Resumen
In this paper we give a proof of convergence of a new numerical method introduced in [N. Besse and E. Sonnendrücker, J. Comput. Phys., 191 (2003), pp. 341-376] for the Vlasov equation. The numerical method is based on the semi-Lagrangian principle and the transport of the gradient of the statistical distribution function in order to get a high-order and stable reconstruction. These kinds of new schemes have been successfully implemented on unstructured meshes of four-dimensional phase space (cf. [N. Besse, Etude mathématique et numérique de l'equation de Vlasov sur des maillages non structurés de l'espace des phases, thèse de l'Université Louis Pasteur, Strasbourg, France, 2003; N. Besse, J. Segré, and E. Sonnendrücker, Transport Theory Statist. Phys., 34 (2005), pp. 311-332]). In order to make a rigorous proof of convergence of this method and simplify the convergence analysis, we have considered the periodic one-dimensional Vlasov-Poisson system in phase space on a grid. The distribution $f(t,x,v)$ and the electric field are shown to converge to the exact solution values in $H^1$ norm. The rate of convergence is of ${\mathcal{O}(\Delta t^2 +\frac{\Delta x^{4-|\alpha|}}{\Delta t}+ \frac{\Delta v^{4-|\alpha|}} {\Delta t})}$, $\alpha \in \N^2, |\alpha|\leq 1$.