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 are positive, bounded, continuous, and have their first moment bounded, then the numerical approximation converges to the weak solution of the system for the weak topology of L. Moreover, if the initial data belong to BV , the convergence is strong in C(0, T ;L loc ). To prove the convergence of the discrete electric field, we obtain an estimation in W (ΩT ). Then we have fh(t, x, v) ⇀ f(t, x, v) in L (QT ) weak-? as h → 0, Eh(t, x) → E(t, x) in C (ΩT ) as h → 0, where (E, f) is the unique weak solution of the Vlasov–Poisson system.