Discretization of the Multiscale Semiconductor Boltzmann Equation by Diffusive Relaxation Schemes

In this paper we derive diffusive relaxation schemes for the linear semiconductor Boltzmann equation that work in both the kinetic and diffusive regimes. Similar to our earlier approach for multiscale transport equations, we use the even- and odd-parity formulation of the kinetic equation, and then reformulate it into the diffusive relaxation system (DRS). In order to handle the implicit anisotropic collision term efficiently, we utilize a suitable power series expansion based on the Wild sum, which yields a time discretization uniformly stable with any desired order of accuracy, yet is explicitly solvable with the correct drift-diffusion limit. The velocity discretization is done with the Gauss-Hermite quadrature rule equivalent to a moment expansion method. Asymptotic analysis and numerical experiments show that the schemes have the usual advantages of a diffusive relaxation scheme for multiscale transport equations and are asymptotic-preserving.