On Deterministic Particle Methods for Solving Vlasov--Poisson--Fokker--Planck Systems

Abstract

We devise and study a deterministic method for approximating Vlasov--Poisson--Fokker--Planck systems. Such a proposed scheme is a splitting method, whereby particle methods are used to treat the convective part and the diffusion is simulated by convolving the particle approximation with the field-free Fokker--Planck kernel. The states of the particles are not affected by the diffusion per se, but the charge or mass on the particles in their previous states is redistributed via the diffusion. Because of this redistribution of mass or charge, it is necessary to monitor the growth in time of the velocity moments of the approximate distribution. Convergence of the errors in both the density and the fields is shown to be first order in time with respect to both the uniform and Lp- senses. This treatment is the first application of the velocity moment analysis by P.L. Lions and B. Perthame [Invent. Math., 105 (1991), pp. 415--430] in a numerical analysis of Vlasov-type kinetic equations. Our study is made feasible by some formulas by F. Bouchut [J. Funct. Anal., 111 (1993), pp. 239--258] concerning the field-free fundamental solution and recent extensions to the periodic setting. The splitting procedure we employ in this work is related to the viscous splitting or fractional step procedure of G.H. Cottet and S. Mas-Gallic [Numer. Math., 57 (1990), pp. 805--827] for treating Navier--Stokes equations modeling viscous, incompressible flow.