Convergence of the Finite Volume Method for Multidimensional Conservation Laws

We establish the convergence of the nite volume method applied to multidimensional hyperbolic conservation laws, and based on monotone numerical ux-functions. Our technique applies under a fairly unrestrictive assumption on the triangulations (\\at elements" are allowed), and to Lipschitz continuous ux-functions. We treat the initial and boundary value problem, and obtain the strong convergence of the scheme to the unique entropy discontinuous solution in the sense of Kruzkov. The proof of convergence is based on a convergence framework due to Coquel and LeFloch (Math. of >From a convex decomposition of the scheme, we derive a new estimate for the rate of entropy dissipation, and a new formulation of the discrete entropy inequalities. These estimates are shown to be suucient for the passage to the limit in the discrete equation. Convergence follows from DiPerna's uniqueness result in the class of entropy measure-valued solutions.