing diffeomorphic representations for the groupwise analysis of non-rigid registrations of medical images[END_REF]

we have

Ẽm = T 0 (∂ xT E)ϕ t αt + (∂ aT E) F tT (∂ α g t ) + T t F sT (∂ x g s ) ds ϕ t αt dt = T 0 (∂ xT E) + T t (∂ aT E)F sT (∂ x g s ) ds =:η * t ϕ t + (∂ aT E)F tT =:ζ * t (∂ α g t ) αt dt T t (∂ a g s ) * F * sT ds (∇ aT E) = ∇ aT E + T t (∂ a g s ) * F * sT (∇ aT E) ds = ∇ aT E + T t (∂ a g s ) * ζ s ds (

14

)

η t = ∇ xT E + T t (∂ x g s ) * T t

(∂ x g s ) * ζ s ds .

[START_REF] Morita | Geometry of differential forms[END_REF]

Acknowledgement. We thank Stanley Durrleman from the University of Utah for the fruitful discussion and helpful suggestions at the initial phase of this paper. We further acknowledge support by DFG-Matheon Project F2. Moreover we thank the referees for their comments which helped to improve the quality of this paper.

Appendix

In the following we prove Theorem 2.

1 http://www-sop.inria.fr/asclepios/projects/Health-e-Child/ShapeAnalysis/


R Eymard

M Roussignol

A Tordeux

Convergence of a misanthrope process to the entropy solution of 1D problems

Keywords: misanthrope stochastic process, non linear scalar hyperbolic equation, entropy Young measure solution, traffic flow simulation, weak BV inequality

We prove the convergence, in some strong sense, of a Markov process called "a misanthrope process" to the entropy weak solution of a one-dimensional scalar nonlinear hyperbolic equation. Such a process may be used for the simulation of traffic flows. The convergence proof relies on the uniqueness of entropy Young measure solutions to the nonlinear hyperbolic equation, which holds for both the bounded and the unbounded cases. In the unbounded case, we also prove an error estimate. Finally, numerical results show how this convergence result may be understood in practical cases.

Introduction

Some recent models of traffic flows [START_REF] Dirk | Traffic and related self-driven many-particle systems[END_REF] are based on the notion of misanthrope process, introduced in [START_REF] Cocozza-Thivent | Processus des misanthropes[END_REF][START_REF] Andjel | Quelques compléments sur le processus des misanthropes et le processus zéro-range[END_REF]. Such a process is a stochastic Markov jump process (η t ) t≥0 , usually defined on N Z , which models the time evolution of occupation of discrete sites by a collection of identical objects. The random variable η t (n) represents the number of objects at site n ∈ Z at time t. Jumps of this process consist of jumps of an object from one site n ∈ Z to the next one n + 1 with a rate denoted b(n, n + 1) ≥ 0. The aim of this paper is to prove that the values of a misanthrope process at time t may approximate the entropy weak solution u(x, t) of a non linear hyperbolic equation, first on a bounded interval (A, B) of R (the bounded case), then on R (the unbounded case). In the bounded case, the non linear equation is

∂ t u(x, t) + ∂ x (f (u))(x, t) = 0 x ∈ (A, B), t ∈ R + , (1)
for given reals A < B, where the partial derivatives of u with respect to time and space are respectively denoted by ∂ t u and ∂ x u, with initial data u(x, 0) = u ini (x), x ∈ (A, B),

and some boundary conditions which are formally:

u(A, t) = u(t), t ∈ R + , (3)
and u(B, t) = u(t), t ∈ R + .

In ( 1)-( 4), we denote by u ini a bounded measurable function defined for all x ∈ (A, B), and by u, u bounded measurable functions defined for all t ∈ R + . We assume, without restricting the generality, that these three functions are a.e. valued in [0, U ], for a given U ∈ R + . Note that the weak sense of the boundary conditions ( 3)-( 4), deduced from Otto's works [START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF], is precised below (in this paper, the regularity on the limit is not sufficient for using the strong sense for these boundary conditions given in [START_REF] Bardos | First order quasilinear equations with boundary conditions[END_REF]).

In the unbounded case the non linear equation is

∂ t u(x, t) + ∂ x (f (u))(x, t) = 0 x ∈ R, t ∈ R + , (5)
with initial data u(x, 0) = u ini (x), x ∈ R,

where u ini denotes a bounded measurable function defined for all x ∈ R, a.e. valued in [0, U ].

Such studies of limit of stochastic process to the solution of hyperbolic equations are called "Euler hydrodynamics of attractive particle systems" in [START_REF] Andjel | Hydrodynamic equations for attractive particle systems on Z[END_REF], [START_REF] Kipnis | Scaling limits of interacting particle systems[END_REF], [START_REF] Bahadoran | Euler hydrodynamics of one-dimensional attractive particle systems[END_REF], [START_REF] Gobron | Couplings, attractiveness and hydrodynamics for conservative particle systems[END_REF]. Indeed, as described in [START_REF] Bahadoran | Euler hydrodynamics of one-dimensional attractive particle systems[END_REF], a misanthrope process (η t ) t≥0 on E = [[0, K]] Z is said to have Euler hydrodynamic limit u(x, t) if, for all continuous function Ψ from R to R + with compact support, we have, for all t ≥ 0 lim

N →∞ E   1 N y∈Z η N t (y) Ψ y N   = R Ψ(x) u(x, t) dx. (7)
We may remark that, defining the real process ν N (x, t) by ν N (x, t) = η N t (y), for all x ∈ [y/N, (y + 1)/N ), y ∈ Z and t ∈ R + , ( 7) is equivalent to the convergence for the weak-⋆ topology of L ∞ (R), and for all t ∈ R + , of the function E(ν N (•, t)) to u(•, t) as N → ∞.

It is then proved that the Euler hydrodynamic limit u(x, t) of a misanthrope process is the entropy weak solution of [START_REF] Champier | Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh[END_REF] such that the function f must check the condition

f (ρ) = b(η(0), η(1)) dµ ρ (η), ∀ρ ∈ R,
where R is the largest subset of [0, K] such that, for any ρ ∈ R, the set of all measures µ on E, which are invariant for the process and shift-invariant with η(0)dµ(η) = ρ, has an extremal element denoted by µ ρ (recall that a measure µ is invariant for the process if the process with initial distribution µ has the distribution µ for all t > 0); note that R is closed and necessarily contains 0 and K, but, as mentioned in [START_REF] Bahadoran | Euler hydrodynamics of one-dimensional attractive particle systems[END_REF], the relation [0, K] = R remains an open problem.

In this paper, although our starting point is also the notion of misanthrope process, we use a different way to approximate equations [START_REF] Andjel | Quelques compléments sur le processus des misanthropes et le processus zéro-range[END_REF] and [START_REF] Champier | Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh[END_REF], with different definitions for both the limit sense and the function f . Let us first emphasize that we will handle the bounded case (A, B) as well as the unbounded one (in the first case, the study of the boundary terms has to be done). In the bounded case, we define, for N ∈ N ⋆ , h = (B -A)/N and we discretize the interval [A, B] in constant size intervals (A + (n -1)h, A + nh), n ∈ {1, . . . , N }, each interval corresponding to a site for the stochastic process. In the unbounded case we discretize R in intervals ((n -1)h, nh), n ∈ Z of size h, each interval corresponding to a site for the stochastic process. In both cases, we discretize the continuous interval of possible values

[0, Ū ] of u(x, t) in discrete values (ki, i = 0, ..., K) with K ∈ N ⋆ and k ∈ R such that k = U K . For a given function g such that • g is Lipschitz continuous from R 2 to R + ,
hence it is assumed to be nonnegative (we denote in this paper M the smallest Lipschitz constant of g)

• g(x, y) = 0 for all (x, y) ∈ (] -∞, 0] × R) ∪ (R × [U , +∞[),
• (x, y) → g(x, y), from [0, U ] 2 to R + , is nondecreasing with respect to x and nonincreasing with respect to y,

• the function f occurring in the nonlinear hyperbolic equation ( 1) is such that f (x) = g(x, x), for all x ∈ R, we define the jump rate of the process by

b(x, y) = 1 h k g(x, y), ∀x, y ∈ R. ( 8
)
Thanks to the hypotheses on the function g, we get b(0, .) = 0 and b(., kK) = 0.

