Deterministic and random particle methods applied to Vlasov-Poisson-Fokker-Planck kinetic equations
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We devise and study two different particle methods for approximating Vlasov-Poisson-Fokker-Planck systems. We first consider a random particle method. Such a proposed scheme takes into account the fact that the trajectories of a particle, undergoing Brownian motion due to collisions with the medium or background particles, can be obtained as the solutions of stochastic differential equations, i.e., the Langevin equations. These equations are the precise analogs of the (deterministic) Hamiltonian system in the collisionless model. The particle approximation, in particular, simulates the action of viscosity by the use of independent Wiener processes (Brownian motions). The analysis relies heavily on the machinery developed by K. Ganguly and H.D. Victory, Jr. [SIAM J. Numer. Anal., 26 (1989), pp. 249-288] to treat the sampling errors due to random motions of the particles. For example, the key idea in the consistency error analysis is to separate the moment and discretization errors - accounting for the deterministic portions of the error - from the sampling errors introduced by the random motion of the particles. The latter errors constitute the dominant component of the overall consistency error in terms of order and are gauged by applying Bennett's Inequality utilized to estimate tail probabilities for standardized sums of independent random variables with zero means. Moreover, the stability estimates for the particle approximations to the collisionless model are extended to the Vlasov-Poisson-Fokker-Planck setting by means of this inequality. We then consider a deterministic method. 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 fieldfree 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 L^- senses. This treatment is the first application of the velocity moment analysis by P.L. Lions and B. Perthame [Invent. Math., 105 (1991), 415- 30] in a numerical analysis of Vlasov-type kinetic equations. Our study is made feasible by some formulas by F. Bouchut [J. Fund Anal, 111 (1993), 239-258] concerning the field-free fundamental solution and recent extensions to the periodic setting. The splitting procedure we employ is related to the viscous splitting or fractional step procedure of G.H. Cottet and S. Mas-Gallic [Numer. Math., 57 (1990), 805-827] for treating Navier-Stokes equations modeling viscous, incompressible flow.