Title: Asymptotic-Preserving schemes for the multiscale semiconductor Boltzmann equation
Abstract: Kinetic equations often contain multiple scales that lead to various asymptotic regimes, in which classical numerical methods become prohibitively expensive. Asymptotic-Preserving (AP) scheme is one efficient way to treat such multiscale problems. It is a unified kinetic solver that preserves the asymptotic limit at the discrete level. In this talk, I will present AP schemes for the semiconductor Boltzmann equation in a diffusive regime with two-scale collisions that leads to an energy-transport system as the mean free path goes to zero. Our scheme is based on a BGK (Bhatnagar-Gross-Krook) penalization together with a spatially dependent threshold on the stiffer collision operator such that the evolution of the solution resembles a Hilbert expansion at the continuous level. An alternative approach via a splitting strategy will also be presented, which can systematically treat the collisions at different scales separately. Formal asymptotic analysis and numerical results confirm the efficiency and accuracy of the schemes. In the end, I will illustrate the promise of these ideas in treating the hierarchy of macroscopic models in semiconductors and other applications.
