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

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1 vailable online at Stochastic Processes and their pplications 122 (212) Convergence of a misanthrope process to the entropy solution of 1D problems. Eymard a, M. oussignol a,,. Tordeux b a Laboratoire d nalyse et de Mathématiques ppliquées, Université Paris-Est, Cité Descartes - Champs-sur-Marne - F Marne-la-Vallée, Cedex 2, France b Laboratoire Ville Mobilité Transport, Université Paris-Est, Cité Descartes - Champs-sur-Marne - F Marne-la-Vallée, Cedex 2, France eceived 12 ugust 211; received in revised form 5 July 212; accepted 8 July 212 vailable online 2 July 212 bstract 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. c 212 Elsevier B.V. ll rights reserved. Keywords: Misanthrope stochastic process; Non linear scalar hyperbolic equation; Entropy Young measure solution; Traffic flow simulation; Weak BV inequality 1. Introduction mong a large number of models for the traffic flow, the continuous nonlinear hyperbolic transport equation t u(x, t) + x ( f (u))(x, t) =, denoting by u(x, t) the flow density at position x and at time t, and by f ( ) the flow volume with respect to the density level, is shown in the literature to be particularly relevant (see for example [7,23,27,3]). These traffic Corresponding author. addresses: robert.eymard@univ-mlv.fr (. Eymard), michel.roussignol@univ-mlv.fr (M. oussignol), antoine.tordeux@enpc.fr (. Tordeux) /$ - see front matter c 212 Elsevier B.V. ll rights reserved. doi:1.116/j.spa

2 . Eymard et al. / Stochastic Processes and their pplications 122 (212) Fig. 1. Traffic flow volume as a function of the flow density on a section of 1 m, for real data (left) and data obtained by simulation using a misanthrope process (right). Performances averaged on 1 min. flow models are related as macroscopic since they describe the evolution of the macroscopic quantities of flow density, flow volume or vehicles mean speed by road sections. On the other hand, interactive particles processes on a lattice, such that the exclusion or the zero-range processes [29], can model one-lane traffic flow [16,28,18]. These models describe the evolution of vehicle positions, vehicle distance spacings or vehicle platoons, by using different interpretations of sites and particles. In all cases, these approaches are microscopic since vehicles evolve individually. The interactive particle misanthrope process, introduced in [6,1], is also used to model traffic flow [17,31]. The models based on the misanthrope process describe the evolution of the number of vehicles by road section. site is a section of an uni-directional road (with potentially several lanes) and a particle is a vehicle. This approach can be seen as mesoscopic since it studies the evolution of the number of vehicles by section and not the evolution of each vehicle position. The interest of such a model, for the simulation of traffic flow at the scale of road networks, is that the intrinsic stochasticity of the process allows to reproduce the variability of the flow performances observed within real (congested) traffic data [19]. Indeed, in Fig. 1, we show in the left part the flow volume as a function of the density as measured for real data (extracted for the merican project NGSIM [12]), and on the right part, we present the simulated results given by the misanthrope traffic model introduced in [31], with fitting the model parameters on these data. The function f used in the nonlinear hyperbolic equation is represented by the solid line in Fig. 1. This paper is devoted to the proof that the misanthrope traffic model converges to the macroscopic continuous nonlinear hyperbolic transport equation, letting the discretization parameters tend to zero. s a result, traffic flow models based on the nonlinear hyperbolic equation may be interestingly approximated by misanthrope stochastic processes, which can take into account the dispersion observed in the traffic data Non linear hyperbolic equation This section is devoted to a formal presentation of the bounded and unbounded nonlinear scalar hyperbolic equations. In the bounded case, the strong form of the nonlinear equation is given by t u(x, t) + x ( f (u))(x, t) = x (, B), t +, (1)

3 365. Eymard et al. / Stochastic Processes and their pplications 122 (212) for given reals < 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, ) = u ini (x), x (, B), (2) and some boundary conditions (in some sense discussed below) and u(, t) = u(t), t +, u(b, t) = u(t), t +. In (1) (4), we denote by u ini a bounded measurable function defined for all x (, B), and by u, u bounded measurable functions defined for all t +. We assume, without restricting the generality, that these three functions are a.e. valued in [, U], for a given U +. In the definition of the entropy weak sense of this equation (see Section 2), the sense of the boundary conditions (3) (4) is deduced from Otto s works [25], since the regularity of the limit obtained in this paper is not sufficient for allowing the use of the stronger sense given in [4]. The strong sense of the unbounded case may be written as t u(x, t) + x ( f (u))(x, t) = x, t +, (5) with initial data u(x, ) = u ini (x), x, where u ini denotes a bounded measurable function defined for all x, a.e. valued in [, U]. s previously, we deal with the more classical entropy weak sense of this equation, introduced in Section Misanthrope stochastic process We now turn to the mathematical description of the stochastic model, whose convergence to the deterministic Eqs. (1) and (5) is the object of the present paper. misanthrope process is a stochastic Markov jump process (η t ) t, 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 by b(η t (n), η t (n+1)). We first discretize the continuous interval of possible values [, Ū] of u(x, t) in discrete values (ki, i =,..., K ) with K N and k such that k = U K. For a given discretization parameter h >, the jump rates of the process are defined by b(x, y) = 1 g(x, y), x, y, (7) h k where the given function g satisfies the following assumptions (denoted by H in the following): g is Lipschitz continuous from 2 to + (we denote in this paper M the smallest Lipschitz constant of g), g(x, y) = for all (x, y) (], ] ) ( [U, + [), (x, y) g(x, y), from [, U] 2 to +, is nondecreasing with respect to x and nonincreasing with respect to y, (3) (4) (6)

