Haut PDF Implicit Kinetic Schemes for Scalar Conservation Laws

Implicit Kinetic Schemes for Scalar Conservation Laws

Implicit Kinetic Schemes for Scalar Conservation Laws

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Convergence of the Finite Volume Method for scalar conservation laws with multiplicative noise: an approach by kinetic formulation

Convergence of the Finite Volume Method for scalar conservation laws with multiplicative noise: an approach by kinetic formulation

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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Scalar conservation laws with rough (stochastic) fluxes

Scalar conservation laws with rough (stochastic) fluxes

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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Adaptive Anisotropic Spectral Stochastic Methods for Uncertain Scalar Conservation Laws

Adaptive Anisotropic Spectral Stochastic Methods for Uncertain Scalar Conservation Laws

FOR UNCERTAIN SCALAR CONSERVATION LAWS J. TRYOEN † ‡ , O. LE MAˆITRE ‡ , AND A. ERN † Abstract. This paper deals with the design of adaptive anisotropic discretization schemes for conservation laws with stochastic parameters. A Finite Volume scheme is used for the deter- ministic discretization, while a piecewise polynomial representation is used at the stochastic level. The methodology is designed in the context of intrusive Galerkin projection methods with Roe-type solver. The adaptation aims at selecting the stochastic resolution level based on the local smoothness of the solution in the stochastic domain. In addition, the stochastic features of the solution greatly vary in the space and time so that the constructed stochastic approximation space depends on space and time. The dynamically evolving stochastic discretization uses a tree-structure representation that allows for the efficient implementation of the various operators needed to perform anisotropic multiresolution analysis. Efficiency of the overall adaptive scheme is assessed on the stochastic traffic equation with uncertain initial conditions and velocity leading to expansion waves and shocks that propagate with random velocities. Numerical tests highlight the computational savings achieved as well as the benefit of using anisotropic discretizations in view of dealing with problems involving a larger number of stochastic parameters.
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A Bhatnagar-Gross-Krook approximation to scalar conservation laws with discontinuous flux

A Bhatnagar-Gross-Krook approximation to scalar conservation laws with discontinuous flux

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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Solutions of kinetic equations related to non-local conservation laws

Solutions of kinetic equations related to non-local conservation laws

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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L2 formulation of multidimensional scalar conservation laws

L2 formulation of multidimensional scalar conservation laws

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].

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Scalar conservation laws with rough (stochastic) fluxes; the spatially dependent case

Scalar conservation laws with rough (stochastic) fluxes; the spatially dependent case

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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Semi-discretization for Stochastic Scalar Conservation Laws with Multiple Rough Fluxes

Semi-discretization for Stochastic Scalar Conservation Laws with Multiple Rough Fluxes

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

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Scalar conservation laws with rough (stochastic) fluxes; the spatially dependent case

Scalar conservation laws with rough (stochastic) fluxes; the spatially dependent case

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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A BGK approximation to scalar conservation laws with discontinuous flux

A BGK approximation to scalar conservation laws with discontinuous flux

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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Finite volume schemes for constrained conservation laws

Finite volume schemes for constrained conservation laws

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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Stability estimates for non-local scalar conservation laws

Stability estimates for non-local scalar conservation laws

For this type of equations, general existence and uniqueness results have been established in [4, 7] for specific classes of scalar equations in one space-dimension, and in [1] for multi- dimensional systems of equations coupled through the non-local term. In particular, existence is usually proved by providing suitable compactness estimates on a sequence of approximate solutions constructed by finite volume schemes, while L 1

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Optimal Regularizing Effect for Scalar Conservation Laws

Optimal Regularizing Effect for Scalar Conservation Laws

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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Microscopic local conservation laws for classical fluids

Microscopic local conservation laws for classical fluids

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

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Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result

Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result

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:

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Coupling schemes for incompressible fluid-structure interaction: implicit, semi-implicit and explicit

Coupling schemes for incompressible fluid-structure interaction: implicit, semi-implicit and explicit

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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High frequency waves and the maximal smoothing effect for nonlinear scalar conservation laws

High frequency waves and the maximal smoothing effect for nonlinear scalar conservation laws

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.

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Validity of nonlinear geometric optics for entropy solutions of multidimensional scalar conservation laws

Validity of nonlinear geometric optics for entropy solutions of multidimensional scalar conservation laws

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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Exact controllability of scalar conservation laws with strict convex flux

Exact controllability of scalar conservation laws with strict convex flux

(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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