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)$$ρ$$(x, t)$, called the drift-diffusion limit, where $M$ is a Maxwellian and the physical density $ρ$ 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 $ε$, where 1/$ε$ is the scale of the collision frequency, and asymptotic preserving. Here in particular, we discuss what properties the discrete Maxwellian must satisfy in order for the schemes to converge in $ε$ 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 $ε$ and the spacial resolution are also included.. Authors: DeCaria, Victor P. [Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)] (ORCID:0000000220457337); Hauck, Cory D. [Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States); Univ. of Tennessee, Knoxville, TN (United States)] (ORCID:000000015559502X); Schnake, Stefan R. [Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)] (ORCID:0000000215183538). DOE Contract: AC05-00OR22725. Subjects: 97 MATHEMATICS AND COMPUTING; asymptotic preserving; discontinuous Galerkin; drift-diffusion; semiconductor models