4 . Eymard et al. / Stochastic Processes and their pplications 122 (212) Under these assumptions, the function g is nonnegative and we get b(,.) = and b(., kk ) =. In the bounded case, we define h = (B )/N for a given N N, and we discretize the interval [, B] in constant size intervals ( + (n 1)h, + nh), n {1,..., N}, each interval corresponding to a site for the stochastic process. We define a non-homogeneous Markov jump process on E = (k[[, K ]]) [[1,N]] with the following possible jumps and the associated rates defined at time t by: jump from η to T,1 (η) with T,1 (η)(1) = η(1) + k and T,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 () = u(t) and η t (N + 1) = u(t), t +, (8) the corresponding Chapman Kolmogorov equation reads: deψ(η t ) N = E b(η t (n), η t (n + 1)) ψ(t n,n+1 (η t )) ψ(η t ), dt n= t +, ψ D, where D is the set of all functions ψ from E to. In the unbounded case, we discretize in intervals ((n 1)h, nh), n Z of size h, each interval corresponding to a site for the stochastic process. We define a homogeneous Markov jump process on E = (k[[, 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 t +, ψ D, b(η t (n), η t (n + 1)) ψ(t n,n+1 (η t )) ψ(η t ), n Z where D is the set of all bounded functions ψ from E to depending on a finite number of coordinates Numerical flux We show in this paper that, letting the parameters h and k tend to zero in some sense, the misanthrope processes defined by (7), for a given function g, converge to the entropy weak solution of the nonlinear hyperbolic equations (1) or (5) where the function f is given by: f (x) = g(x, x) for all x. (11) (9) (1)

5 3652. Eymard et al. / Stochastic Processes and their pplications 122 (212) The first step of numerical studies is then to adjust on actual data, similar to those given in the left part of Fig. 1, a Lipschitz continuous function f such that f (x) for all x and f (x) = for all x (, ] [U, + ). The second step is then to derive an expression of g(.,.) which simultaneously matches (11) and hypotheses (H). Let us provide such expressions (see for example [14,22] and references therein): the Godunov numerical flux [15], given by min{ f (ξ), ξ [a, b]} if a b, g(a, b) = max{ f (ξ), ξ [b, a]} if b a. the modified Lax Friedrichs or usanov numerical flux, defined by f (a) + f (b) g(a, b) = max + D(a b),, (13) 2 with D such that 2D max{ f (s), s [, U]}. These two examples are considered in the numerical examples. emark 1.1. We could extend, for a given Lipschitz continuous function f ( ) such that f (x) for all x and f (x) = for all x (, ] [U, + ), the framework of this paper to the following numerical fluxes g which match (11) but not all the hypotheses (H): the splitting numerical flux: assume f = f 1 + f 2, where the Lipschitz continuous functions f 1, f 2 are such that f 1 (s) and f 2 (s) for a.e. s [, U] (define for example f 1 as a primitive of max( f, ) and f 2 = f f 1 ), and take g(a, b) = f 1 (a) + f 2 (b), the modified Lax Friedrichs numerical flux with no imposed positivity, g(a, b) = f (a) + f (b) 2 + D(a b), with D again chosen such that 2D max{ f (s), s [, U]}. Then the functions g(x, y) such defined are Lipschitz continuous, nondecreasing with respect to x and nonincreasing with respect to y, and are therefore such that g(y, ), g(u, y), g(, y) and g(y, U) for all y [, U]. We can then extend the definition of the misanthrope process by assuming that, if g(η t (n), η t (n + 1)) <, then b(η t (n), η t (n + 1)) = h 1 k g(η t(n), η t (n + 1)) denotes the rate of jump of an object from the site n + 1 to the site n. We may then prove that η t (n) remains included in [, U], and all the results of this paper still hold Main results We will prove that the values of the misanthrope process defined below may approximate the entropy weak solution u(x, t) of the nonlinear hyperbolic equation, first on a bounded interval (, B) of (the bounded case), then on (the unbounded case). In the bounded case, we associate to the process η t the real process ν(x, t) defined from (, B) + to [, Ū] by ν(x, t) = η t (n) x [ + (n 1)h, + nh), n [[1, N]], t +. (14) (12)

6 . Eymard et al. / Stochastic Processes and their pplications 122 (212) The final result for the bounded case (Theorem 2.2) is the convergence of T B = T E ( ν(x, t) u(x, t) ) dxdt 1 n N [+(n 1)h,+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 Eq. (1), assuming that ν(x, ) 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 (18), since it implies that E(ν(x, t)) converges in L 1 ((, B) (, T )) to u (although it would be possible, we do not consider here the framework C, T ; L 1 ((, B)) for the sake of simplicity). The proofs of these results are closely related to the methods involved in the convergence proofs for numerical schemes [5,9,24] (further works for providing an error estimate might also be done). In the unbounded case, we associate to the process η t the real process ν(x, t) defined from + to [, Ū] by ν(x, t) = η t (n) x [(n 1)h, nh), n Z, t +. (15) We prove a similar convergence result to the bounded case, and moreover prove in Theorems 3.4 and 3.5 that, for given and T, for a suitable initial value ν(x, ) 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 Eq. (5), we have T E ( ν(x, t) u(x, t) ) dxdt C e h r, (16) where r = 1/4 in the general case, r = 1/2 if the function g is such that g C 2 ( 2 ), and C e only depends on, T, g and u ini. Inequality (16) can be equivalently written as T E ( η t (n) u(x, t) ) dxdt C e h r (17) {n Z/ nh, (n 1)h } [(n 1)h,nh) which brings into play values η t (n) of the process at a finite number of sites n Comparison with Euler hydrodynamic limit Classical studies of limit of stochastic process to the solution of hyperbolic equations deal with the Euler hydrodynamic limit for attractive particle systems [2,2,3,13,26]. s described in [3], a misanthrope process (η t ) t on E = [[, K ]] Z is said to have Euler hydrodynamic limit u(x, t) for some initial configuration if, for all continuous function Ψ from to + with compact support, we have, for all t lim N 1 N y η Nt (y) Ψ = N y Z Ψ(x) u(x, t) dx in probability. (18) It is proved that the Euler hydrodynamic limit u(x, t) of a misanthrope process for some initial distributions is the entropy weak solution of (5) such that the function f must check the condition f (ρ) = b(η(), η(1)) dµ ρ (η), ρ,

