SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1067-1097, June 2024.
Abstract. A key property of the linear Boltzmann semiconductor model is that as the collision frequency tends to infinity, the phase space density [math] converges to an isotropic function [math], called the drift-diffusion limit, where [math] is a Maxwellian and the physical density [math] 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 [math], where 1/[math] 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 [math] to an accurate [math]-approximation of the drift-diffusion limit. Discrete versions of the drift-diffusion equation and error estimates in several norms with respect to [math] and the spacial resolution are also included.

中文翻译：

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线性玻尔兹曼半导体模型的渐近保断间伽辽金法

SIAM 数值分析杂志，第 62 卷，第 3 期，第 1067-1097 页，2024 年 6 月
。摘要。线性玻尔兹曼半导体模型的一个关键特性是，当碰撞频率趋于无穷大时，相空间密度 [math] 收敛到各向同性函数 [math]，称为漂移扩散极限，其中 [math] 是麦克斯韦函数，物理密度 [数学] 满足称为漂移扩散方程的二阶抛物线偏微分方程。反映这一性质的数值近似被认为是渐近保持的。在本文中，我们对半导体模型建立了一种不连续伽辽金方法，并证明该方案在[数学]上一致稳定，其中1/[数学]是碰撞频率的尺度，并且渐近保持。特别是，我们讨论了离散麦克斯韦必须满足哪些属性，以便方案在[数学]上收敛到漂移扩散极限的精确[数学]近似。还包括漂移扩散方程的离散版本以及关于[数学]和空间分辨率的几个规范的误差估计。