In the bounded case, we define a non-homogeneous Markov jump process on E = (k[[0, K]]) [[1,N ]] with the following possible jumps and the associated rates defined at time t by:

• jump from η to T 0,1 (η) with T 0,1 (η)(1) = η(1) + k and T 0,1 (η)(j) = η(j) for all j = 2, . . . , N , and rate b(u(t), η(1)),

• jump from η to T n,n+1 (η)

for n = 1, . . . , N -1, with T n,n+1 (η)(n) = η(n)-k, T n,n+1 (η)(n+1) = η(n+ 1)
+ k and T n,n+1 (η)(j) = η(j) for all j = 1, . . . , N different from n and n + 1, and rate b(η(n), η(n + 1)),

• jump from η to T N,N +1 (η) with T N,N +1 (η)(N ) = η(N ) -k and T N,N +1 (η)(j) = η(j) for all j = 1, . . . , N -1, and rate b(η(N ), u(t)).
Note that the process is a homogeneous Markov jump process for almost everywhere constant boundary conditions. Using the following notation

η t (0) = u(t) and η t (N + 1) = u(t), ∀t ∈ R + , (9)
the corresponding Chapman-Kolmogorov equation reads:

dEψ(η t ) dt = E N n=0 b(η t (n), η t (n + 1)) ψ(T n,n+1 (η t )) -ψ(η t ) , ∀t ∈ R + , ∀ψ ∈ D, (10)
where D is the set of all functions ψ from E to R + . We associate to the process η t the real process ν(x, t) defined from (A,

B) × R + to [0, Ū ] by ν(x, t) = η t (n) ∀x ∈ [A + (n -1)h, A + nh), ∀n ∈ [[1, N ]], ∀t ∈ R + . ( 11
)
The final result for the bounded case (Theorem 2.2) is the convergence of

T 0 B A E (|ν(x, t) -u(x, t)|) dxdt = T 0 1≤n≤N [A+(n-1)h,A+nh) E (|η t (n) -u(x, t)|) dxdt
to zero as h and k/h simultaneously tend to zero, where u(x, t) is the entropy weak solution of equation ( 1), assuming that ν(x, 0) converges in some sense to u ini (x) as h and k/h simultaneously tend to zero. Note that this limit sense is stronger than that provided by [START_REF] Diperna | Measure-valued solutions to conservation laws[END_REF], since it implies that E(ν(x, t)) converges in L 1 ((A, B) × (0, T )) to u (although it would be possible, we do not consider here the framework C 0 0, T ; L 1 ((A, B)) for the sake of simplicity). The proof of these results are closely related to the methods involved in the convergence proofs for numerical schemes [START_REF] Champier | Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh[END_REF][START_REF] Eymard | Finite volume methods[END_REF][START_REF] Merlet | Error estimate for finite volume scheme[END_REF] (further works for providing an error estimate might also be done).

In the unbounded case, we define an homogeneous Markov jump process on

E = (k[[0, K]]) Z with the possible jumps from η to T n,n+1 (η) for n ∈ Z, with T n,n+1 (η)(n) = η(n) -k, T n,n+1 (η)(n + 1) = η(n + 1) + k and T n,n+1 (
η)(j) = η(j) for all j different from n and n + 1, and rate b(η(n), η(n + 1)).

The corresponding Chapman-Kolmogorov equation reads:

dEψ(η t ) dt = E n∈Z b(η t (n), η t (n + 1)) ψ(T n,n+1 (η t )) -ψ(η t ) , ∀t ∈ R + , ∀ψ ∈ D, ( 12
)
where D is the set of all bounded functions ψ from E to R + depending on a finite number of coordinates. We associate to the process η t the real process ν(x, t)

defined from R × R + to [0, Ū ] by ν(x, t) = η t (n) ∀x ∈ [(n -1)h, nh), ∀n ∈ Z, ∀t ∈ R + . (13)
In the unbounded case, we prove a similar convergence result to the bounded case, and moreover prove in Theorem 3.4 that, for given R and T , for a suitable initial value ν(x, 0) close from u ini (x) (assumed to have bounded variations) and for k = h 2 , if u(x, t) is the entropy weak solution of equation ( 5), we have

T 0 R -R E (|ν(x, t) -u(x, t)|) dxdt ≤ C e h 1 4 , (14)
where C e only depends on R, T , g and u ini . Inequality ( 14) can be equivalently written

T 0 {n∈Z/|nh|≤R,|(n-1)h|≤R} [(n-1)h,nh) E (|η t (n) -u(x, t)|) dxdt ≤ C e h 1 4 (15)
which brings into play values η t (n) of the process at a finite number of sites n.

So we obtain an approximation of the entropy solution of ( 1) and ( 5) with the help of a misanthrope process in a different way than Euler hydrodynamic. The sense of the limit is different and the relation between the function f in the hyperbolic equation and the misanthrope process is explicitly given by f (x) = g(x, x). Note that different functions g(., .) could match a given function f (.) with respecting the above hypotheses (see also [START_REF] Godlewski | Numerical approximation of hyperbolic systems of conservation laws[END_REF], [START_REF] Leveque | Numerical methods for conservation laws[END_REF] and references therein):

• the Godunov numerical flux: introduced in [START_REF] Godounov | Résolution numérique des problèmes multidimensionnels de la dynamique des gaz[END_REF], it may be summarized by the following expression.

g(a, b) = min{f (ξ), ξ ∈ [a, b]} if a ≤ b, max{f (ξ), ξ ∈ [b, a]} if b ≤ a. (16)
• the splitting numerical flux: assume

f = f 1 + f 2 , with f 1 , f 2 ∈ C 1 (R, R), f ′ 1 (s) ≥ 0 and f ′ 2 (s)
≤ 0 for all s ∈ [0, U ] (such a decomposition for f is always possible, see the modified Lax-Friedrichs numerical flux below), and take

g(a, b) = f 1 (a) + f 2 (b).
Note that if f ′ ≥ 0, taking f 1 = f and f 2 = 0, the splitting numerical flux boils down to the upwind numerical flux, i.e. g(a, b) = f (a).

• the modified Lax-Friedrichs numerical flux, also called the Rusanov numerical flux in different frameworks: take

g(a, b) = f (a) + f (b) 2 + D(a -b), (17)
with

D ∈ R such that 2D ≥ max{|f ′ (s)|, s ∈ [U m , U ]}.
We remark that in the last two examples the functions defined above can be negative. We could easily extend the definition of the misanthrope process by assuming that, if b(n, n + 1) < 0, then -b(n, n + 1) denotes the rate of jump of such an object from the site n + 1 to the site n. Then all the results of this paper would hold as well. But, for the sake of simplicity, we only consider here positive jump rates.

This paper is organized as follows: the second section is devoted to the bounded case, while the third section deals with the unbounded case; numerical computations on the Riemann problem are presented in section 4. Some tracks for further research are finally shown in a short conclusion.

The bounded case

In this section, our aim is to prove that the misanthrope process, defined by the Chapman-Kolmogorov equation [START_REF] Eymard | Limit boundary conditions for finite volume approximations of some physical problems[END_REF] and a given initial probability measure on D, converges to an entropy weak solution in some sense. Let us recall this notion in the bounded case as given in [START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF].

Let us denote a⊤b the maximum of a and b, a⊥b the minimum of a and b, for all real values a, b, and A,B)), and u, u ∈ L ∞ (R + ) be given functions. We say that u is a weak entropy solution of problem (1)-( 4) if:

C 1 c ([A, B] × R + , R + ) the set of the restriction to [A, B] × R + of the non-negative C 1 functions with compact support from R 2 to R + . Definition 2.1 (Entropy weak solution) Let f ∈ C 1 (R, R) (or f : R → R Lipschitz continuous) be given, let u ini ∈ L ∞ ((
• u ∈ L ∞ ((A, B) × (0, ∞)),
• there exists M > 0 such that, for all functions ζ ⊤ (s) = s⊤κ -κ and

F ⊤ (s) = f (s⊤κ) -f (κ) with κ ∈ R, we have for all ϕ ∈ C 1 c ([A, B] × R + , R + ) ∞ 0 B A (ζ ⊤ (u)∂ t ϕ(x, t) + F ⊤ (u) ∂ x ϕ(x, t)) dxdt +M ∞ 0 (ζ ⊤ (u(t))ϕ(A, t) + ζ ⊤ (u(t))ϕ(B, t)) dt + B A ζ ⊤ (u ini )ϕ(x, 0) dx ≥ 0, (18)
• the same inequality as [START_REF]On kinematic waves II: a theory of traffic flow on long, crowded roads[END_REF]

holds replacing ζ ⊤ (s) by ζ ⊥ (s) = κ -s⊥κ and F ⊤ (s) by F ⊥ (s) = f (κ) - f (s⊥κ).
We have the following fundamental theorem [START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF].

Theorem 2.1 Let f ∈ C 0 (R) be Lipschitz continuous , u ini ∈ L ∞ ((A, B)), u, u ∈ L ∞ (R + )
, then there exists a unique entropy weak solution in the sense of Definition (2.1) to Problem [START_REF] Andjel | Quelques compléments sur le processus des misanthropes et le processus zéro-range[END_REF].

Remark 2.1 The solution of [START_REF]On kinematic waves II: a theory of traffic flow on long, crowded roads[END_REF] does not depends on the choice of M (the value M is chosen in this paper as the Lipschitz constant of g). A uniqueness result on a larger class of objects (Young measures instead of measurable functions) is used in this paper for the proof of convergence (this result is proved in [START_REF] Vovelle | Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains[END_REF] for the purpose of the convergence study of a numerical scheme [START_REF] Eymard | Limit boundary conditions for finite volume approximations of some physical problems[END_REF]). It is interesting to remark that if one replaces in [START_REF]On kinematic waves II: a theory of traffic flow on long, crowded roads[END_REF], the set of function ζ ⊤ or ζ ⊥ by the set of all entropies |u -κ| (as done in the unbounded case, see Definition 3.1), one has an existence result (since |u -κ| = s⊤κ -κ + κ -s⊥κ) but no uniqueness result, see [START_REF] Vovelle | Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains[END_REF] for a counter-example to uniqueness.

Estimates

The first step is to obtain a discrete entropy inequality for the misanthrope process.