7 3654. Eymard et al. / Stochastic Processes and their pplications 122 (212) where is the largest subset of [, K ] such that, for any ρ, the set of all measures µ on E, which are invariant for the process and shift-invariant with η()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 > ); note that is closed and necessarily contains and K, but, as mentioned in [3], the relation [, K ] = remains an open problem. In this paper, we obtain an approximation of (1) and (5) with the help of a misanthrope process in a different way from Euler hydrodynamic. First, the relation between the function f in the hyperbolic equation and the misanthrope process is explicitly given by f (x) = g(x, x) and does not depend on the invariant distributions of the process. Second, the sense of the limit is different: we are dealing with a sequence of processes, whereas in the Euler hydrodynamic limit, there is only one process observed at different scales of space and time Organization of the paper This paper is organized as follows: Section 2 is devoted to the bounded case, while Section 3 deals with the unbounded case; numerical computations on the iemann problem are presented in Section 4. Some tracks for further research are finally shown in a short conclusion. 2. The bounded case In this section, our aim is to prove that the misanthrope process, defined by the Chapman Kolmogorov equation (9) 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 [25]. 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 C 1 c ([, B] +, + ) the set of the restrictions to [, B] + of the non-negative C 1 functions with compact support from 2 to +. Definition 2.1 (Entropy Weak Solution). Let f C 1 (, ) (or f : Lipschitz continuous) be given, let u ini L ((, B)), and u, u L ( + ) be given functions. We say that u is a weak entropy solution of problem (1) (4) if: u L ((, B) (, )), there exists M > such that, for all functions ζ (s) = s κ κ and F (s) = f (s κ) f (κ) with κ, we have for all ϕ Cc 1([, B] +, + ) B (ζ (u) t ϕ(x, t) + F (u) x ϕ(x, t)) dxdt + M + ζ (u(t))ϕ(b, t)) dt + B (ζ (u(t))ϕ(, t) ζ (u ini )ϕ(x, ) dx, (19) the same inequality as (19) 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 [25]. Theorem 2.1. Let f C () be Lipschitz continuous, u ini L ((, B)), u, u L ( + ), then there exists a unique entropy weak solution in the sense of Definition 2.1 to Problem (1).

8 . Eymard et al. / Stochastic Processes and their pplications 122 (212) emark 2.1. The solution of (19) does not depends on the choice of M (the value M is chosen in this paper as the Lipschitz constant of g). 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 [32] for the purpose of the convergence study of a numerical scheme [1]). It is interesting to remark that if one replaces in (19), 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 [32] 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 () be a convex function (i.e. ζ (κ) for all κ ). Let η t be the misanthrope process defined by the Chapman Kolmogorov equation (9) 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, ) max ζ (s), n {1,..., N}, (2) s [,U] denoting G ζ = G ζ or G ζ, for given X < < U < Y, where Y G ζ (x, y) = ζ (s)(g(x s, y s) f (s)) ds + g(x, y)ζ (X), x, y [X, Y ], X Y G ζ (x, y) = ζ (s)( f (s) g(x s, y s)) ds + g(x, y)ζ (Y ), x, y [X, Y ]. X (21) Proof of Lemma 2.1. Since we have ζ C 2 (), we get for all η E and for a given n {1,..., N}, that there exists sn (η) (η(n) k, η(n)) such that 1 ζ(η(n) k) ζ(η(n)) = ζ (η(n)) + k k 2 ζ (sn (η)), and that there exists s n + (η) (η(n), η(n) + k) such that 1 ζ(η(n) + k) ζ(η(n)) = ζ (η(n)) + k k 2 ζ (s n + (η)). Hence, from Chapman Kolmogorov equation (9), 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}.

9 3656. Eymard et al. / Stochastic Processes and their pplications 122 (212) Thanks to the monotonicity properties of g, we have the properties, for x, y, z, s (defining sign + (s) = 1 if s >, otherwise): and sign + (y s)(g(y, z) g(s, s)) g(y s, z s) g(s, s), 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 (21) and ζ (s) for all s, ζ (y)(g(y, z) g(x, y)) G ζ (y, z) G ζ (x, y), thanks to definition (21) of G ζ, which concludes the proof of (2) in this case. Turning to the case G ζ = G ζ, we write, for x, y, z, s (defining sign (s) = 1 if s <, otherwise): and sign (y s)(g(y, z) g(s, s)) g(s, s) g(y s, z s), 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 (21) and ζ (s) for all s, ζ (y)(g(y, z) g(x, y)) G ζ (y, z) G ζ (x, y), thanks to definition (21) of G ζ, which concludes the proof of (2) 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 [5,9,11] for the use of such inequalities). Lemma 2.2. Let T >. Let η t be the misanthrope process defined by the Chapman Kolmogorov equation (9) and a given initial probability measure on D. Then there exists C, only depending on T, U, g such that, assuming k h 1, T N E (H(η t (n), η t (n + 1))) dt C, (22) h n=

10 where H is defined by. Eymard et al. / Stochastic Processes and their pplications 122 (212) 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, (23) denoting, for all a, b, by C(a, b) = {(c, d) [a b, a b] 2 ; (d c)(b a) }. Proof of Lemma 2.2. In this proof, we shall denote by C i (i N) various quantities only depending on g, U, T. pplying the Cauchy Schwarz inequality to the left hand side of (22) yields 2 T N E (H(η t (n), η t (n + 1))) dt with n= 2T (N + 1) T N E (H 2 (η t (n), η t (n + 1))) dt n= H 2 (a, b) = max (g(c, d) f (c,d) C(a,b) (c))2 + max (g(c, d) f (c,d) C(a,b) (d))2, a, b. 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. (24) Let us remark that the following property d (θ(s) θ(c)) ds 1 2G (θ(d) θ(c))2, c, d. (25) c holds for all monotone, Lipschitz continuous function θ :, with a Lipschitz constant G >. 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: and d c (θ(s) θ(c)) ds d pplying (25), we can notice that d c d c ( f (s) g(c, d)) ds ( f (s) g(c, d)) ds c (ϕ(s) θ(c)) ds = l 1 (θ(d) θ(c)) = 2 2G (θ(d) θ(c))2. d c d c (g(c, s) g(c, d)) ds 1 2M ( f (c) g(c, d))2, (26) (g(s, d) g(c, d)) ds 1 2M ( f (d) g(c, d))2. (27) Multiplying (26) and (27) by 1/2, taking the maximum for (c, d) C(a, b), and adding the two equations yield, with (24), b a ( f (s) g(a, b)) ds 1 4M H 2(a, b). (28)

