In this paper, we develop an asymptotic-preserving and positivity-preserving discontinuous Galerkin (DG) method for solving the semiconductor Boltzmann equation in the diffusive scaling. We first formulate the diffusive relaxation system based on the even-odd decomposition method, which allows us to split into one relaxation step and one transport step. We adopt a robust implicit scheme that can be explicitly implemented for the relaxation step that involves the stiffness of the collision term, while the third-order strong-stability-preserving Runge-Kutta method is employed for the transport step. We couple this temporal scheme with the DG method for spatial discretization, which provides additional advantages including high-order accuracy, $h$-$p$ adaptivity, and the ability to handle arbitrary unstructured meshes. A positivity-preserving limiter is further applied to preserve physical properties of numerical solutions. The stability analysis using the even-odd decomposition is conducted for the first time. We demonstrate the accuracy and performance of our proposed scheme through several numerical examples.

翻译：本文针对扩散尺度下的半导体玻尔兹曼方程，提出了一种渐近保持且正性保持的间断伽辽金（DG）方法。我们首先基于奇偶分解法构建了扩散松弛系统，该系统可将求解过程分解为一个松弛步和一个输运步。对于涉及碰撞项刚性的松弛步，我们采用了一种可显式执行的鲁棒隐式格式；而对于输运步，则采用了三阶强稳定性保持的Runge-Kutta方法。我们将此时间离散格式与空间离散的DG方法相结合，该方法具有高阶精度、$h$-$p$自适应能力以及处理任意非结构网格的优势。为进一步保持数值解的物理特性，我们还引入了正性保持限制器。基于奇偶分解的稳定性分析在本文中首次得到系统阐述。通过多个数值算例，我们验证了所提格式的精确性与计算性能。