Abstract:
We design asymptotic-preserving schemes for the semiconductor Boltzmann equation with two-scale stiff collisions — a leading order elastic collision and a lower-order inter-particle collision. When the mean free path is small, it leads to an energy-transport system for electron mass and internal energy. Numerically solving this equation is prohibitively expensive due to the stiffness. Since the equilibrium solution is a zero-momentum Fermi-Dirac distribution resulting from joint action of both collisions, the simple BGK penalization designed for the one-scale collision cannot capture the correct diffusion limit. We propose to set up a spatially dependent threshold on the penalization of the stiffer collision operator such that the evolution of the solution resembles a Hilbert expansion at the continuous level. We also present an alternative approach based on a splitting strategy that can automatically treat the collisions at different scales separately. Formal asymptotic analysis and numerical results confirm the efficiency and accuracy of the schemes. This is a joint work with Jingwei Hu and Shi Jin. |