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.

翻译：线性 Boltzmann 半导体模型的一个关键性质是：当碰撞频率趋于无穷时，相空间密度 $f = f(x,v,t)$ 收敛于各向同性函数 $M(v)\rho(x,t)$，即漂移-扩散极限，其中 $M$ 是麦克斯韦分布，物理密度 $\rho$ 满足称为漂移-扩散方程的二阶抛物型偏微分方程。能够再现这一性质的数值逼近方法被称为渐近保持型。本文针对半导体模型构建了两种间断 Galerkin 方法：一种采用标准迎风通量，另一种采用含 $\varepsilon$ 缩放的 Lax-Friedrichs 通量，其中 $1/\varepsilon$ 表示碰撞频率的尺度。我们证明了这些格式在 $\varepsilon$ 意义上一致稳定且具有渐近保持性。特别地，我们讨论了离散麦克斯韦分布需满足何种性质，才能使格式在 $\varepsilon$ 极限下精确逼近漂移-扩散极限的 $h$ 阶近似。文中还包含了漂移-扩散方程的离散版本以及关于 $\varepsilon$ 和空间分辨率的多种范数下的误差估计。