Asymptotic preserving schemes for the Wigner–Poisson–BGK equations in the diffusion limit

Abstract

This work focuses on the numerical simulation of the Wigner–Poisson–BGK equation in the diffusion asymptotics. Our strategy is based on a “micro–macro” decomposition, which leads to a system of equations that couple the macroscopic evolution (diffusion) to a microscopic kinetic contribution for the fluctuations. A semi-implicit discretization provides a numerical scheme which is stable with respect to the small parameter ε (mean free path) and which possesses the following properties: (i) it enjoys the asymptotic preserving property in the diffusive limit; (ii) it recovers a standard discretization of the Wigner–Poisson equation in the collisionless regime. Numerical experiments confirm the good behavior of the numerical scheme in both regimes. The case of a spatially dependent ε(x) is also investigated.