ON BOUNDARY CONDITIONS FOR MULTIDIMENSIONAL HYPERBOLIC SYSTEMS OF CONSERVATION LAWS IN THE FINITE VOLUME FRAMEWORK

This report presents a method for imposing boundary conditions in the context of hyperbolic systems of conservation laws in the finite volume framework. This method is particularly well suited for approximations in the framework of Finite Volume Methods in the sense that it computes directly the normal flux at the boundary with using just the hyperbolic nature of the system and nothing else. We discuss both linear and non linear problems. In the first group, we consider the wave equation, the Maxwell system and the linear elasticity problem. In the second group, we firstly study conservative systems as the magneto-hydrodynamic system, the Euler equations together with its classical reduced versions : isentropic, isothermal and shallow-water approximations and finally we consider complex (non conservative) models arising in the numerical computation of two fluid models. These latter systems initially motivated our approach. For each application, we analyze the hyperbolicity and write the eigensystem, then we present the so-called VFFC finite volume approach and provide at the discrete level a general theory for the boundary condition treatment. We analytically and numerically compare our treatment with the incomplete Riemann invariant technique. Finally we address practical issues and we present some numerical results. A last section is devoted to the widely studied one dimensional Euler equations for inviscid flow.