Asymptotic-preserving numerical schemes for the semiconductor boltzmann equation efficient in the high field regime

We present asymptotic-preserving numerical schemes for the semiconductor Boltzmann equation efficient in the high field regime. A major challenge in this regime is that there may be no explicit expression of the local equilibrium which is the main component of classical asymptoticpreserving schemes. Inspired by [F. Filbet and S. Jin, J. Comput. Phys., 229 (2010), pp. 7625-7648] and [F. Filbet, J. Hu, and S. Jin, Math. Model. Numer. Anal., 46 (2012), pp. 443-463], our idea is to penalize the stiff collision term with a classical Bhatnagar-Gross-Krook operator-which is not the local equilibrium in the high field limit-while treating the stiff force term implicitly with the spectral method. These schemes, despite being implicit, can be inverted easily with a stability independent of the physically small parameter. We design these schemes for both nondegenerate and degenerate cases and show their asymptotic properties. We present several numerical examples to validate the efficiency, accuracy, and asymptotic properties of these schemes.