Numerical methods for hyperbolic equations of Saint-Venant type.

The purpose of the dissertation is to contribute to the numerical study of hyperbolic conservation laws with source terms, motivated by the application to the Saint-Venant equations for shallow waters. The first part deals with usual questions in the analysis of numerical approximations for scalar conservation laws. We focus on semi-discrete finite volume schemes, in the general case of a nonuniform spatial mesh. To define appropriate discretizations of the source term, we introduce the formalism peculiar to the "Upwind Interface Source" method and we establish conditions on the numerical functions so that the discrete solver preserves the stationary solutions. Then we formulate a rigorous definition of consistency, adapted to "well-balanced schemes", for which we are able to prove a Lax-Wendroff type convergence theorem. The method first considered is essentially first order. To improve accuracy, we develop high resolution approaches for the "Upwind Interface Source" method and we show that these are efficient ways to derive higher order schemes with suitable properties. We prove an error estimate in $L^p$, $1\le p < +\infty$, which is an optimal result in the case of a uniform mesh. We thus conclude that the same convergence rates $O(h)$ and $O(h^2)$ hold as for the corresponding homogeneous systems. The second part presents a numerical scheme to compute Saint-Venant equations, with a geometrical source term, which satisfies the following theoretical properties: it preserves the steady states of still water, it satisfies a discrete entropy inequality, it preserves the non-negativity of the height of water and remains stable with a discontinuous bottom. This is achieved by means of a kinetic approach to the system ; in this context, we use a natural description of the microscopic behaviour of the system to define numerical fluxes at the interfaces of an unstructured mesh. We also use the concept of cell-centered conservative quantities (typical of the finite volume method) and upwind interfacial sources. Finally, we present some numerical simulations of the Saint-Venant system modified by including small friction and viscosity, in order to recover the results of experimental studies. An application to the numerical modelling of friction terms for debris avalanches is proposed in the Appendix.