AN ASYMPTOTIC PRESERVING DISCONTINUOUS GALERKIN METHOD FOR A LINEAR BOLTZMANN SEMICONDUCTOR MODEL

A key property of the linear Boltzmann semiconductor model is that as the collision frequency tends to infinity, the phase space density f = f(x,v,t) converges to an isotropic function M(v)\rho(x,t), called the drift-diffusion limit, where M is a Maxwellian and the physical density \rho satisfies a second-order parabolic PDE known as the drift-diffusion equation. Numerical approximations that mirror this property are said to be asymptotic preserving. In this paper we build a discontinuous Galerkin method to the semiconductor model, and we show this scheme is both uniformly stable in \varepsilon, where 1/\varepsilon is the scale of the collision frequency, and asymptotic preserving. In particular, we discuss what properties the discrete Maxwellian must satisfy in order for the schemes to converge in \varepsilon to an accurate h-approximation of the drift-diffusion limit. Discrete versions of the drift-diffusion equation and error estimates in several norms with respect to \varepsilon and the spacial resolution are also included.