Implicit Asymptotic Preserving Schemes for Semiconductor Boltzmann Equation in the Diffusive Regime

We design several implicit asymptotic-preserving schemes for the linear semiconductor Boltzmann equation with a diffusive scaling, which lead asymptotically to the implicit discretizations of the drift-diffusion equation. The constructions are based on a stiff relaxation step and a stiff convection step obtained by splitting the system equal to the model equation. The one space dimensional schemes are given with the uniform grids and the staggered grids, respectively. The uniform grids are considered only in two space dimension. The relaxation step is evolved with the BGK-penalty method of Filbet and Jin [F. Filbet and S. Jin, J. Comp. Phys. 229(20), 7625-7648, 2010], which avoids inverting the complicated nonlocal anisotropic collision operator. The convection step is performed with a suitable implicit approximation to the convection term, which gives a banded matrix easy to invert. The von-Neumman analysis for the Goldstein-Taylor model show that the one space dimensional schemes are unconditionally stable. The heuristic discussions suggest that all the proposed schemes have the correct discrete drift-diffusion limit. The numerical results verify that all the schemes are asymptotic-preserving. As far as we know, they are the first class of asymptotic-preserving schemes ever introduced for the Boltzmann equation with a diffusive scaling that lead to an implicit discretization of the diffusion limit, thus significantly relax to stability condition.