Lemma 2.1 (Discrete entropy inequalities) Let ζ ∈ C 2 (R) be a convex function (i.e. ζ ′′ (κ) ≥ 0 for all κ ∈ R)
. Let η t be the misanthrope process defined by the Chapman-Kolmogorov equation [START_REF] Eymard | Limit boundary conditions for finite volume approximations of some physical problems[END_REF] and a given initial probability measure on D. Then the following inequalities hold:

h dE (ζ(η t (n))) dt + E (G ζ (η t (n), η t (n + 1)) -G ζ (η t (n -1), η t (n))) ≤ k g(U , 0) max s∈[0,U ] ζ ′′ (s), ∀n ∈ {1, . . . , N }, ( 19
)
denoting G ζ = G ⊤ ζ or G ⊥ ζ , for given X < 0 < U < Y , where G ⊤ ζ (x, y) = Y X ζ ′′ (s)(g(x⊤s, y⊤s) -f (s)) ds + g(x, y)ζ ′ (X), ∀x, y ∈ [X, Y ], G ⊥ ζ (x, y) = Y X ζ ′′ (s)(f (s) -g(x⊥s, y⊥s)) ds + g(x, y)ζ ′ (Y ), ∀x, y ∈ [X, Y ]. ( 20
)
Proof of Lemma 2.1

Since we have ζ ∈ C 2 (R), we get for all η ∈ E and for a given n ∈ {1, . . . , N }, that there exists

s - n (η) ∈ (η(n) -k, η(n)) such that 1 k ζ(η(n) -k) -ζ(η(n)) = -ζ ′ (η(n)) + k 2 ζ ′′ (s - n (η)),
and that there exists

s + n (η) ∈ (η(n), η(n) + k) such that 1 k ζ(η(n) + k) -ζ(η(n)) = ζ ′ (η(n)) + k 2 ζ ′′ (s + n (η)).
Hence, from Chapman-Kolmogorov equation [START_REF] Eymard | Limit boundary conditions for finite volume approximations of some physical problems[END_REF], we obtain, choosing, for n ∈ {1, . . . , N }, ψ ∈ D such that

ψ(η) = ζ(η(n)) for all η ∈ E, h dEζ(η t (n)) dt + E (ζ ′ (η t (n))(g(η t (n), η t (n + 1)) -g(η t (n -1), η t (n)))) = k 2 E ζ ′′ (s - n (η t ))g(η t (n), η t (n + 1)) + ζ ′′ (s + n (η t ))g(η t (n -1), η t (n)) , ∀n ∈ {1, . . . , N }.
Thanks to the monotonicity properties of g, we have the properties, for x, y, z, s ∈ R (defining sign

+ (s) = 1 if s > 0, 0 otherwise): sign + (y -s)(g(y, z) -g(s, s)) ≥ g(y⊤s, z⊤s) -g(s, s),
and sign + (y -s)(g(s, s) -g(x, y)) ≥ g(s, s) -g(x⊤s, y⊤s), and therefore sign + (y -s)(g(y, z) -g(x, y)) ≥ g(y⊤s, z⊤s) -g(s, s) + g(s, s) -g(x⊤s, y⊤s), easily checked by considering all cases. Since we may write,

ζ ′ (y) = Y X ζ ′′ (s)sign + (y -s) ds + ζ ′ (X),
we get, using [START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF] and

ζ ′′ (s) ≥ 0 for all s ∈ R, ζ ′ (y)(g(y, z) -g(x, y)) ≥ G ⊤ ζ (y, z) -G ⊤ ζ (x, y), thanks to Definition (20) of G ⊤ ζ , which concludes the proof of (19) in this case. Turning to the case G ζ = G ⊥ ζ , we write, for x, y, z, s ∈ R (defining sign -(s) = -1 if s < 0, 0 otherwise): sign -(y -s)(g(y, z) -g(s, s)) ≥ g(s, s) -g(y⊥s, z⊥s), and sign -(y -s)(g(s, s) -g(x, y)) ≥ g(x⊥s, y⊥s) -g(s, s),
and therefore

sign -(y -s)(g(y, z) -g(x, y)) ≥ g(s, s) -g(y⊥s, z⊥s) + g(x⊥s, y⊥s) -g(s, s), again verified by considering all cases. Since we may write

ζ ′ (y) = Y X ζ ′′ (s)sign -(y -s) ds + ζ ′ (Y ),
we get, using [START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF] and ζ ′′ (s) ≥ 0 for all s ∈ R,

ζ ′ (y)(g(y, z) -g(x, y)) ≥ G ⊥ ζ (y, z) -G ⊥ ζ (x, y),
thanks to Definition (20) of G ⊥ ζ , which concludes the proof of ( 19) in this case. Let us write the following lemma, which provides an inequality used in the convergence proof playing the same role as the so-called "weak BV inequalities" in the case of deterministic numerical schemes (see [START_REF] Champier | Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh[END_REF][START_REF] Eymard | Finite volume methods[END_REF][START_REF] Eymard | An implicit finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise deterministic markov processes[END_REF] for the use of such inequalities). Lemma 2.2 Let T > 0. Let η t be the misanthrope process defined by the Chapman-Kolmogorov equation [START_REF] Eymard | Limit boundary conditions for finite volume approximations of some physical problems[END_REF] and a given initial probability measure on D. Then there exists C ∈ R, only depending on T , U , g such that,

assuming k h ≤ 1, T 0 N n=0 E (H(η t (n), η t (n + 1))) dt ≤ C √ h , ( 21
)
where H is defined by

H(a, b) = max (c,d)∈C(a,b) |g(c, d) -f (c)| + max (c,d)∈C(a,b) |g(c, d) -f (d)|, ∀a, b ∈ R, (22)
denoting, for all a, b ∈ R, by C(a, b) = {(c, d) ∈ [a⊥b, a⊤b] 2 ; (d -c)(b -a) ≥ 0}. Proof of Lemma 2.2
In this proof, we shall denote by C i (i ∈ N) various quantities only depending on g, U , T .

Applying the Cauchy-Schwarz inequality to the left hand side of (21) yields

T 0 N n=0 E (H(η t (n), η t (n + 1))) dt 2 ≤ 2T (N + 1) T 0 N n=0 E (H 2 (η t (n), η t (n + 1))) dt with H 2 (a, b) = max (c,d)∈C(a,b) (g(c, d) -f (c)) 2 + max (c,d)∈C(a,b) (g(c, d) -f (d)) 2 , ∀a, b ∈ R.
Thanks to the monotonicity properties of g (and using the fact that g(s, s) = f (s)), the following inequality holds, for any

(c, d) ∈ C(a, b): b a (f (s) -g(a, b)) ds ≥ d c (f (s) -g(a, b)) ds ≥ d c (f (s) -g(c, d)) ds ≥ 0. ( 23
)
Let us remark that the following property

| d c (θ(s) -θ(c)) ds| ≥ 1 2G (θ(d) -θ(c)) 2 , ∀c, d ∈ R. ( 24
)
holds for all monotone, Lipschitz continuous function θ : R → R, with a Lipschitz constant G > 0. Indeed, let us assume, for instance, that θ is nondecreasing and c < d (the other cases are similar). Then, one has θ(s) ≥ ϕ(s),

for all s ∈ [c, d], where ϕ(s) = θ(c) for s ∈ [c, d -l] and ϕ(s) = θ(c) + (s -d + l)G for s ∈ [d -l, d], with lG = θ(d) -θ(c)
, and therefore:

d c (θ(s) -θ(c)) ds ≥ d c (ϕ(s) -θ(c)) ds = l 2 (θ(d) -θ(c)) = 1 2G (θ(d) -θ(c)) 2 .
Applying (24), we can notice that

d c (f (s) -g(c, d)) ds ≥ d c (g(c, s) -g(c, d))ds ≥ 1 2M (f (c) -g(c, d)) 2 , (25)
(f (s) -g(a, b)) ds ≥ 1 4M H 2 (a, b) (27)
So we have to find a bound for

T 1 = T 0 N n=0 E ηt(n+1) ηt(n) (f (s) -g(η t (n), η t (n + 1))) ds dt If Φ is a primitive of the function (•)f ′ (•), an integration by parts yields, for all (a, b) ∈ R 2 , Φ(η t (n + 1)) -Φ(η t (n)) = ηt(n+1) ηt(n) sf ′ (s) ds = η t (n + 1)(f (η t (n + 1)) -g(η t (n), η t (n + 1)) -η t (n)(f (η t (n)) -g(η t (n), η t (n + 1)) - ηt(n+1) ηt(n) (f (s) -g(η t (n), η t (n + 1)) ds (28)
and we can write

T 1 = T 3 + T 2 with T 3 = T 0 N n=0 E η t (n)(g(η t (n), η t (n + 1)) -f (η t (n))) +η t (n + 1)(f (η t (n + 1)) -g(η t (n), η t (n + 1))) dt,
and

T 2 = T 0 N n=0 E Φ(η t (n)) -Φ(η t (n + 1)) dt = T 0 E Φ(u(t)) -Φ(u(t)) dt. It is clear that T 2 ≤ C 2 .
Using Chapman-Kolmogorov formula [START_REF] Eymard | Limit boundary conditions for finite volume approximations of some physical problems[END_REF] with ψ(η) = η(n) 2 for n ∈ {1, . . . , N }, we have

h N n=1 E η t (n) 2 -h N n=1 E η 0 (n) 2 = T 0 N n=1 E ((k -2η t (n))g(η t (n), η t (n + 1)) + (k + 2η t (n))g(η t (n -1), η t (n))) dt.
Then we get

h N n=1 E η t (n) 2 -h N n=1 E η 0 (n) 2 = 2 T 0 k N n=0 E (g(η t (n), η t (n + 1))) dt - T 0 E ((2η t (N + 1) + k)g(η t (N ), η t (N + 1)) + (k -2η t (0))g(η t (0), η t (1))) dt +2 T 0 (η t (N + 1)f (η t (N + 1)) -η t (0)f (η t (0))) dt -2T 3 .
(recall that we denote by η t (0) = u(t) and η t (N + 1) = u(t)). We have

T 0 k N n=0 E (g(η t (n), η t (n + 1))) dt ≤ C 5 k(N + 1) ≤ C 4 k h .
This gives, using simple bounds for η(n), u, u that

T 3 ≤ C 1 + C4k h .
We can then deduce :

1 4M T 0 N n=0 E (H 2 (η t (n), η t (n + 1))) dt ≤ C 1 + C 4 k h + C 2 ≤ C 3 . ( 29
)
This completes the proof of Lemma 2.2.

