Poisson-Vlasov: stochastic representation and numerical codes

Abstract

A stochastic representation for the solutions of the Poisson-Vlasov equation, with several charged species, is obtained. The representation involves both an exponential and a branching process and it provides an intuitive characterization of the nature of the solutions and its fluctuations. Here, the stochastic representation is also proposed as a tool for the numerical evaluation of the solutions

Domain decomposition solution of nonlinear two-dimensional parabolic problems by random trees

“…The error made in doing so is statistical in nature and of the order of N À1=2 . However, it turns out that when we evaluate numerically functionals of a branching process, the fluctuations around the mean are often non-Gaussian (see [18,35], e.g.). For this reason, a large deviation analysis was conducted to assess the reliability of our numerical results.…”

“…5, a comparison between the analytical results obtained in (18) and the measured computational time is shown as function of the final time, t in logarithmic scale (for the y-axis). Here the computational time, spent in solving probabilistically the corresponding initial value problem for the nonlinear PDE in (8), at a single point x ¼ 0, has been measured.…”

“…Subsequent theoretical advances have been accomplished recently, aiming to generalize the results of [25] to other nonlinear PDEs of interest to Physics and Engineering [36]. In [18,34], a theoretical investigation on kinetic equations of the Vlasov-Poisson type has been accomplished and preliminary numerical simulations conducted, showing the computational feasibility of this approach.…”

Poisson-Vlasov in a strong magnetic field: A stochastic solution approach

“…The exit measures provided by diffusion plus branching processes [6] [7] [8] [9] [10] [11] as well as the stochastic representations recently constructed for the Navier-Stokes [12] [13] [14] [15] [16] [17], the Poisson-Vlasov [18] [19], the Euler [20] and a nonlinear fractional differential equation [21] define initial condition-independent processes for which the mean values of some functionals are solutions to these equations. Therefore, they are exact stochastic solutions.…”

“…In this paper, pursuing the work on kinetic equations initiated in [18] and [19], solutions are obtained for the Maxwell-Vlasov equation in the approximation where magnetic field fluctuations are neglected and the electrostatic potential is used to compute the electric field. This is a reasonable approximation for plasmas in a strong external magnetic field.…”