Asymptotic Preserving Discontinuous Galerkin Methods for a Linear Boltzmann Semiconductor Model

TL;DR

This paper develops and analyzes two discontinuous Galerkin numerical schemes for the linear Boltzmann semiconductor model, demonstrating their stability, asymptotic preserving property, and convergence to the drift-diffusion limit as collision frequency increases.

Contribution

The paper introduces two novel DG schemes with different fluxes for the semiconductor model, proving their uniform stability and asymptotic preservation in the high collision frequency limit.

Findings

Schemes are uniformly stable in psilon

Schemes converge to the drift-diffusion limit as psilon rrow 0

Error estimates provided in various norms

Abstract

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 two discontinuous Galerkin methods to the semiconductor model: one with the standard upwinding flux and the other with a $\varepsilon$-scaled Lax-Friedrichs flux, where 1/$\varepsilon$ is the scale of the collision frequency. We show that these schemes are uniformly stable in $\varepsilon$ and are 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.