论文标题
渐近保留线性螺栓曼型半导体模型的不连续的Galerkin方法
Asymptotic Preserving Discontinuous Galerkin Methods for a Linear Boltzmann Semiconductor Model
论文作者
论文摘要
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方程式。据称是渐近保存该特性的数值近似值。在本文中，我们为半导体模型构建了两种不连续的Galerkin方法：一个具有标准的向上通量，另一个带有$ \ varepsilon $ scal的lax-friedrichs Flux，其中1/$ \ varepsilon $是碰撞频率的规模。我们表明，这些方案在$ \ varepsilon $中均匀稳定，并且是渐近保存的。特别是，我们讨论了离散的Maxwellian必须满足的属性，以便这些方案以$ \ varepsilon $收敛到精确的$ h $ h $ - approximation drift Edrifusion listimation。还包括有关$ \ varepsilon $和空间分辨率的几种规范的漂移扩散方程和错误估计的离散版本。
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 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.
