We present here some numerical schemes for general multidimensional systems of conservation laws based on a class of discrete kinetic approximations, which includes the relaxation schemes by S. Jin and Z. Xin. These schemes have a simple formulation even in the multidimensional case and do not need the solution of the local Riemann problems. For these approximations we give a suitable multidimensional generalization of the Whitham's stability subcharacteristic condition. In the scalar multidimensional case we establish the rigorous convergence of the approximated solutions to the unique entropy solution of the equilibrium Cauchy problem.

Abstract. We consider a simple model case of stiff source terms in hyperbolic conservation laws, namely, the case of scalar conservation laws with a zeroth order source with low regularity. It is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory results for both the reduced CFL condition and refined meshes required because of the lack of accuracy on equilibrium states. The source term should be taken into account in the upwinding and discretized at the nodes of the grid. In order to solve numerically the problem, we introduce a so-called equilibrium schemes with the properties that (i) the maximum principle holds true; (ii) discrete entropy inequalities are satisfied; (iii) steady state solutions of the problem are maintained. One of the difficulties in studying the convergence is that there are no BV estimates for this problem. We therefore introduce a kinetic interpretation of upwinding taking into account the source terms. Based on the kinetic formulation we give a new convergence proof that only uses property (ii) in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions. The computational efficiency of our equilibrium schemes is demonstrated by numerical tests that show that, in comparison with an usual upwind scheme, the corresponding equilibrium version is far more accurate. Furthermore, numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients. 1.

We develop a well-posedness theory for solutions in L to the Cauchy problem of general degenerate parabolic-hyperbolic equations with non-isotropic nonlinearity. A new notion of entropy and kinetic solutions and a corresponding kinetic formulation are developed which extends the hyperbolic case. The notion of kinetic solutions applies to more general situations than that of entropy solutions; and its advantage is that the kinetic equations in the kinetic formulation are well defined even when the macroscopic fluxes are not locally integrable, so that L is a natural space on which the kinetic solutions are posed. Based on this notion, we develop a new, simpler, more e#ective approach to prove the contraction property of kinetic solutions in L , especially including entropy solutions. It includes a new ingredient, a chain rule type condition, which makes it di#erent from the isotropic case. Resume Nous developpons une theorie d'existence et unicite pour les solutions L seulement du probleme de Cauchy pour un probleme de Cauchy hyperbolique-parabolique avec di#usion nonisotrope generale. Des notions de formulations entropique et cinetique sont introduites qui incorporent un nouvel ingredient, une condition de type derivation composee, qui montre la di#erence fondamentale avec le cas d'une di#usion isotrope. L'avantage de la notion de solution cinetique est de travailler directement dans l'espace naturel L . Key-words: Kinetic solutions, entropy solutions, kinetic formulation, degenerate parabolic equations, convection-di#usion, non-isotropic di#usion, stability, existence, well-posedness Subj. numbers: 35K65, 35K10, 35B30, 35D05 1.

The motion of a collection of vertical strings subject to horizontal linear vibrations in the plane can be described by a system of first order nonlinear conservations laws. This system-that we call the Chaplygin-Born-Infeld (CBI) system- is related to Magnetohydrodynamics and more specifically to its shallow water version. Then, each vibrating string can be interpreted as a magnetic line. The CBI system is also related to the Born-Infeld theory for the electromagnetic field, a nonlinear correction to the classical Maxwell’s equations. Due to the linearity of vibrations, there is a priori no mechanism to prevent the strings to cross each other, at least for sufficiently large initial impulse. These crossings generate concentration singularities in the CBI system. A numerical scheme is introduced to maintain order preserving strings beyond singularities. This order preserving scheme is shown to be convergent to a distinguished limit, which can be interpreted, through maximal monotone operator theory, as a vanishing viscosity limit of the CBI system. Finally, models of pressureless gas with sticky particles are revisited and a new formulation is provided.

This article is devoted to the proof of the hydrodynamical limit from kinetic equations (including B.G.K. like equations) to multidimensional isentropic gas dynamics. It is based on a relative entropy method, hence the derivation is valid only before shocks appear on the limit system solution. However, no a priori knowledge on high velocities distributions for kinetic functions is needed. The case of the Saint-Venant system with topography (where a source term is added) is included. Key-words: Hydrodynamic limit, Entropy method, B.G.K. equation, Isentropic gas dynamics, Saint-Venant system.

monotone operator

Abstract. We consider here second-order finite volume methods for onedimensional scalar conservation laws. We give a method to determine a slope reconstruction satisfying all the exact numerical entropy inequalities. It avoids inhomogeneous slope limitations and, at least, gives a convergence rate of ∆x 1/2. It is obtained by a theory of second-order entropic projections involving values at the nodes of the grid and a variant of error estimates, which also gives new results for the first-order Engquist-Osher scheme. 1.

Abstract. Averaging lemmas consist in a regularizing effect on the average of the solution to a linear kinetic equation. Some of the main results are reviewed and their proofs presented in as self contained a way as possible. The use of kinetic formulations for the well posedness of scalar conservation laws is eventually explained as an example of application. Key words. Regularizing effects, averaging lemmas, dispersion estimates, conservation laws. Mathematics Subject Classification 35B65, 82C40, 47G10.

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Kinetic/Fluid coupling