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 asymptotic-preserving schemes. Inspired by [14] and [13], our idea is to penalize the stiff collision term by a ‘classical’ BGK operator – which is not the local equilibrium in the high field limit – while treat the stiff force term implicitly by 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.