Let us recall that ν(x, t) is a function from R × R + to [0, Ū ] associated to η t by ν(x, t) = η t (n) if x ∈ [nh, (n + 1)h).
The following lemma proves an entropy inequality associated to ν(x, t).

Lemma 2.3 Let η t be the misanthrope process defined by the Chapman-Kolmogorov equation [START_REF] Eymard | Limit boundary conditions for finite volume approximations of some physical problems[END_REF] and a given initial probability measure on D. Let ν be the real process defined by [START_REF] Eymard | An implicit finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise deterministic markov processes[END_REF].

Let κ ∈ [0, U ] be given and ζ ⊤ (s) = s⊤κ -κ and F ⊤ (s) = f (s⊤κ) -f (κ).
Then the following inequality holds:

+∞ 0 B A E ζ ⊤ (ν(x, t)) ∂ t ϕ(x, t) dxdt + B A ζ ⊤ (u ini (x))ϕ(x, 0) dx +M +∞ 0 ζ ⊤ (u(t))ϕ(A, t) dt + M +∞ 0 ζ ⊤ (u(t))ϕ(B, t) dt + +∞ 0 B A E(F ⊤ (ν(x, t))) ∂ x ϕ(x, t) dxdt ≥ - +∞ 0 ϕ(A, t)dµ(t) - +∞ 0 ϕ(B, t)dµ(t) - B A ϕ(x, 0)dµ ini (x) - (A,B)×R + ϕ(x, t)dµ 0 (x, t) - (A,B)×R + (|∂ t ϕ(x, t)| + |∂ x ϕ(x, t)|)dµ 1 (x, t), (30) for all ϕ ∈ C ∞ c (R × R, R + ) (recall that M is a Lipschitz constant for g). The same inequality holds replacing ζ ⊤ (s) by ζ ⊥ (s) = κ -s⊥κ and F ⊤ (s) by F ⊥ (s) = f (κ) -f (s⊥κ).
In (30), the measures µ 0 , µ 1 , µ, µ and µ ini verify the following properties:

1. For all T > 0, there exists C 0 depending only on T , g and U such that

µ 0 ([A, B] × [0, T ]) + µ([0, T ]) + µ([0, T ]) ≤ C 0 (k + k h ). ( 31
)
2. For all T > 0, there exists C 1 depending only on g, u ini , U and T such that, for h < R,

µ 1 ([A, B] × [0, T ]) ≤ C 1 ( √ h + k h ). ( 32
)
3. The measure µ ini is the measure of density E|u ini (.) -ν(., 0)| + C ini k h with respect to the Lebesgue measure, where C ini only depends on T , g and U .

Proof of Lemma 2.3

We remark that we cannot directly apply Lemma 2.

1 to ζ = ζ ⊤ since Lemma 2.1 involves C 2 (R) convex functions.
Therefore, we approximate the function ζ ⊤ , for a given κ ∈ [0, U ], by a regular function ζ ⊤ ε , defined, for a value ε ∈ (0, 1) which will be chosen later, and for a mollifier ρ (defined as a nonnegative element of C ∞ c (R) with support included in [-1, 1] and integral equal to 1), by

ζ ⊤ ε (x) = R (y⊤κ -κ) 1 ε ρ( x -y ε ) dy.
We then get that, for given X < -1

< U + 1 < Y , we have (ζ ⊤ ε ) ′ (X) = 0 and (ζ ⊤ ε ) ′ (Y ) = 1, and
|ζ ⊤ (x) -ζ ⊤ ε (x)| ≤ R |y -x| 1 ε ρ( x -y ε ) dy ≤ ε, ∀x ∈ R. ( 33
) Let us denote G ⊤ ε the function G ⊤ ζ ⊤ ε
as defined by [START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF]. We have, for

x ∈ [X, Y ], (ζ ⊤ ε ) ′′ (x) = 1 ε ρ( x-κ ε ), which leads to G ⊤ ε (x, y) = Y X 1 ε ρ( s -κ ε )(g(x⊤s, y⊤s) -f (s)) ds, ∀x, y ∈ [X, Y ].
We define G ⊤ (x, y) by

G ⊤ (x, y) = g(x⊤κ, y⊤κ) -f (κ), ∀x, y ∈ [X, Y ],
and we get, since M is a Lipschitz constant for g,

|G ⊤ (x, y) -G ⊤ ε (x, y)| ≤ 2M Y X |s -κ| 1 ε ρ( s -κ ε ) ds ≤ 2M ε, ∀x, y ∈ [X, Y ]. (34)
We have, for

x, x ′ ∈ [X, Y ], that ζ ⊤ ε (x) -ζ ⊤ ε (x ′ ) = R (y⊤κ -κ) 1 ε ρ( x -y ε ) dy - R (y⊤κ -κ) 1 ε ρ( x ′ -y ε ) dy,
Hence, changing y in the second integral in y -

x + x ′ , |ζ ⊤ ε (x) -ζ ⊤ ε (x ′ )| ≤ R (y⊤κ -κ) -((y -x + x ′ )⊤κ -κ) 1 ε ρ( x -y ε ) dy ≤ R |x ′ -x| 1 ε ρ( x -y ε ) dy = |x ′ -x|. (35)
It leads to

Y X 1 ε ρ( y -κ ε )(f (x⊤y) -f (x⊤κ)) dy ≤ Y X 1 ε ρ( y -κ ε )M |y -κ|dy ≤ M ε, (36)
(recall that M is also a Lipschitz constant for f ).

Let ϕ ∈ C ∞ c (R×R, R + ) be given. Applying Lemma 2.1 to ζ = ζ ⊤ ε , we multiply Inequality (19) by 1 h nh (n-1)h ϕ(A+ x, t) dx, sum over n = 1, . . . , N and integrate the resulting equation with respect to t. This gives

T 1 + T 2 ≤ T 3 , (37)
with

T 1 = +∞ 0 N n=1 h dE ζ ⊤ ε (η t (n)) dt 1 h nh (n-1)h ϕ(A + x, t) dxdt, T 2 = +∞ 0 N n=1 E G ⊤ ε (η t (n), η t (n + 1)) -G ⊤ ε (η t (n -1), η t (n)) 1 h nh (n-1)h ϕ(A + x, t) dxdt,
and

T 3 = C N n=1 +∞ 0 max s∈[X,Y ] (ζ ⊤ ε ) ′′ (s)kg(U , 0) 1 h nh (n-1)h ϕ(A + x, t) dxdt.
We may write

T 1 = +∞ 0 B A dE ζ ⊤ ε (ν(x, t)) dt ϕ(x, t) dxdt,
which provides, thanks to an integrate by parts with respect to time,

T 1 = - +∞ 0 B A E ζ ⊤ ε (ν(x, t)) ∂ t ϕ(x, t) dxdt - B A E ζ ⊤ ε (ν(x, 0)) ϕ(x, 0) dx. ( 38
)
We have

T 1 = - +∞ 0 B A E(ζ ⊤ (ν(x, t)))∂ t ϕ(x, t) dxdt - B A ζ ⊤ (u ini (x))ϕ(x, 0) dx + T 4 + T 5 + T 6 ,
with T 4 , T 5 and T 6 such that

T 4 = +∞ 0 B A E ζ ⊤ (ν(x, t)) -ζ ⊤ ε (ν(x, t)) ∂ t ϕ(x, t) dxdt. T 5 = B A E ζ ⊤ ε (u ini (x)) -ζ ⊤ ε (ν(x, 0)) ϕ(x, 0) dx. T 6 = B A (ζ ⊤ (u ini (x)) -ζ ⊤ ε (u ini (x))ϕ(x, 0) dx.
We get from (33) that

|T 4 | ≤ ε +∞ 0 B A |∂ t ϕ(x, t)|dxdt,
and

|T 6 | ≤ ε B A ϕ(x, 0) dx,
and, from (35), we may write

|T 5 | ≤ B A E|u ini (x) -ν(x, 0)| ϕ(x, 0) dx.
Turning to the study of T 2 , we can write

T 2 = T 7 -T 8 + (T 2 -T 9 ) + (T 9 -T 10 + T 8 ) + (T 10 -T 7 ), with
T 7 = - +∞ 0 B A E(f (ν(x, t)⊤κ) -f (κ))∂ x ϕ(x, t) dxdt, T 8 = +∞ 0 (E(G ⊤ (η t (0), η t (1))ϕ(A, t) -G ⊤ (η t (N ), η t (N + 1))ϕ(B, t)) dt, T 9 = +∞ 0 N n=1 E G ⊤ ε (η t (n), η t (n + 1)) -F ⊤ ε (η t (n)) ϕ(A + nh, t) dt + +∞ 0 N n=1 E F ⊤ ε (η t (n)) -G ⊤ ε (η t (n -1), η t (n)) ϕ(A + (n -1)h, t) dt,
and

