Convergence of a finite volume scheme for the Vlasov-Poisson system

We propose a finite volume scheme to discretize the one-dimensional Vlasov-Poisson system, we prove that, if the initial data is bounded, continuous, and has its first moment bounded, then the numerical approximation weakly converges to the solution of the system. Moreover if the initial data belongs to BV, the convergence is strong in C-0([0,T]; L-loc(1)). (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.