11 3658. Eymard et al. / Stochastic Processes and their pplications 122 (212) So we have to find a bound for T N ηt (n+1) T 1 = E ( f (s) g(η t (n), η t (n + 1))) ds dt. n= η t (n) If Φ is a primitive of the function ( ) f ( ), an integration by parts yields, for all (a, b) 2, Φ(η t (n + 1)) Φ(η t (n)) = ηt (n+1) η t (n) s f (s) ds = η t (n + 1)( f (η t (n + 1)) g(η t (n), η t (n + 1))) η t (n)( f (t(n)) g(η t (n), η t (n + 1))) ηt (n+1) η t (n) ( f (s) g(η t (n), η t (n + 1))) ds (29) and we can write T 1 = T 3 + T 2 with T N ηt (n)(g(η T 3 = E t (n), η t (n + 1)) f (η t (n))) dt, + η t (n + 1)( f (η t (n + 1)) g(η t (n), η t (n + 1))) and T 2 = T n= N n= T E Φ(η t (n)) Φ(η t (n + 1)) dt = E Φ(u(t)) Φ(u(t)) dt. It is clear that T 2 C 2. Using Chapman Kolmogorov formula (9) with ψ(η) = η(n) 2 for n {1,..., N}, we have h N E η t (n) 2 h n=1 T = Then we get h N n=1 = 2 N n=1 E η (n) 2 N 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. n=1 E η t (n) 2 h T k n= T + 2 T N n=1 E η (n) 2 N E (g(η t (n), η t (n + 1))) dt E((2η t (N + 1) + k)g(η t (N), η t (N + 1)) + (k 2η t ())g(η t (), η t (1))) dt (η t (N + 1) f (η t (N + 1)) η t () f (η t ())) dt 2T 3. (recall that we denote by η t () = u(t) and η t (N + 1) = u(t)). We have T k N n= E (g(η t (n), η t (n + 1))) dt C 5 k(n + 1) C 4k h.

12 . Eymard et al. / Stochastic Processes and their pplications 122 (212) This gives, using simple bounds for η(n), u, u that T 3 C 1 + C 4k h. We can then deduce: 1 4M T N n= E (H 2 (η t (n), η t (n + 1))) dt C 1 + C 4k h + C 2 C 3. (3) This completes the proof of Lemma 2.2. Let us recall that ν(x, t) is a function from + to [, Ū] associated to η t by ν(x, t) = η t (n) if x [(n 1)h, nh). 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 (9) and a given initial probability measure on D. Let ν be the real process defined by (14). Let κ [, U] be given and ζ (s) = s κ κ and F (s) = f (s κ) f (κ). Then the following inequality holds: + B B E ζ (ν(x, t)) t ϕ(x, t) dxdt + ζ (u ini (x))ϕ(x, ) dx + M B + B ζ (u(t))ϕ(, t) dt + M + E(F (ν(x, t))) x ϕ(x, t) dxdt ϕ(, t)dµ(t) + ϕ(b, t)dµ(t) ζ (u(t))ϕ(b, t) dt ϕ(x, )dµ ini (x) ϕ(x, t)dµ (x, t) (,B) + ( t ϕ(x, t) + x ϕ(x, t) )dµ 1 (x, t), (31) (,B) + for all ϕ C c (, + ) (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 (31), the measures µ, µ 1, µ, µ and µ ini verify the following properties: 1. For all T >, there exists C depending only on T, g and U such that k µ ([, B] [, T ]) + µ([, T ]) + µ([, T ]) C k +. (32) h 2. For all T >, there exists C 1 depending only on g, u ini, U and T such that, for h <, µ 1 ([, B] [, T ]) C 1 h + k h. (33) 3. The measure µ ini is the measure of density E u ini ( ) ν(, ) + C ini k h with respect to the Lebesgue measure, where C ini only depends on T, g and U.

13 366. Eymard et al. / Stochastic Processes and their pplications 122 (212) 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 () convex functions. Therefore, we approximate the function ζ, for a given κ [, U], by a regular function ζ, defined, for a value (, 1) which will be chosen later, and for a mollifier ρ (defined as a nonnegative element of Cc () with support included in [ 1, 1] and whose integral is equal to 1), by ζ (x) = (y κ κ) 1 x y ρ dy. We then get that, for given X < 1 < U + 1 < Y, we have (ζ ) (X) = and (ζ ) (Y ) = 1, and ζ (x) ζ (x) y x 1 x y ρ dy, x. (34) Let us denote G the function G as defined by (21). We have, for x [X, Y ], (ζ ζ ) (x) = 1 ρ( x κ ), which leads to Y G (x, y) = 1 s κ ρ (g(x s, y s) f (s)) ds, x, y [X, Y ]. We define G (x, y) by X 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 (x, y) 2M Y x, y [X, Y ]. X s κ 1 ρ s κ We have, for x, x [X, Y ], that ζ (x) ζ (x ) = (y κ κ) 1 x y ρ ds 2M, dy (y κ κ) 1 x ρ y dy. Hence, changing y in the second integral in y x + x, ζ (x) ζ (x ) (y κ κ) ((y x + x ) κ κ) 1 x y ρ dy x x 1 x y ρ dy = x x. (36) It leads to Y X 1 ρ y κ Y 1 ( f (x y) f (x κ)) dy X ρ y κ M y κ dy (35) M, (37) (recall that M is also a Lipschitz constant for f ). Let ϕ Cc (, + ) be given. pplying Lemma 2.1 to ζ = ζ, we multiply inequality (2) by h 1 nh (n 1)h ϕ( + x, t) dx, sum over n = 1,..., N and integrate the resulting equation