T 10 = - +∞ 0 B A E F ⊤ ε (ν(x, t)) ∂ x ϕ(x, t) dxdt,
where F ⊤ ε is the function defined by

F ⊤ ε (x) = Y X 1 ε ρ( y -κ ε )(f (x⊤y) -f (y))dy, ∀x ∈ R. (39)
In order to compare T 2 with T 9 we write

T 2 = +∞ 0 N n=1 E G ⊤ ε (η t (n), η t (n + 1)) -F ⊤ ε (η t (n)) 1 h nh (n-1)h ϕ(A + x, t) dxdt + +∞ 0 N n=1 E F ⊤ ε (η t (n)) -G ⊤ ε (η t (n -1), η t (n)) 1 h nh (n-1)h ϕ(A + x, t) dxdt.
Using the inequalities

|ϕ(A + nh, t) - 1 h nh (n-1)h ϕ(A + x, t) dx| ≤ nh (n-1)h |∂ x ϕ(A + x, t)|dx, |ϕ(A + (n -1)h, t) - 1 h nh (n-1)h ϕ(A + x, t) dx| ≤ nh (n-1)h |∂ x ϕ(A + x, t)|dx, |G ⊤ ε (x, y) -F ⊤ ε (y)| ≤ 2H(x, y), |G ⊤ ε (x, y) -F ⊤ ε (x)| ≤ 2H(x, y) ∀x, y ∈ [X, Y ],
we get

|T 9 -T 2 | ≤ 2 +∞ 0 N n=1 E (H(η t (n), η t (n + 1)) + H(η t (n -1), η t (n))) nh (n-1)h |∂ x ϕ(A + x, t)|dxdt.
We define the measure µ 1 by

R×R + ψ(x, t) dµ 1 (x, t) = C +∞ 0 N n=1 E (H(η t (n), η t (n + 1)) + H(η t (n -1), η t (n))) nh (n-1)h |∂ x ψ(x, t)|dxdt,
which is, thanks to Lemma 2.2, such that (32) holds. We have

T 10 -T 9 = +∞ 0 (E(G ⊤ ε (η t (0), η t (1))ϕ(A, t) -G ⊤ ε (η t (N ), η t (N + 1))ϕ(B, t)) dt.
then using (34), we get

|T 9 -T 10 + T 8 | ≤ 2M ε +∞ 0 (ϕ(A, t) + ϕ(B, t)) dt.
We now remark that

G ⊤ (a, b) = g(a⊤κ, b⊤κ) -f (κ) ≤ g(a⊤κ, κ) -f (κ) ≤ M (a⊤κ -κ),
and that

-G ⊤ (a, b) = f (κ) -g(a⊤κ, b⊤κ) ≤ f (κ) -g(κ, b⊤κ) ≤ M (b⊤κ -κ).
This leads to

T 8 ≤ M +∞ 0 ζ ⊤ (u(t))ϕ(A, t) + ζ ⊤ (u(t))ϕ(B, t) dt.
Besides, we have, using (36),

|T 7 -T 10 | ≤ 2M ε +∞ 0 B A |∂ x ϕ(x, t)|dxdt.
Turning to the study of T 3 , we finally write that

|T 3 | ≤ Ckg(U , 0) εh +∞ 0 B A ϕ(x, t) dxdt.
We then define the measure µ 0 by

R×R + ψ(x, t) dµ 0 (x, t) = ε + Ckg(U , 0) εh +∞ 0 B A ψ(x, t) dxdt,
and we choose ε = k h . Gathering all the above results, and doing similarly for the "bottom" case, this completes the proof of Lemma 2.3.

Convergence study

We now state and prove a convergence result. Theorem 2.2 Let us consider a sequence (h i , k i ) i∈N with h i → 0 and k i /h i → 0 as i → ∞. Let us denote by ν i (x, t) the process ν(x, t) associated by [START_REF] Eymard | An implicit finite volume scheme for a scalar hyperbolic problem with measure data related to piecewise deterministic markov processes[END_REF] to the misanthrope process, defined by the Chapman-Kolmogorov equation [START_REF] Eymard | Limit boundary conditions for finite volume approximations of some physical problems[END_REF] and a given initial probability measure on D, with parameters h i , k i . If we assume that

lim i→∞ E B A u ini (x) -ν i (x, 0) dx = 0,
the process ν i (x, t) converges to the unique entropy weak solution u(x, t) of the equation [START_REF] Andjel | Quelques compléments sur le processus des misanthropes et le processus zéro-range[END_REF], in the sense that for all T > 0 lim

i→∞ [A,B]×[0,T ] E (|ν i (x, t) -u(x, t)|) dxdt = 0 Proof of Theorem 2.2 Let us define the Young measure µ i , for any i ∈ N, by R ζ(s) dµ i (x, t)(s) = E (ζ(ν i (x, t))) , t ∈ R + , x ∈ R, ∀ζ ∈ C 0 (R).
We first remark that there exists a subsequence, again denoted µ i , and a Young measure µ limit for the nonlinear weak-⋆ topology, that is

lim i→∞ [A,B]×R + ϕ(x, t) R ζ(s) dµ i (x, t)(s) dxdt = [A,B]×R + ϕ(x, t) R ζ(s) dµ(x, t)(s) dxdt, ∀ϕ ∈ C ∞ c (R × R, R + ), ∀ζ ∈ C 0 (R).
In order to justify this, let us develop in the framework of this paper an argument which is classical in the L ∞ framework [START_REF] Diperna | Measure-valued solutions to conservation laws[END_REF]. There exists a sequence (ζ j ) j∈N of elements of C 0 ([0, Ū ]), dense in C 0 ([0, Ū ]) for the uniform convergence topology. Then the sequence R ζ 1 (s) dµ i (x, t)(s) i∈N is bounded in L ∞ ((A, B)×R + ). Then there exists a subsequence and g 1 ∈ L ∞ ((A, B) × R + ) such that R ζ 1 (s) dµ i (x, t)(s) converges to g 1 for the weak-⋆ topology of L ∞ ((A, B) × R + ). By a diagonal process, we may extract a subsequence again denoted (µ i ) i∈N such that for all j ∈ N the sequence of functions (x, t) → R ζ j (s) dµ i (x, t)(s) tends to g j for the weak-⋆ topology of L ∞ ((A, B) × R + ). By density, for all ζ ∈ C 0 ([0, Ū ]), there exists a function

g ζ ∈ L ∞ ((A, B) × R + ) such that R ζ(s) dµ i (x, t)(s) converges to g ζ (x, t)
, as i → ∞, for the weak-⋆ topology of L ∞ ((A, B) × R + ). Now, by considering the Lebesgue points of all functions (g j ) j∈N , we build a subset of (A, B) × R + whose complementary in (A, B)×R + has zero Lebesgue measure. For (x, t) in this subset, we may consider the application ζ → g ζ (x, t), checking that (x, t) is a Lebesgue point as well for any function ζ ∈ C 0 ([0, Ū ]). This application defines a Young measure on (A, B) × R + since it is continuous. For a given κ ∈ R, we pass to the limit i → ∞ in (30). Hence we get that µ is such that

+∞ 0 B A R (s⊤κ -κ) dµ(x, t)(s) ∂ t ϕ(x, t) dxdt + B A (u ini (x)⊤κ -κ)ϕ(x, 0) dx +M +∞ 0 ϕ(A, t)(u(t)⊤κ -κ) dt + M +∞ 0 ϕ(B, t)(u(t)⊤κ -κ) dt + +∞ 0 B A R (f (s⊤κ) -f (κ))dµ(x, t)(s) ∂ x ϕ(x, t) dxdt ≥ 0.
We have also the same inequality with functions κ -s⊥κ. Then µ is an entropy Young measure solution (also called "process" solution in [START_REF] Vovelle | Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains[END_REF]) of the problem. Thanks to the uniqueness result, given as Theorem 2 in [START_REF] Vovelle | Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains[END_REF], we know that this entropy Young measure solution resumes to the entropy weak solution u(x, t). We then have

lim i→∞ [A,B]×R + ϕ(x, t) R s 2 dµ i (x, t)(s) dxdt = [A,B]×R + ϕ(x, t)u(x, t) 2 dxdt,
and lim

i→∞ [A,B]×R + ϕ(x, t)u(x, t) R sdµ i (x, t)(s) dxdt = [A,B]×R + ϕ(x, t)u(x, t)dxdt, which shows that lim i→∞ [A,B]×R + ϕ(x, t) R (s -u(x, t)) 2 dµ i (x, t)(s) dxdt = 0.
Hence we conclude the proof of the theorem taking ϕ = 1 on ((A, B) × [0, T ]) and using the Cauchy-Schwarz inequality.

The unbounded case

In this section, we now aim to prove that the misanthrope process, defined by the Chapman-Kolmogorov equation ( 12) and a given initial probability measure, converges to some entropy weak solution that we have to define in the unbounded case (this is simpler than in the bounded case).

