CONVERGENCE OF A HIGH-ORDER SEMI-LAGRANGIANSCHEME WITH PROPAGATION OF GRADIENTS FOR THEONE-DIMENSIONAL VLASOV—POISSON SYSTEM

摘要

In this paper we give a proof of convergence of a new numerical method introduced in[N. Besse and E. Sonnendrficker, 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 ofthe statistical distribution function in order to get a high-order and stable reconstruction. These kindsof new schemes have been successfully implemented on unstructured meshes of four-dimensional phasespace (cf. [N. Besse, Etude mathematique et numerique de l'equation de Vlasov sur des maillagesnon structures de l'espace des phases, these de ('Universite Louis Pasteur, Strasbourg, France, 2003;N. Besse, J. Segre, and E. Sonnendrficker, Transport Theory Statist. Phys., 34 (2005), pp. 311-332]). In order to make a rigorous proof of convergence of this method and simplify the convergenceanalysis, we have considered the periodic one-dimensional Vlasov–Poisson system in phase space ona grid. The distribution f (t, x, v) and the electric field are shown to converge to the exact solution values in Ht norm. The rate of convergence is of C9(At2 6'1 t) a E N2, la] < 1.