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Abstract
Under a standard CFL condition, we prove the convergence of the explicit-in- time Finite Volume method with monotone fluxes **for** the approximation of **scalar** first-order **conservation** **laws** with multiplicative, compactly supported noise. In [9], a framework **for** the analysis of the convergence of approximations to stochastic **scalar** first-order **conservation** **laws** has been developed, on the basis of a **kinetic** formulation. Here, we give a **kinetic** formulation of the numerical method, analyse its properties, and explain how to cast the problem of convergence of the numerical scheme into the framework of [9]. This uses standard estimates (like the so-called “weak BV estimate”, **for** which we give a proof using specifically the **kinetic** formu- lation) and an adequate interpolation procedure.

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Our approach is based on the concepts and methods introduced by Lions and Souganidis [16, 17] and extended by the same authors in [21, 20, 19, 18] **for** the theory of pathwise stochastic viscosity solution of fully nonlinear first- and second-order stochastic pde including stochastic Hamilton-Jacobi equations. One of the fundamental tools of this theory is the class of test functions constructed by inverting locally, and at the level of test functions, the flow of the characteristics corresponding to the stochastic first-order part of the equation and smooth initial data. Such approach is best implemented **for** **conservation** **laws** using the **kinetic** formulation which we follow here.

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Our purpose here is to apply the **kinetic** formulation of [3] to show the convergence of the BGK approximation. To this end we ﬁrst study the BGK equation itself in § 2. In § 3 we introduce the **kinetic** formulation **for** the limit problem. We also introduce a notion of the generalized (**kinetic**) solution in deﬁnition 3.3. We show that any generalized solution reduces to a mere solution, i.e. a solution in the sense of deﬁnition 3.1. This theorem of ‘reduction’ is theorem 3.4. Then, in § 4 we show that the BGK model converges to a generalized solution of (1.4) and, using theorem 3.4, deduce the strong convergence of the BGK model to a solution of (1.4), theorem 4.1.

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1.3 Main results and organization of the paper
The organization of the paper is as follows. In the section 2, we consider the assumptions the **kinetic** model have to satisfy in order that our study works. First, we need two assumptions (2.1)-(2.2) to assure the consistance between the **kinetic** equation and the non-local **scalar** equation. Then, we expose the hypotheses necessary to obtain the existence of solutions **for** the **kinetic** equa- tion according to the method used. We will present two proof methods, each one requiring specific hypotheses. **For** the first existence result, we need (2.3). **For** the second existence result, we need (2.4)-(2.10). This section ends with a formal proof that justifies the need **for** consistence assumptions.

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but **for** p = 2 [Brn]. However, as shown in the present work, L 2 is a perfectly
suitable space **for** entropy solutions to multidimensional **scalar** **conservation** **laws**, provided a different formulation is used, based on a combination of level- set, **kinetic** and transport-collapse approximations, in the spirit of previous works by Giga, Miyakawa, Osher, Tsai and the author [Br1, Br2, Br3, Br4, GM, TGO].

The **kinetic** formulation **for** smooth paths. We review here the basic concepts of the **kinetic** theory of **scalar** **conservation** **laws** and we show that it allows to define a change of variable along the “**kinetic**” characteristics which turns out to be a very convenient tool **for** the study of sscl.
Although we use the notation of the introduction, here we assume that W ∈ C 1 ([0, ∞); R), in which case du stands **for** the usual derivative and ◦ is the usual multiplication and, hence, should be ignored. The entropy inequalities (see, **for** example, [3, 30]), which yield (8) and guarantee the uniqueness of the entropy solutions, are

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In the general inhomogeneous case, that is, **for** (1.1), no bounded variation estimates are known either **for** the solution u or **for** the approximations u ∆t . In addition, due to the spatial dependence, we cannot
use averaging techniques. To circumvent these difficulties, we devise a new method of proof based on the concept of generalized **kinetic** solutions and new energy estimates (see Lemma 3.3 below). The result (see Theorem 3.1) is that, if u 0 ∈ (L 1 ∩ L 2 )(R N ), then

The **kinetic** formulation **for** smooth paths. We review here the basic concepts of the **kinetic** theory of **scalar** **conservation** **laws** and we show that it allows to define a change of variable along the “**kinetic**” characteristics which turns out to be a very convenient tool **for** the study of sscl.
Although we use the notation of the introduction, here we assume that W ∈ C 1 ([0, ∞); R), in which case du stands **for** the usual derivative and ◦ is the usual multiplication and, hence, should be ignored. The entropy inequalities (see, **for** example, [3, 30]), which yield (8) and guarantee the uniqueness of the entropy solutions, are

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Our purpose here is to apply the **kinetic** formulation of [BV06] to show the convergence of the BGK approximation. To this aim, we first study the BGK equation in itself in Section 2. In Section 3, we introduce the **kinetic** formula- tion **for** the limit problem. We also introduce a notion of generalized (**kinetic**) solution, Definition 6. We show that any generalized solution reduces to a mere solution, i.e. a solution in the sense of Def. 4. This theorem of “reduction” is Theorem 7. Then in Section 4, we show that the BGK model converges to a generalized solution of (4) and, using Theorem 7, deduce the strong convergence of the BGK model to a solution of (4), Theorem 11.

