Discretization of the multiscale semiconductor Boltzmann equation by diffusive relaxation schemes

In this paper we derive diffusive relaxation schemes for the linear semicon ductor Boltzmann equation that work in both the kinetic and diffusive regim es. 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 hand le the implicit anisotropic collision term efficiently, we utilize a suitab le power series expansion based on the Wild sum, which yields a time discre tization uniformly stable with any desired order of accuracy, yet is explic itly solvable with the correct drift-diffusion limit. The velocity discreti zation is done with the Gauss-Hermite quadrature rule equivalent to a momen t 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. (C) 2000 Aca demic Press.