Definition 3.1 (Entropy weak solution) Let f ∈ C 1 (R, R) and u ini ∈ L ∞ (R).
The entropy weak solution to Problem ( 5) is a function u such that

• u ∈ L ∞ (R × R + ),
• the following inequality holds

|u(x, t) -κ|∂ t ϕ(x, t) dtdx + f (u(x, t)⊤κ) -f (u(x, t)⊥κ) ∂ x ϕ(x, t) dtdx + R |u ini (x) -κ|ϕ(x, 0) dx ≥ 0, ∀ϕ ∈ C 1 c (R × R + , R + ), ∀κ ∈ R. ( 40
)
We have the following fundamental theorem [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF] (let us observe that the unbounded case has been solved a long time before the bounded one).

Theorem 3.1 Let f ∈ C 0 (R) be Lipschitz continuous , u ini ∈ L ∞ (R)
, then there exists a unique entropy weak solution to Problem (5).

Estimates

Estimates in the unbounded case are obtained in a similar way as in the bounded case. The next lemma is similar to Lemma 2.1 and is proved using

G ζ = 1 2 (G ⊤ ζ + G ⊥ ζ ).
Lemma 3.1 (Discrete entropy inequalities) Let ζ ∈ C 2 (R) be a convex function. Let η t be the misanthrope process defined by the Chapman-Kolmogorov equation ( 12) and a given initial probability measure on D.

Then the following inequality holds:

h dE (ζ(η t (n))) dt + E (G ζ (η t (n), η t (n + 1)) -G ζ (η t (n -1), η t (n))) ≤ k g(U , 0) max κ∈[0,U ] ζ ′′ (κ), ∀n ∈ Z, (41) denoting, for given X < 0 < U < Y , G ζ (x, y) = 1 2 Y X ζ ′′ (κ)(g(x⊤κ, y⊤κ) -g(x⊥κ, y⊥κ))dκ + g(x, y) ζ ′ (X) + ζ ′ (Y ) 2 , ∀x, y ∈ [X, Y ]. ( 42
)
The next lemma is similar to Lemma 2.2. The differences come from the fact that, in the unbounded case, one needs in introduce arbitrary bounds, related to the support of test functions in the entropy formulation.

Lemma 3.2 Let η t be the misanthrope process defined by the Chapman-Kolmogorov equation [START_REF] Gobron | Couplings, attractiveness and hydrodynamics for conservative particle systems[END_REF] and a given initial probability measure on D. Let T > 0, R > 0, A, B ∈ N such that R < Bh < R + 1 and -R > Ah > -R -1. Then there exists C ∈ R, only depending on T , R, U , g such that, for h < R, assuming k h ≤ 1,

T 0 B-1 n=A E (H(η t (n), η t (n + 1))) dt ≤ C √ h , ( 43
)
with H defined by [START_REF] Vovelle | Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains[END_REF].

One more time, the next lemma is similar to Lemma 2.3.

Lemma 3.3 Let η t be the misanthrope process defined by the Chapman-Kolmogorov equation [START_REF] Gobron | Couplings, attractiveness and hydrodynamics for conservative particle systems[END_REF] and a given initial probability measure on D. Let ν be the real process defined by [START_REF] Godlewski | Numerical approximation of hyperbolic systems of conservation laws[END_REF]. Let κ ∈ R be given. Then the following inequality holds:

+∞ 0 R E (|ν(x, t) -κ|) ∂ t ϕ(x, t) dxdt + R |u ini (x) -κ|ϕ(x, 0) dx + +∞ 0 R E(f (ν(x, t)⊤κ) -f (ν(x, t)⊥κ)) ∂ x ϕ(x, t) dxdt ≥ - R ϕ(x, 0)dµ ini (x) - R×R + ϕ(x, t)dµ 0 (x, t) - R×R + (|∂ t ϕ(x, t)| + |∂ x ϕ(x, t)|)dµ 1 (x, t), (44)
for all ϕ ∈ C ∞ c (R × R, R + )
, and where the measures µ 0 , µ 1 and µ ini verify the following properties:

1. For all R > 0 and T > 0, there exists C 0 depending only on R and T , g and U such that

µ 0 ([-R, R] × [0, T ]) ≤ C 0 k h . ( 45
)
2. For all R > 0 and T > 0, there exists C 1 depending only on g, u ini , U , R and T such that, for h < R,

µ 1 ([-R, R] × [0, T ]) ≤ C 1 ( √ h + k h ). ( 46
)
3. The measure µ ini is the measure of density E|u ini (.)-ν(., 0)|+ C ini √ Kh with respect to the Lebesgue measure, where C ini only depends on R and T , g and U .

Convergence study

The following convergence result may be proved as in the bounded case, its proof relies on the preceding lemma and on the uniqueness theorem given in [START_REF] Diperna | Measure-valued solutions to conservation laws[END_REF].

Theorem 3.2 Let us consider a sequence h i , k i with h i → 0 and k i /h i → 0. Let us denote by ν i (x, t) the process ν(x, t) associated by [START_REF] Godlewski | Numerical approximation of hyperbolic systems of conservation laws[END_REF] to the misanthrope process defined by the Chapman-Kolmogorov equation [START_REF] Gobron | Couplings, attractiveness and hydrodynamics for conservative particle systems[END_REF] and a given initial probability measure on D with parameters h i , k i . If we assume that for all A < B

lim i→∞ E B A u ini (x) -ν i (x, 0) dx = 0,
the process ν i (x, t) converges to the unique entropy weak solution u(x, t) of the equation ( 5), in the sense that for all T > 0 and all A < B lim

i→∞ [A,B]×[0,T ] E (|ν i (x, t) -u(x, t)|) dxdt = 0

Error estimate

Let us finally provide an error estimate in the unbounded case, the proof of which is inspired by that of error estimates for deterministic numerical schemes [START_REF] Eymard | Finite volume methods[END_REF]. To this purpose, we assume that u ini has locally bounded variations (that we denote by u ini ∈ BV loc (R)), which simply means that its derivative in the distribution sense is a measure which is not necessarily finite (for example, u ini = 0 on [2k, 2k + 1) and

u ini = 1 on [2k + 1, 2k + 2), for all k ∈ Z). Lemma 3.4 Let u ini ∈ BV loc (R) ∩ L ∞ (R).
Let ν be a stochastic process on R × R + , valued in in a bounded subset of R, such that there exists measures µ 0 , µ

1 on R × R + and µ ini on R such that +∞ 0 R E |ν(x, t) -κ|∂ t ϕ(x, t) + (f (ν(x, t)⊤κ) -f (ν(x, t)⊥κ))∂ x ϕ(x, t) dxdt + R |u ini (x) -κ|ϕ(x, 0)dx ≥ - R×R + |∂ t ϕ(x, t)| + |∂ x ϕ(x, t)| dµ 1 (x, t) - R×R + ϕ(x, t)dµ 0 (x, t) - R ϕ(x, 0)dµ ini (x), ∀κ ∈ R, ∀ϕ ∈ C ∞ c (R × R + , R + ). ( 47
)
Let u be the unique entropy weak solution of (5) in the sense of Definition 3.

1. Let ψ ∈ C ∞ c (R × R + , R + ) be given, and let S = {ψ = 0} = {(x, t) ∈ R × R + ; ψ(x, t) = 0} and S 0 = {ψ(•, 0) = 0} = {x ∈ R; ψ(x, 0) = 0}. Then there exists C only depending on ψ L ∞ (R×R + ) , ∂ t ψ L ∞ (R×R + ) , ∂ x ψ L ∞ (R×R + ) , f , S, S 0 and u ini , such that +∞ 0 R E |ν(x) -u(x, t)|∂ t ψ(x, t) + f (ν(x)⊤u(x, t)) -f (ν(x)⊥u(x, t)) (∂ x ψ(x, t)) dxdt ≥ -C(µ ini (S 0 ) + (µ 1 (S)) 1 2 + (µ 1 + µ 0 )(S)). (48)
Proof of Lemma 3.4

The main tool of the proof (the so-called Krushkov's "double variable technique") is to take κ = u(y, s) in (47), κ = ν(x, t) in (40) and to introduce mollifiers in order to have y close to x and s close to t. Let ρ ∈ C ∞ c (R, R) satisfying the following properties:

supp(ρ) = {x ∈ R; ρ(x) = 0} ⊂ [-1, 0], (49) ρ
(x) ≥ 0, ∀x ∈ R, R ρ(x)dx = 1.
For example, one might define ρ by ρ(x) = exp(-1/(1 -(2x + 1) 2 )) for x ∈ (-1, 0) and 0 elsewhere, and

ρ(x) = ρ(x)/ 0 -1 ρ(y)dy. For r ∈ R, r ∈ [1, +∞), one defines ρ r (x) = rρ(rx), for all x ∈ R. Let ψ ∈ C ∞ c (R × R + , R + ), and let ϕ : (R × R + ) 2 → R + be defined by: ϕ(x, t, y, s) = ψ(x, t)ρ r (x -y)ρ r (t -s).
Note that, for any (y,

s) ∈ R × R + , one has ϕ(•, •, y, s) ∈ C ∞ c (R × R + , R + ) and, for any (x, t) ∈ R × R + , one has ϕ(x, t, •, •) ∈ C ∞ c (R × R + , R + ). Let us consider, in (40)
(in which we denote the space and time variables by (y, s)), ϕ(x, t, •, •) as test function, and κ = ν(x, t). We then take the mean value and integrate for (x, t) ∈ R × R + . We thus get, remarking that, thanks to (49), ϕ(x, t, y, 0) = 0 for all t ≥ 0,