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the kind (A 2 ), **for** the traces of G-entropy process solution. This prevents us
from mimicking the proof of uniqueness of entropy solutions; as a matter of fact, we are unable to give a sign to the term coming from the comparison of two G-entropy process solutions at the interface {x = 0} (note that in [?], Bachmann and Vovelle propose to use an even weaker notion of solution, based on a **kinetic** interpretation of the problem; it could be interesting to extend their idea to our framework). However, because we know the existence of a G -entropy solution, we can compare a G -entropy process solution with a G -entropy solution and thus deduce the uniqueness and the reduction principle **for** G-entropy solutions: Proposition 3.3. Let u 0 ∈ L ∞ (R; [0, 1]). If u is the G -entropy solution and if

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The regularizing effect **for** this type of solutions of (2) has been studied so far by using a **kinetic** formulation of the **scalar** **conservation** law: see [27] **for** the original contribution, [22] **for** an improved regularity result, and [32] **for** a detailed presen- tation of **kinetic** formulations and their properties. The tool **for** establishing the regularizing effect **for** **kinetic** formulations is a class of results known as velocity av- eraging, introduced independently in [1] and [19], with subsequent generalizations and improvements described **for** instance in chapter 1 of [7] — see also the list of references given in section 3.

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In the 1960’s, Jacques Yvon was professor of physics at the University of Paris. Beforehand, he had directed the Department of physics and nuclear reactors at Saclay. He therefore asked Cirano De Dominicis, then myself, who worked in this department, to become his assistant **for** his course of statistical mechanics. He focused on classical fluids, modelled as an assembly of N point particles with mass m interacting through a two-body potential W (|r j − r k |). His lectures started from the most fundamental

The time-dependent variable y denotes the bus position. When the traffic conditions allow it, the bus moves at its own maximal speed denoted by V b < V .
When the surrounding traffic is too dense, the bus adapts its velocity accord- ingly, therefore it is not possible **for** the bus to overtake the cars, see Figure 2 . From a mathematical point of view, the velocity of the bus can be described by a traffic density dependent speed of the form:

The work summarized in this review was devoted to the approximation and numerical resolution of the mechanical interaction between a viscous incompressible fluid and an elastic structure, with a strong added-mass effect. In this framework, standard explicit (or loosely coupled) **schemes** are known to be unstable, irrespectively of the discretization parameters. In the context of **implicit** coupling, we have seen that the exact evaluation of the cross-derivative of the Jacobian (shape terms) leads to robust Newton iterations. Yet, these procedures remain computationally expensive in real applications.

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loc convergence in [5]. Here, results of [5] are specified in C 1 **for** a well
chosen phase and proved **for** a particular sequence of smooth solutions (without shocks on a strip). This allows us to prove that, necessarily, the best uniform Soblev exponent s, **for** entropy solutions and **for** positive time, satisfies s ≤ α **for** a ball of L ∞ initial data. Notice that we look **for** the best uniform Sobolev exponent **for** a set of solutions. The smoothness of any individual solution is not studied in this paper. This point is discussed later.

Let u := u ε be the Krushkov solution of (1.1)–(1.2). Consider the geometric optics asymptotic expansion of the solution (1.4). Then the new approach in Section 4 **for** one-dimensional **conservation** **laws** requires further refinement **for** solving the general nonlinear geometric optics **for** multidimensional **scalar** **conservation** **laws**. We need a general scaling of variables to recover all the numerous cases. We will perform that with a “quasi” LU factorization depending on the magnitude of all frequencies. We will also use Lemma 3.1 to preserve the L 1

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(b m,k (t) dt ≤ ∧ |ρ(j + A) − ρ( k + A) | . (3.27) Under the above notations we have
Proof of Lemma 2.1 Let ρ satisfies (2.7), then **for** ρ( j + A) < T and from (3.25), (3.26), **for** each j, {f 0 (b m,j ) } m ∈N is bounded in total variation norm. Therefore from super linearity of f, {b m,j } m ∈N is uniformly bounded in L ∞ **for** all j, m. Hence from Helly’s theorem and Cantors diagonalization, we can extract a subsequence still de- noted by {b m,j } such that **for** every j, f 0 (b m,j ) → f 0 (b j ) as m → ∞ in L 1 and **for** a.e. t. Since (f 0 ) −1 exist and hence b m,j → b j a.e. t and by dominated convergence Theo- rem, b m,j → b j in L 1 . Let ρ m,j (x) = ρ(x, P m,n j , j ), then from (3.14) ρ m,j (x) → ρ ε j (x)

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