We consider the Cauchy problem for the 2 2 nonstrictly hyperbolic system u t + f(a; u) x g(a; u)a x = 0 (a; u)(0; ) = (a o ; u o ): For possibly large, discontinuous and resonant data, the generalized solution to the Riemann problem is introduced, interaction estimates are carried out using a new change of variables and the convergence of Godunov approximations is shown. Uniqueness is addressed relying on a suitable extension of Kruzkov's techniques. 1

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.

We are concerned with efficient numerical simulation of the radiative transfer equations...

incline, when the Froude number is above two. The goal of this paper is to analyze the behavior of numerical approximations to a model roll wave equation u(x; 0) = u 0 (x); which arises as a weakly nonlinear approximation of the shallow water equations. The main difficulty associated with the numerical approximation of this problem is its linear instability. Numerical round-off error can easily overtake the numerical solution and yields false roll wave solution at the steady state.

We derive and study Well-Balanced schemes for quasimonotone discrete kinetic models. By means of a rigorous localization procedure, we reformulate the collision terms as nonconservative products and solve the resulting Riemann problem whose solution is self-similar. The construction of an Asymptotic Preserving (AP) Godunov scheme is straightforward and various compactness properties are established within different scalings. At last, some computational results are supplied to show that this approach is realizable and ecient on concrete 2 × 2 models.

This paper presents a very efficient numerical strategy for computing the weak solutions of scalar conservation laws which fail to be genuinely nonlinear. We concentrate on the typical situation of either concave-convex or convex-concave flux functions. In such a situation, nonclassical shocks violating the classical Oleinik entropy criterion must be taken into account since they naturally arise as limits of certain diffusive-dispersive regularizations to hyperbolic conservation laws. Such discontinuities play an important part in the resolution of the Riemann problem and their dynamics turns out to be driven by a prescribed kinetic function which acts as a selection principle. It aims at imposing the entropy dissipation rate across nonclassical discontinuities, or equivalently their speed of propagation. From a numerical point of view, the serious difficulty consists in enforcing the kinetic criterion, that is in controling the numerical entropy dissipation of the nonclassical shocks for any given discretization. This is known to be a very challenging issue. By means of an algorithm made of two steps, namely an Equilibrium step and a Transport step, we show how to force the validity of the kinetic criterion at the discrete level. The resulting scheme provides in addition sharp profiles. Numerical evidences illustrate the validity of our approach. 1

Abstract. We propose and prove convergence of a front tracking method for scalar conservation laws with source term. The method is based on writing the single conservation law as a 2 × 2 quasilinear system without a source term, and employ the solution of the Riemann problem for this system in the front tracking procedure. In this way the source term is processed in the Riemann solver, and one avoids using operator splitting. Since we want to treat the resonant regime, classical arguments for bounding the total variation of numerical solutions do not apply here. Instead compactness of a sequence of front tracking solutions is achieved using a variant of the singular mapping technique invented by Temple [69]. The front tracking method has no CFL–condition associated with it, and it does not discriminate between stiff and non-stiff source terms. This makes it an attractive approach for stiff problems, as is demonstrated in numerical examples. In addition, the numerical examples show that the front tracking method is able to preserve steady–state solutions (or achieving them in the long time limit) with good accuracy. 1.

In this paper we deal with the numerical approximation of integro-differential equations arising in financial applications in which jump processes act as the underlying stochastic processes. Our aim is to find finite differences schemes which are high-order accurate for large time regimes. Therefore, we study the asymptotic time behavior of such equations and we define as asymptotic high-order schemes those schemes that are consistent with this behavior. Numerical tests are presented to investigate the efficiency and the accuracy of such approximations.

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Hyperbolic conservation laws with source terms arise in many applications, especially as a model for geophysical ows because of the gravity, and their numerical approximation leads to speci c diculties. In the context of nite volume schemes, many authors have proposed to Upwind Sources at Interfaces, i.e. the \U. S. I." method, while a cell-centered treatment seems more natural. This note gives a general mathematical formalism for such schemes. We de ne consistency and give a stability condition for the \U. S. I." method. We relate the notion of consistency to the \well-balanced" property, but its stability remains open, and we also study second order approximations as well as error estimates. The general case of a nonuniform spatial mesh is particularly interesting, motivated by two dimensional problems set on unstructured grids.