14 . Eymard et al. / Stochastic Processes and their pplications 122 (212) with respect to t. This gives with and T 1 + T 2 T 3, T 1 = T 2 = T 3 = C We may write T 1 = + N n=1 + 1 h N n=1 N n=1 nh (n 1)h + + B h de ζ (η t(n)) dt 1 h nh (n 1)h ϕ( + x, t) dxdt, E G (η t(n), η t (n + 1)) G (η t(n 1), η t (n)) ϕ( + x, t) dxdt, max s [X,Y ] (ζ ) (s)kg(u, ) 1 h nh de ζ (ν(x, t)) ϕ(x, t) dxdt, dt (n 1)h ϕ( + x, t) dxdt. which provides, thanks to an integration by parts with respect to time, T 1 = We have T 1 = + B B E + B B E ζ (ν(x, t)) t ϕ(x, t) dxdt (38) ζ (ν(x, )) ϕ(x, ) dx. (39) E(ζ (ν(x, t))) t ϕ(x, t) dxdt ζ (u ini (x))ϕ(x, ) dx + T 4 + T 5 + T 6, with T 4, T 5 and T 6 such that + B T 4 = E ζ (ν(x, t)) ζ (ν(x, t)) t ϕ(x, t) dxdt. T 5 = T 6 = B B E We get from (34) that T 4 ζ (uini (x)) ζ (ν(x, )) ϕ(x, ) dx. ζ (u ini (x)) ζ (uini (x)) ϕ(x, ) dx. + B t ϕ(x, t) dxdt,

15 3662. Eymard et al. / Stochastic Processes and their pplications 122 (212) and B T 6 ϕ(x, ) dx, and, from (36), we may write with and T 5 B E u ini (x) ν(x, ) ϕ(x, ) dx. Turning to the study of T 2, we can write T 2 = T 7 T 8 + (T 2 T 9 ) + (T 9 T 1 + T 8 ) + (T 1 T 7 ), T 7 = T 8 = T 9 = + B + + E( f (ν(x, t) κ) f (κ)) x ϕ(x, t) dxdt, E(G (η t (), η t (1))ϕ(, t) G (η t (N), η t (N + 1))ϕ(B, t)) dt, N n=1 + + T 1 = E G (η t(n), η t (n + 1)) F (η t(n)) ϕ( + nh, t) dt N n=1 + B E F (η t(n)) G (η t(n 1), η t (n)) ϕ( + (n 1)h, t) dt, E F (ν(x, t)) x ϕ(x, t) dxdt, where F is the function defined by Y F (x) = 1 y κ ρ ( f (x y) f (y)) dy, x. (4) X In order to compare T 2 with T 9 we write T 2 = + N n= E G (η t(n), η t (n + 1)) F (η t(n)) h N n=1 Using the inequalities ϕ( + nh, t) 1 nh h ϕ( + (n 1)h, t) 1 h nh (n 1)h ϕ( + x, t) dxdt 1 nh E F (η t(n)) G (η t(n 1), η t (n)) ϕ( + x, t) dxdt. h (n 1)h (n 1)h nh nh ϕ( + x, t) dx x ϕ( + x, t) dx, (n 1)h nh ϕ( + x, t) dx x ϕ( + x, t) dx, (n 1)h (n 1)h G (x, y) F (y) 2H(x, y), G (x, y) F (x) 2H(x, y) x, y [X, Y ],

16 we get. Eymard et al. / Stochastic Processes and their pplications 122 (212) T 9 T n=1 nh (n 1)h We define the measure µ 1 by + ψ(x, t) dµ 1 (x, t) = C N E (H(η t (n), η t (n + 1)) + H(η t (n 1), η t (n))) x ϕ( + x, t) dxdt. + N E(H(η t (n), η t (n + 1)) n=1 + H(η t (n 1), η t (n))) which is, thanks to Lemma 2.2, such that (33) holds. We have T 1 T 9 = + then using (35), we get T 9 T 1 + T 8 2M We now remark that and that nh (n 1)h ψ(x, t) dxdt, E(G (η t(), η t (1))ϕ(, t) G (η t(n), η t (N + 1))ϕ(B, t)) dt + (ϕ(, t) + ϕ(b, t)) dt. G (a, b) = g(a κ, b κ) f (κ) g(a κ, κ) f (κ) M(a κ κ), 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 + Besides, we have, using (37), T 7 T 1 2M ζ (u(t))ϕ(, t) + ζ (u(t))ϕ(b, t) dt. + B x ϕ(x, t) dx dt. Turning to the study of T 3, we finally write that + B Ckg(U, ) T 3 ϕ(x, t) dx dt. h We then define the measure µ by + ψ(x, t) dµ (x, t) = k h. + Ckg(U, ) h + B ψ(x, t) dx dt, and we choose = Gathering all the above results, and doing similarly for the bottom case, this completes the proof of Lemma 2.3.

17 3664. Eymard et al. / Stochastic Processes and their pplications 122 (212) 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 = B i and k i = K U i such that ik i as i. Let us denote by ν i (x, t) the process ν(x, t) associated by (14) to the misanthrope process, defined by the Chapman Kolmogorov equation (9) and a given initial probability measure on D, with parameters h i, k i. If we assume that B lim E u ini (x) ν i (x, ) dx =, i the process ν i (x, t) converges to the unique entropy weak solution u(x, t) of Eq. (1), in the sense that for all T > E ( ν i (x, t) u(x, t) ) dx dt =. lim i [,B] [,T ] Proof of Theorem 2.2. Let us define the Young measure µ i, for any i N, by ζ(s) dµ i (x, t)(s) = E (ζ(ν i (x, t))), t +, x, ζ C (). We first remark that there exists a subsequence, again denoted by µ i, and a Young measure µ limit for the nonlinear weak- topology, that is lim ϕ(x, t) ζ(s) dµ i (x, t)(s) dx dt i [,B] + = ϕ(x, t) ζ(s) dµ(x, t)(s) dx dt, ϕ C [,B] + c (, + ), ζ C (). In order to justify this, let us develop in the framework of this paper an argument which is classical in the L framework [8]. There exists a sequence (ζ j ) j N of elements of C ([, Ū]), dense in C ([, Ū]) for the uniform convergence topology. Then the sequence ζ 1(s) dµ i (x, t)(s) i N is bounded in L ((, B) + ). Then there exists a subsequence and g 1 L ((, B) + ) such that ζ 1(s) dµ i (x, t)(s) converges to g 1 for the weak- topology of L ((, B) + ). By a diagonal process, we may extract a subsequence again denoted by (µ i ) i N such that for all j N the sequence of functions (x, t) ζ j(s) dµ i (x, t)(s) tends to g j for the weak- topology of L ((, B) + ). By density, for all ζ C ([, Ū]), there exists a function g ζ L ((, B) + ) such that ζ(s) dµ i(x, t)(s) converges to g ζ (x, t), as i, for the weak- topology of L ((, B) + ). Now, by considering the Lebesgue points of all functions (g j ) j N, we build a subset of (, B) + whose complementary in (, B) + 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 ([, Ū]). This application defines a Young measure on (, B) + since it is continuous. For a given κ, we pass to the limit i in (31). Hence we get that µ is such that + B B (s κ κ) dµ(x, t)(s) t ϕ(x, t) ddx dt + (u ini (x) κ κ)ϕ(x, ) dx

