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.

WestEC here te comput tmp of shallow-wath equat ons wi ttN:LE&BN y byFinit Volume metmeN , in a one-dimensional framework(tNL&3 allmetB: sint oduced may benatEfiLLN extEfiLL in t o dimensions) . AllmetA:3 are based on a discretcrNB on of tN tBCEAEN(B by a piecewisefunctE n constEC on each cell of tN mesh, from an original idea of Le Rouxet al. Whereaste Well-Balanced scheme of Le Roux is based on tN exact resol utol of each Riemann problem, we consider here approximat Riemann solvers. Several singlestg metleN are derived from tom formalism, and numerical result are comparedt a fract ionalsta metN4 . Some tme cases arepresent& : convergencetn ardsstsN: stsN: in subcritECC and supercriterc configuratfiE s, occurrence of dry area by a drain over a bump and occurrence of vacuum by a double rarefactNL wave over astAA Numerical schemes, combined wi t an appropriat high-order extNfi ion, provideaccurat e and convergent approxim atximN # 2003 Elsevier ScienceLtc Allright s reserved. 1. I5964V139 We stE4 intNL paper some approximat Godunov schemest comput shallow-watN equatlow wit a sourcetur oftBAL4fiN(LA in a one-dimensional framework. Allmetfi3L presentL may beextB3L4 natB3L4N t tt tB3L4N(&CL4 al model.

This paper deals with the diusion limit of a kinetic equation where the collisions are modeled by a Lorentz type operator. The main aim is to construct a discrete scheme to approximate this equation which gives for any value of the Knudsen number, and in particular at the diusive limit, the right discrete diusion equation with the same value of the diusion coecient as in the continuous case. We are also naturally interested with a discretization which can be used with few velocity discretization points, in order to reduce the cost of computation.

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.

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.

Numerical methods with different orders of accuracy are proposed to approximate hyperbolic models for chemosensitive movements. On the one hand, first- and second-order well-balanced finite volume schemes are presented. This approach provides exact conservation of the steady state solutions. On the other hand, a high-order finite difference weighted essentially nonoscillatory (WENO) scheme is constructed and the well-balanced reconstruction is adapted to this scheme to exactly preserve steady states and to retain high-order accuracy. Numerical simulations are performed to verify accuracy and the well-balanced property of the proposed schemes and to observe the formation of networks in the hyperbolic models similar to those observed in the experiments.

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.

We study in this paper some numerical schemes for hyperbolic systems with unilateral constraint. In particular, we deal with the scalar case, the isentropic gas dynamics system and the full-gas dynamics system. We prove the