Abstract not found

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 present a class of first and second order in space and time relaxation schemes for the shallow water (SW) equations. A new approach of incorporating the geometrical source term in the relaxation model is also presented. The schemes are based on classical relaxation models combined with Runge-Kutta time stepping mechanisms. Numerical results are presented for several benchmark test problems with or without the source term present.

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.

Abstract. This paper is devoted to a review of the analysis tools which have been developed for the the mathematical study of cell centred finite volume schemes in the past years. We first recall the general principle of the method and give some simple examples. We then explain how the analysis is performed for elliptic equations and relate it to the analysis of the continuous problem; the lack of regularity of the approximate solutions is overcome by an estimate on the translates, which allows the use of the Kolmogorov theorem in order to get compactness. The parabolic case is treated with the same technique. Next we introduce a co-located scheme for the incompressible Navier–Stokes, which requires the definition of some discrete derivatives. Here again, we explain how the continuous estimates can guide us for the discrete estimates. We then give the basic ideas of the convergence analysis for non linear hyperbolic conservation laws, and conclude with an overview of the recent domains of application. Key words. Finite volume methods, elliptic equations, parabolic equations, Navier-Stokes equations, hyperbolic equations AMS subject classifications. 65M12, 65N12, 76D05, 76D07, 76M12

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.

Flux-gradient and source term balancing for certain high resolution shock-capturing schemes

for the shallow water equations

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.

with high friction