18 . Eymard et al. / Stochastic Processes and their pplications 122 (212) M ϕ(, t)(u(t) κ κ) dt + M ϕ(b, t)(u(t) κ κ) dt + B + ( f (s κ) f (κ))dµ(x, t)(s) x ϕ(x, t) dx dt. We have also the same inequality with functions κ s κ. Then µ is an entropy Young measure solution (also called process solution in [32]) of the problem. Thanks to the uniqueness result, given as Theorem 2 in [32], we know that this entropy Young measure solution resumes to the entropy weak solution u(x, t). We then have and lim i [,B] + ϕ(x, t) s 2 dµ i (x, t)(s) dx dt = [,B] + ϕ(x, t)u(x, t) 2 dx dt, lim ϕ(x, t)u(x, t) sdµ i (x, t)(s) dx dt = ϕ(x, t)u(x, t) dx dt, i [,B] + [,B] + which shows that lim ϕ(x, t) (s u(x, t)) 2 dµ i (x, t)(s) dx dt =. i [,B] + Hence we conclude the proof of the theorem taking ϕ = 1 on ((, B) [, T ]) and using the Cauchy Schwarz inequality. 3. The unbounded case In this section, we now aim to prove that the misanthrope process, defined by the Chapman Kolmogorov equation (1) 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 (, ) and u ini L (). The entropy weak solution to Problem (5) is a function u such that u L ( + ), the following inequality holds u(x, t) κ t ϕ(x, t) dt dx + f (u(x, t) κ) f (u(x, t) κ) x ϕ(x, t) dt dx + u ini (x) κ ϕ(x, ) dx, ϕ C 1 c ( +, + ), κ. (41) We have the following fundamental theorem [21] (let us observe that the unbounded case has been solved a long time before the bounded one). Theorem 3.1. Let f C () be Lipschitz continuous, u ini L (), then there exists a unique entropy weak solution to Problem (5).

19 3666. Eymard et al. / Stochastic Processes and their pplications 122 (212) 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 () be a convex function. Let η t be the misanthrope process defined by the Chapman Kolmogorov equation (1) 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, ) max ζ (κ), n Z, (42) κ [,U] denoting, for given X < < 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 ), x, y [X, Y ]. (43) 2 The next lemma is similar to Lemma 2.2. The differences come from the fact that, in the unbounded case, one needs to 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 (1) and a given initial probability measure on D. Let T >, >,, B N such that < Bh < + 1 and > h > 1. Then there exists C, only depending on T,, U, g such that, for h <, assuming h k 1, T B 1 E (H(η t (n), η t (n + 1))) dt C, (44) h n= with H defined by (23). 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 (1) and a given initial probability measure on D. Let ν be the real process defined by (15). Let κ be given. Then the following inequality holds: + E ( ν(x, t) κ ) t ϕ(x, t) dx dt + u ini (x) κ ϕ(x, ) dx + + E( f (ν(x, t) κ) f (ν(x, t) κ)) x ϕ(x, t) dx dt ϕ(x, )dµ ini (x) ϕ(x, t)dµ (x, t) + ( t ϕ(x, t) + x ϕ(x, t) )dµ 1 (x, t), (45) +

20 . Eymard et al. / Stochastic Processes and their pplications 122 (212) for all ϕ C c (, + ), and where the measures µ, µ 1 and µ ini verify the following properties: 1. For all > and T >, there exists C depending only on and T, g and U such that µ ([, ] [, T ]) C k h. (46) 2. For all > and T >, there exists C 1 depending only on g, u ini, U, and T such that, for h <, µ 1 ([, ] [, T ]) C 1 h + k h. (47) 3. The measure µ ini is the measure of density E u ini ( ) ν(, ) + Cini K h with respect to the Lebesgue measure, where C ini only depends on 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 [8]. Theorem 3.2. Let us consider a sequence h i, k i = K U i with h i and k i /h i. Let us denote by ν i (x, t) the process ν(x, t) associated by (15) to the misanthrope process defined by the Chapman Kolmogorov equation (1) and a given initial probability measure on D with parameters h i, k i. If we assume that for all < B lim E i B u ini (x) ν i (x, ) dx =, the process ν i (x, t) converges to the unique entropy weak solution u(x, t) of Eq. (5), in the sense that for all T > and all < B E ( ν i (x, t) u(x, t) ) dx dt =. lim i [,B] [,T ] 3.3. Error estimates Let us provide error estimates in the unbounded case. To this purpose, we assume that u ini has locally bounded variations (that we denote by u ini BV loc ()), which simply means that its derivative in the distribution sense is a measure which is not necessarily finite (for example, u ini = on [2k, 2k + 1) and u ini = 1 on [2k + 1, 2k + 2), for all k Z). The proof of the next lemma, which provides an error estimate in the general case, is very similar to that of [9, Lemma 29.2 p. 955], based on the so-called Krushkov s double variable technique. The main difference is that a mathematical expectation applies to all the terms issued from the stochastic approximation, in the same way as it occurs in the statement of the lemma. Lemma 3.4. Let u ini BV loc () L (). Let ν be a stochastic process on +, valued in a bounded subset of, such that there exist measures µ, µ 1 on + and µ ini on such that

