Asymptotic-preserving and positivity-preserving discontinuous Galerkin method for the semiconductor Boltzmann equation in the diffusive scaling

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.