∞ 0 R ∞ 0 R E |u(y, s) -ν(x, t)|∂ s ϕ(x, t, y, s) + f (u(y, s)⊤ν(x, t)) -f (u(y, s)⊥ν(x, t)) ∂ y ϕ(x, t, y, s) dydsdxdt ≥ 0, ∀κ ∈ R, ∀ϕ ∈ C ∞ c (R × R + , R + ). (50)
We now consider, in (47), ϕ(•, •, y, s) as test function and κ = u(y, s). We then integrate (47) for (y, s) ∈ R × R + . We then get

∞ 0 R ∞ 0 R E |ν(x, t) -u(y, s)|∂ t ϕ(x, t, y, s) +(f (ν(x, t)⊤u(y, s)) -f (ν(x, t)⊥u(y, s)))∂ x ϕ(x, t, y, s) dxdtdyds + ∞ 0 R R |u ini (x) -u(y, s)|ϕ(x, 0, y, s)dxdyds ≥ - ∞ 0 R R×R + |∂ t ϕ(x, t, y, s)| + |∂ x ϕ(x, t, y, s)| dµ 1 (x, t)dyds - ∞ 0 R R×R + ϕ(x, t, y, s)dµ 0 (x, t)dyds - ∞ 0 R R |ϕ(x, 0, y, s)|dµ ini (x)dyds, (51)
Adding the two inequalities (50) and (51) and using ∂ s ϕ(x, t, y, s) + ∂ t ϕ(x, t, y, s) = ∂ t ψ(x, t)ρ r (x -y)ρ r (t -s) and ∂ x ϕ(x, t, y, s) + ∂ y ϕ(x, t, y, s) = ∂ x ψ(x, t)ρ r (x -y)ρ r (t -s), yield

E 11 + E 12 + E 14 ≥ -E 2 , (52)
where

E 11 = ∞ 0 R ∞ 0 R E |ν(x, t) -u(y, s)|∂ t ψ(x, t)ρ r (x -y)ρ r (t -s) dxdtdyds, E 12 = ∞ 0 R ∞ 0 R E f (ν(x, t)⊤u(y, s)) -f (ν(x, t)⊥u(y, s)) ∂ x ψ(x, t)ρ r (x -y)ρ r (t -s) dxdtdyds, E 14 = R ∞ 0 R |u ini (x) -u(y, s)|ψ(x, 0)ρ r (x -y)ρ r (-s)dydsdx
and

E 2 = ∞ 0 R R×R + |ρ r (x -y)(∂ t ψ(x, t)ρ r (t -s) + ψ(x, t)ρ ′ 1,r (t -s))| +|ρ r (t -s)(∂ x ψ(x, t)ρ r (x -y) + ψ(x, t)ρ ′ r (x -y))| dµ 1 (x, t)dyds + ∞ 0 R R×R + ρ r (x -y)ψ(x, t)ρ r (t -s)dµ 0 (x, t)dyds + ∞ 0 R R |ψ(x, 0)ρ r (x -y)ρ r (-s)|dµ ini (x)dyds. (53)
Let us now study the four terms of (52). In the following, the notation C i (i ∈ N) will refer to various real quantities only depending on

ψ L ∞ (R×R + ) , ∂ t ψ L ∞ (R×R + ) , ∂ x ψ L ∞ (R×R + ) , f , S, S 0 and u ini . Equality (53) leads, thanks to R |ρ ′ r (s)|ds = r R |ρ ′ (s)|ds, to E 2 ≤ (r + 1)C 1 µ 1 (S) + C 2 µ ini (S 0 ) + C 2 µ 0 (S). (54)
Let us handle the term E 11 . For all x ∈ R and for all t ∈ R + , one has, using (49),

R ∞ 0 ρ r (x -y)ρ 1,r (t -s)dsdy = 1.
Then,

|E 11 - +∞ 0 R E |ν(x, t) -u(x, t)|∂ t ψ(x, t) dxdt| ≤ ∞ 0 R ∞ 0 R |u(x, t) -u(y, s)||∂ t ψ(x, t)|ρ r (x -y)ρ r (t -s) dxdtdyds ≤ ∂ t ψ ∞ ε(r, S), with ε(r, S) = sup{ u -u(• + η, • + τ ) L 1 (S) , |η| ≤ 1 r , 0 ≤ τ ≤ 1 r }. (55)
Since u ini ∈ BV (R), the function u belongs to BV (R×(-T, T )), for all T > 0, setting, for instance, u(., t) = u ini for t < 0 (see [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF] where this result is proved passing to the limit on numerical schemes). Then,

ε(r, S) ≤ C 3 r . (56)
Hence,

|E 11 - +∞ 0 R E |ν(x, t) -u(x, t)|∂ t ψ(x, t) dxdt| ≤ C 4 r . (57)
In the same way, using |f (a⊤b

) -f (a⊤c)| ≤ M |b -c| and |f (a⊥b) -f (a⊥c)| ≤ M |b -c| for all a, b, c ∈ [0, U ] where M is the Lipschitz constant of f in [0, U ], |E 12 - +∞ 0 R E f (ν(x, t)⊤u(x, t)) -f (ν(x, t)⊥u(x, t)) ∂ x ψ(x, t)dxdt| ≤ C 5 ε(r, S) ≤ C6 r . (58)
In order to estimate E 14 , let us take in (40) (in which we denote the space and time variables by (y, s)),

for x ∈ R fixed, ϕ = ϕ(x, •, •), with ϕ(x, y, s) = ψ(x, 0)ρ r (x -y) ∞ s ρ r (-τ )dτ,
and κ = u ini (x). Note that ϕ(x, •, •) ∈ C ∞ c (R × R + , R + ).
We then integrate the resulting inequality with respect to x ∈ R. We get

-E 14 + E 15 + E 16 ≥ 0, with E 15 = - ∞ 0 R R ∞ s (f (u(y, s)⊤u ini (x)) -f (u(y, s)⊥u ini (x))) ψ(x, 0)ρ ′ r (x -y)ρ r (-τ )dτ dydxds, E 16 = R R ∞ 0 ψ(x, 0)ρ r (x -y)ρ r (-τ )|u ini (x) -u ini (y)|dτ dydx.
To bound E 15 , one introduces E 15b defined as

E 15b = - ∞ 0 R R ∞ s
(f (u(y, s)⊤u ini (y)) -f (u(y, s)⊥u ini (y)))

ψ(x, 0)ρ ′ r (x -y)ρ r (-τ )dτ dydxds. Integrating by parts for the x variable yields

E 15b = ∞ 0 R R ∞ s (f (u(y, s)⊤u ini (y)) -f (u(y, s)⊥u ini (y))) ∂ x ψ(x, 0)ρ r (x -y)ρ r (-τ )dτ dydxds.
Then, noting that the time support of this integration is reduced to s ∈ [0, 1/r], one has

E 15b ≤ C 11 r . (59)
Furthermore, one has

|E 15 -E 15b | ≤ C 12 ∞ 0 R R ∞ s |u ini (x) -u ini (y)||ρ ′ r (x -y)|ψ(x, 0)ρ r (-τ )dτ dydxds,
which is bounded by C 13 ε 0 (r, S 0 ), since the time support of the integration is reduced to s ∈ [0, 1/r], where ε 0 (r, S 0 ) is defined by

ε 0 (r, S 0 ) = sup{ S0 |u ini (x) -u ini (x + η)|dx; |η| ≤ 1 r }. (60)
Since u ini ∈ BV (R), one has ε 0 (r, S 0 ) ≤ C

Using ( 52), ( 54), ( 57), ( 58), (61), one obtains

+∞ 0 R E |ν(x, t) -u(x, t)|∂ t ψ(x, t) + f (ν(x, t)⊤u(x, t)) -f (ν(x, t)⊥u(x, t)) ∂ x ψ(x, t) dxdt ≥ -C 1 (r + 1)µ 1 (S) -C 2 (µ ini (S 0 ) + µ 0 (S)) - C 18 r , which, taking r = 1/ µ 1 (S) if 0 < µ 1 (S) ≤ 1 (r → ∞ if µ 1 (S) = 0 and r = 1 if µ 1 (S) > 1)
, gives (48). This concludes the proof of Lemma 3.4.

Theorem 3.3 Let u ini ∈ BV loc (R) ∩ L ∞ (R)
, let u be the unique entropy weak solution of the problem. Let us assume the same hypotheses as in Lemma 3.4, Then, for all R > 0 and all T > 0 there exists C e and R, only depending on R, T , f and u ini , such that the following inequality holds:

T 0 R -R E|ν(x, t) -u(x, t)| dxdt ≤ C e (µ ini ([-R, R]) + [µ 1 ([-R, R] × [0, T ])] 1 2 +(µ 1 + µ 0 )([-R, R] × [0, T ])). Proof of Theorem 3.3 Let R > 0 and T > 0. Let M be a Lipschitz constant of f in [0, U ] (indeed, since f ∈ C 1 (R, R), one has M = sup{|f ′ (s)|; s ∈ [0, U ]}). Let ρ ∈ C 1 c (R + , [0, 1]) be a function such that ρ(r) = 1 if r ∈ [0, R + M T ], ρ(r) = 0 if r ∈ [R + M T + 1, ∞
) and ρ ′ (r) ≤ 0, for all r ∈ R + (ρ only depends on R, T , f and u ini ). One takes, in (48), ψ defined by ψ(x, t) = ρ(|x| + M t) T -t T , for x ∈ R and t ∈ [0, T ], ψ(x, t) = 0, for x ∈ R and t ≥ T.