21 3668. Eymard et al. / Stochastic Processes and their pplications 122 (212) E ν(x, t) κ t ϕ(x, t) + ( f (ν(x, t) κ) f (ν(x, t) κ)) x ϕ(x, t) dxdt + u ini (x) κ ϕ(x, )dx t ϕ(x, t) + x ϕ(x, t) dµ 1 (x, t) ϕ(x, t)dµ (x, t) + + ϕ(x, )dµ ini (x), κ, ϕ Cc ( +, + ). (48) Let u be the unique entropy weak solution of (5) in the sense of Definition 3.1. Let ψ Cc ( +, + ) be given, and let S = {ψ } = {(x, t) + ; ψ(x, t) } and S = {ψ(, ) } = {x ; ψ(x, ) }. Then there exists C only depending on ψ L ( + ), t ψ L ( + ), x ψ L ( + ), f, S, S and u ini, such that + 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 µ ini (S ) + (µ 1 (S)) (µ1 + µ )(S). (49) The next theorem is an immediate adaptation of [9, Theorem 29.3 p. 961]. Theorem 3.3. Let u ini BV loc () L (), 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 > and all T > there exist C e and, only depending on, T, f and u ini, such that the following inequality holds: T E ν(x, t) u(x, t) dx dt C e (µ ini ([, ]) + [µ 1 ([, ] [, T ])] (µ1 + µ )([, ] [, T ])). We then deduce the following theorem. Theorem 3.4. Let η t be the misanthrope process defined by the Chapman Kolmogorov equation (1) 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 h 1 nh (n 1)h uini (x) dx. Let ν be the real process defined by (15) and let u be the entropy weak solution of (5). For all > and T >, assuming u ini BV loc () L (), taking k = h 2, then there exists C e, only depending on, T, g, u ini, such that the following inequality holds: T E ν(x, t) u(x, t) dx dt C e h 1 4. (5) The 1/4 exponent in (5) is clearly not sharp (see the numerical results), and the question of improving this result arises. n interesting method, which holds under an additional regularity hypothesis on g, was suggested by the anonymous referee at the revision of this manuscript. Indeed, by comparing the misanthrope stochastic process to the deterministic approximate

22 . Eymard et al. / Stochastic Processes and their pplications 122 (212) solution given by the semi-discrete finite volume scheme (continuous in time), it is possible to avoid the use of the weak BV-inequality Lemma 3.2, whose behavior in h 1/2 is the main reason of the exponent 1/4 in (5). For the sake of simplicity, we assume that the total variation of u ini is globally bounded, and not only locally bounded. Theorem 3.5. Let η t be the misanthrope process defined by the Chapman Kolmogorov equation (1), assuming that g C 2 ( 2 ), 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 β (n) := 1 h nh (n 1)h uini (x) dx. We also assume that u ini BV () L () and k = h 2. Let ν be the real process defined by (14) and let u be the entropy weak solution of (5). Then, for all > and T >, there exists C e, only depending on T,, U, g and u ini such that the following inequality holds: T E ν(x, t) u(x, t) dx dt C e h 1 2. (51) Proof of Theorem 3.5. Since ν and u are bounded, it suffices to consider that h (, 1). We may derive the nonhomogeneous Chapman Kolmogorov equation from (1): deψ(t, η t ) ψ = E dt t (t, η t) + b(η t (n), η t (n + 1)) n Z ψ(t, T n,n+1 (η t )) ψ(t, η t ), t +, ψ D nh, (52) where D nh is the set of all functions ψ from + E to, such that, for all t +, ψ(t, ) D and for all η E, ψ(, η) C 1 ( + ) (it suffices to write the nonhomogeneous function as the limit of time dependent linear combinations of homogeneous ones). Letting the time step in the one dimensional finite volume scheme for Eq. (5) (provided for example in [9, p. 884]) tend to zero, we define the functions β t (n) C 1 ( + ) for all n Z by h dβ t(n) dt = g(β t (n 1), β t (n)) g(β t (n), β t (n + 1)), n Z, t +, (53) with the initial value β (n) given in the statement of this theorem. We associate to the functions β t (n) the function v(x, t) defined from + to [, Ū] by v(x, t) = β t (n) x [(n 1)h, nh), n Z, t +. (54) Denoting by C BV = u ini BV() the total variation of u ini, the following property holds (see for example [9, p. 895]): β t (n) β t (n 1) C BV, t +. (55) n Z Moreover, the following error estimate holds (see for example [9, p. 952]): T v(x, t) u(x, t) dx dt C u h 1 2, (56)

23 367. Eymard et al. / Stochastic Processes and their pplications 122 (212) where C u is only depending on T,, U, g and u ini, and where the h 1 2 error estimate is obtained thanks to the strong BV estimate (55). We define, for all (, 1), the regularization ζ (x) = x y 1 y ρ dy, of the function using a mollifier ρ, in the same sense as in the proof of Lemma 2.3. We have ζ (x) = sign(x y) 1 ρ( y ) dy and ζ (x) = 2 ρ( x ). We then consider the function ψ(t, η t ) defined, for a given n Z, by ψ(t, η t ) = ζ (η t (n) β t (n)). Similarly to the proof of Lemma 2.1, we then obtain that where h de (ζ (η t (n) β t (n))) dt + E(ζ (η t(n) β t (n))(g(η t (n), η t (n + 1)) g(η t (n 1), η t (n)))) E(ζ (η t(n) β t (n))(g(β t (n), β t (n + 1)) g(β t (n 1), β t (n)))) = k t(n), t +, n Z, t (n) := 2g(U, ) max s ρ(s), t +, n Z. (57) We then get that where and h de (ζ (η t (n) β t (n))) dt = k t(n), t +, n Z, + E y, n) + B(t, y, n)) ((t, 1 y ρ dy (t, y, n) = sign(η t (n) β t (n) y)(g(η t (n), η t (n + 1)) g(η t (n 1), η t (n))) + sign(β t (n) + y η t (n))(g(β t (n) + y, β t (n + 1) + y) g(β t (n 1) + y, β t (n) + y)), B(t, y, n) = sign(η t (n) β t (n) y) (g(β t (n) + y, β t (n + 1) + y) g(β t (n), β t (n + 1)) (g(β t (n 1) + y, β t (n) + y) g(β t (n 1), β t (n)))). Thanks to the monotonicity properties of g, we may write for all x, y, z, s, and sign(x z)(g(x, y) g(z, s)) g(x z, y s) g(x z, y s), sign(x z)(g(s, z) g(y, x)) g(y s, x z) g(y s, x z). This implies (t, y, n) (t, y, n) (t, y, n 1),