Note that ρ(|x|

+ M t) = 1, if (x, t) ∈ [-R, R] × [0, T ]. The function ψ is not in C ∞ c (R × R + , R +
), but, using a usual regularization technique, it may be proved that such a function can be considered in (48), in which case Inequality (48) writes, with

R = R + M T + 1, T 0 R E |ν(x, t) -u(x, t)| T -t T M ρ ′ (|x| + M t) - 1 T ρ(|x| + M t) + f (ν(x, t)⊤u(x, t)) -f (ν(x, t)⊥u(x, t)) T -t T ρ ′ (|x| + M t) x |x| ) dxdt ≥ -C(µ ini ([-R, R]) + (µ 1 ([-R, R] × [0, T ])) 1 2 + (µ 1 + µ 0 )([-R, R] × [0, T ])),
where C only depends on R, T , f and u ini . Since ρ ′ ≤ 0, one has

f (ν(x, t)⊤u(x, t)) -f (ν(x, t)⊥u(x, t)) T -t T ρ ′ (|x| + M t)( x |x| ) ≤ |ν(x, t) -u(x, t)| T -t T M (-ρ ′ (|x| + M t)), and therefore, since ρ(|x| + M t) = 1, if (x, t) ∈ [-R, R] × [0, T ], T 0 [-R,R] E|ν(x, t) -u(x, t)|dxdt ≤ CT (µ ini ([-R, R]) + (µ 1 ([-R, R] × [0, T ])) 1 2 +(µ 1 + µ 0 )([-R, R] × [0, T ])).
This completes the proof of Theorem 3.3.

We then deduce the following theorem.

With the aim to model traffic flow, one considers henceforth the two following positive and concave flux functions:

f 1 (u) = u 0 ≤ u ≤ 1/2 1 -u 1/2 < u ≤ 1 (64) and f 2 (u) = u(1 -u). (65)
Hence f 1 ∈ C 0 (R) is a triangular and piecewise linear flux function while f 2 ∈ C ∞ (R) is a regular polynomial one.

One proposes to numerically approximate the solution of Problem ( 1)-( 4) by using the misanthrope process η t defined on

E = (h 2 [[0, N 2 ]]) [[1,N ]]
with N ∈ N ⋆ and h = 1/N (one assumes with the previous notation that k = h 2 , U = 1 and therefore K = U /k = N 2 ). The process is characterized by the Chapman-Kolmogorov equation [START_REF] Eymard | Limit boundary conditions for finite volume approximations of some physical problems[END_REF].

The jump rate of the process, given by ( 8), depends on the numerical flux function. One uses and compares two numerical fluxes in the numerical experiments: the Godunov numerical flux ( 16) and a modified positive Rusanov (or Lax-Friedrichs) numerical flux given by 1 and4).

g(a, b) = 1 2 (f (a) + f (b) + a -b) + , with (x) + = x if x ≥ 0 0 otherwise (see figure
One uses an event-driven evolution scheme to simulate the stochastic process η. Each site occupied by at least one particle has an exponential clock giving the jump time of a particle towards the next site, or creation (resp. deletion) of a particle for the first (resp. last) site. The exponential times are calibrated by the jump rate function b. The process is simulated as follows. At the initial time, one calculates the jump time of each site according to their jump rate. Then, at each step (i.e. at each jump), one has to:

1. Select the site with the minimal time;

2. Transfer a particle from this site to the next one, or create or delete a particle; 3. Update the global time of the system with the jump time of the section selected;

4. Calculated the new jump times of the sections whose jump rate has been modified.

In the simulation experiments, one considers two initial conditions:

Exp. 1 leading to a rarefaction wave, u ini (x) = u l = 0.8 if x < x 0 = 0.5 u r = 0.2 otherwise, (see Figures 2 and5);

Exp. 2 leading to a shock wave, u ini (x) = u l = 0.4 if x < x 0 = 0.8 u r = 0.7 otherwise, (see Figures 3 and6).

One compares the numerical approximations respectively obtained using the Godunov and Rusanov numerical fluxes. The simulation are done for different values of the parameter h (equal to 0.05, 0.02 and 0.01).

One observes that the precision of the approximations is similar for the two numerical fluxes when the flux function is triangular since the jump rate functions are close. The approximation solutions are more accurate when using the Godunov numerical flux than the Rusanov one if the flux function is the polynomial f : u → u(1 -u). As expected, in all the cases, the approximations are more and more accurate as h tends towards 0.

  s) -g(c, d)) ds ≥ d c (g(s, d) -g(c, d))ds ≥ 1 2M (f (d) -g(c, d)) 2 . (26) Multiplying (25) and (26) by 1/2, taking the maximum for (c, d) ∈ C(a, b), and adding the two equations yields, with (23), b a

1 Figure 1 :Figure 2 :

 112 Figure 1: Jump rate function for the triangular flux (64). Left, case of the Godunov numerical flux and, right, case of the modified Rusanov one.

Figure 3 : 2 Figure 4 :

 324 Figure 3: Exp. 2 -case of the triangular flux (64). Solution (in gray dotted lines) and the misanthrope process plot at time t = {0, 0.8, 1.6} for h = {0.05, 0.02, 0.01}. Top, one uses the Godunov numerical flux and bottom the modified Rusanov one.

Figure 6 :

 6 Figure 6: Exp. 2 -case of the polynomial flux (65). Solution (in gray dotted lines) and the misanthrope process plot at time t = {0, 2.2, 4.4} for h = {0.05, 0.02, 0.01}. Top, one uses the Godunov numerical flux and bottom the modified Rusanov one.

Conclusion

We show the convergence of a misanthrope process to the entropy solution of a hyperbolic equation, both on a bounded and a non-bounded space. The proof of the convergence is based on the Chapman-Kolmogorov equation and on the uniqueness of the entropy Young measure solution. In the non-bounded case, we provide an error estimate through a method close to those involved in the convergence proofs for deterministic numerical schemes. The use of this framework enables a strong limit sense of the stochastic process to the entropy weak solution of the problem (note that, even in the linear case, one should use the nonlinear framework for proving such a convergence). From a numerical point of view, the stochastic continuous-time misanthrope process might be compared to first order deterministic discrete-time numerical schemes, in particular from the points of view of the computing time and the accuracy. It seems clear that the error estimate provided in this paper is not sharp, in a similar way to that obtained for deterministic numerical schemes. The error estimate including boundary terms remains to be studied. Finally, the results given in this paper show the close relation between modeling traffic flows using misanthrope processes and using the limit nonlinear scalar hyperbolic equation. Large work remains to be done for improving the accuracy of these models.

Theorem 3.4 Let η t be the misanthrope process defined by the Chapman-Kolmogorov equation [START_REF] Gobron | Couplings, attractiveness and hydrodynamics for conservative particle systems[END_REF] and the initial probability measure given by the Dirac measure on the function defined, for all n ∈ Z, by the closest element of kZ to 1 h nh (n-1)h u ini (x) dx. Let ν be the real process defined by [START_REF] Godlewski | Numerical approximation of hyperbolic systems of conservation laws[END_REF]. For all R > 0 and T > 0, assuming

then there exists C e , only depending on R, T , g, u ini , such that the following inequality holds:

(62)

Numerical computations on the Riemann problem

For a given U > 0, one considers Problem ( 1)-( 4) with (A, B) = (0, 1), assuming that u ini is defined, for given real values u l , u r ∈ [0, U ] and x 0 ∈ (0, 1), by

and that the functions u and u are respectively defined by u(t) = u l and u(t) = u r for all t ∈ R + (this problem is the generalization of the Riemann problem to the bounded setting since this boundary condition allows to reproduce the unbounded solution at least for a finite time which is not precisely given in the examples below).

In the linear case f (u) = v u, v ∈ R, the entropy weak solution to Problem ( 1)-( 4) (which is also in this case the unique weak solution) is then obtained by translation of the initial condition:

If the flux function f is non-linear, the entropy weak solution shows shock or rarefaction waves: shock waves are discontinuity lines between two density levels, that propagate over the time. Rarefaction waves are regular transitions between density levels. The entropy condition allows to specify the unique physical solution. When f is strictly convex or concave, the solution is deduced from the sign of u l -u r . Assuming f strictly convex (resp. strictly concave), u l > u r (resp. u l < u r ) and denoting

then the entropy weak solution u of Problem ( 1)-( 4) is the shock wave starting at x 0 and propagating at the constant speed σ

The solution is a shock wave starting at x 0 and propagating at the constant speed σ.

If one assumes f strictly convex (resp. strictly concave) and u l < u r (resp. u l > u r ) then the entropy weak solution of Problem ( 1)-( 4) is the rarefaction wave given by

with G is the reciprocal function to f ′ , which means that, for a given value u 0 ∈ [u l ⊥u r , u l ⊤u r ], then u(x 0 + tf ′ (u 0 ), t) = u 0 .

In the application of this model to traffic flow [START_REF]On kinematic waves II: a theory of traffic flow on long, crowded roads[END_REF][START_REF]Shock waves on a highway[END_REF], the flux function f is assumed to be concave. The nondecreasing part of the function corresponds to a free traffic state into which traffic characteristics propagate down-stream. The nonincreasing part describes an interactive or congested traffic state. For this traffic state, the characteristics propagate up-stream. These aspects are observed on real traffic data [START_REF] Dirk | Traffic and related self-driven many-particle systems[END_REF].