24 with. Eymard et al. / Stochastic Processes and their pplications 122 (212) (t, y, n) = g(η t (n) (β t (n) + y), η t (n + 1) (β t (n + 1) + y)) g(η t (n) (β t (n) + y), η t (n + 1) (β t (n + 1) + y)). Since g is Lipschitz continuous with constant M, we have (t, y, n) M( η t (n) β t (n) y + η t (n + 1) β t (n + 1) y ), t +, n Z, y. (58) We let δ(y) = g(x + y, z + y) g(x + y,z + y), for given x, z,x,z [, U] and y [, +]. Since δ C 2 (), we may write This gives δ(y) = δ() + yδ (y), for some value y y. g(x + y, z + y) g(x + y,z + y) g(x, z) + g(x,z) = y( 1 g(x + y, z + y) + 2 g(x + y, z + y) 1 g(x + y,z + y) 2 g(x + y,z + y)), and therefore, denoting by G 2 a bound of the second order derivatives of g on [ 1, U + 1], for all y [, +], g(x + y, z + y) g(x + y,z + y) g(x, z) + g(x,z) G 2 ( x x + z z ). We then get that, for y [, +], B(t, y, n) G 2 ( β t (n) β t (n 1) + β t (n + 1) β t (n) ). We then obtain, using (57), h de (ζ (η t (n) β t (n))) dt + E y, n) (t, y, n 1)) ((t, 1 y ρ dy k + G 2 ( β t (n) β t (n 1) + β t (n + 1) β t (n) ), t +, n Z. Let us now multiply the above equation by exp( nh bt) T T t, for b > chosen later and t [, T ], sum on n N and integrate for t [, T ]. We get F 1 + F 2 F 3, with and F 1 = n Z h F 2 = n Z F 3 = n Z T T de (ζ (η t (n) β t (n))) exp( nh bt) T t dt T E y, n) (t, y, n 1)) ((t, 1 y ρ dy exp( nh bt) T t T T dt, dt, k + G 2( β t (n) β t (n 1) + β t (n + 1) β t (n) ) exp( nh bt) T t T dt.

25 3672. Eymard et al. / Stochastic Processes and their pplications 122 (212) n integration by parts provides with and F 1 = 1 h T n Z T E (ζ (η t (n) β t (n))) exp( nh bt) dt + F 11 F 12, (59) F 11 = b T h E (ζ (η t (n) β t (n))) exp( nh bt) T t n Z T F 12 = n Z he (ζ (η (n) β (n))) exp( nh ). We get, thanks to a discrete integration by parts, F 2 = T E (t, y, n) 1 y n Z ρ dy (exp( nh bt) exp( (n + 1)h bt)) T t dt, T which gives, thanks to (58) F 2 M n Z T dt, E (ζ (η t (n) β t (n)) + ζ (η t (n + 1) β t (n + 1))) exp( nh bt) exp( (n + 1)h bt) T t dt. T Using, for all n Z, the bounds and we get exp( nh bt) exp( (n + 1)h bt) h exp(h) exp( nh bt), exp( nh bt) exp( (n + 1)h bt) h exp(h) exp( (n + 1)h bt), T E ζ (η t (n) β t (n)) exp( nh bt) F 2 Mh exp(h) n Z + ζ (η t (n + 1) β t (n + 1)) exp( (n + 1)h bt) T t T which provides 2M exp(h) F 2 F 11. b It now suffices to take b = 2Me, to get F 11 + F 2. We have 2 exp( nh ) = 1 exp( h) 1 2e h, n Z dt, (6)

26 . Eymard et al. / Stochastic Processes and their pplications 122 (212) using 1 exp( h) h e for h 1. Therefore, using η (n) β (n) k, we get F 12 2e(k + ). Finally, we have, using (55), F 3 T k 2e h + 2T G 2 C BV. (62) It now suffices to take = k/h = h for obtaining, from (59) (62), the existence of C 1, only depending on T, g, U, and u ini, such that exp( 1 bt ) T k E (ζ (ν(x, t) v(x, t))) dx dt C 1 T h + k, since exp( nh bt) exp( 1 bt ) for all n Z such that [(n 1)h, nh] [, ] and all t [, T ]. We then conclude (51), using ζ (x) x for all x, and using (56). 4. Numerical computations on the iemann problem For a given U >, one considers Problem (1) (4) with (, B) = (, 1), assuming that u ini is defined, for given real values u l, u r [, U] and x (, 1), by u ini ul if x < x (x) = (63) otherwise u r and that the functions u and u are respectively defined by u(t) = u l and u(t) = u r for all t + (this problem is the generalization of the iemann 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, with v, 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: u(x, t) = u ini ul if x tv < x (x tv) = u r if x tv > x. 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. arefaction 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. ssuming f strictly convex (resp. strictly concave), u l > u r (resp. u l < u r ) and denoting σ = f (u r ) f (u l ) u r u l, then the entropy weak solution u of Problem (1) (4) is the shock wave starting at x and propagating at the constant speed σ ul if (x x u(x, t) = )/t < σ, u r if (x x )/t > σ. The solution is a shock wave starting at x and propagating at the constant speed σ. (61)

27 3674. Eymard et al. / Stochastic Processes and their pplications 122 (212) Fig. 2. Jump rate function for the triangular flux (64). Left, case of the Godunov numerical flux (12) and, right, case of the modified usanov one (13). 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 u l if (x x )/t < f (u l ), u(x, t) = G((x x )/t) if f (u l ) (x x )/t f (u r ), u r if (x x )/t > f (u r ), with G is the reciprocal function to f, which means that, for a given value u [u l u r, u l u r ], then u(x + t f (u ), t) = u. With the aim to model traffic flow, the flux function f is henceforth assumed to be positive and unimodal. 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 [19,16]. The two following positive, unimodal and concave flux functions are considered: f 1 (u) = u u 1/2 1 u 1/2 < u 1 (64) and f 2 (u) = u(1 u). (65) Hence f 1 C () is a triangular and piecewise linear flux function while f 2 C () 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 [[, 1/h 2 ]]) [[1,1/h]] with h > such that 1/h N (one assumes with the previous notation that k = h 2, U = 1 and K = 1/h 2 ). The process is characterized by the Chapman Kolmogorov equation (9). The jump rate of the process, given by (7), depends on the numerical flux function. One uses and compares two numerical fluxes in the numerical experiments: the Godunov numerical flux (12) and the modified positive usanov numerical flux given by (13) (see Figs. 2 and 5). 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.