An asymptotic-preserving scheme for the semiconductor Boltzmann equation toward the energy-transport limit

TL;DR

This paper introduces an asymptotic-preserving numerical scheme for the semiconductor Boltzmann equation that accurately captures the energy-transport limit as mean free path approaches zero, overcoming stiffness challenges.

Contribution

The paper develops a novel asymptotic-preserving scheme with a thresholded BGK penalization to handle stiff collision terms in the semiconductor Boltzmann equation.

Findings

The scheme effectively captures the energy-transport limit.

Numerical results demonstrate high accuracy and efficiency.

The method overcomes limitations of traditional penalization techniques.

Abstract

We design an asymptotic-preserving scheme for the semiconductor Boltzmann equation which leads to an energy-transport system for electron mass and internal energy as mean free path goes to zero. To overcome the stiffness induced by the convection terms, we adopt an even-odd decomposition to formulate the equation into a diffusive relaxation system. New difficulty arises in the two-scale stiff collision terms, whereas the simple BGK penalization does not work well to drive the solution to the correct limit. We propose a clever variant of it by introducing a threshold on the stiffer collision term such that the evolution of the solution resembles a Hilbert expansion at the continuous level. Formal asymptotic analysis and numerical results are presented to illustrate the efficiency and accuracy of the new scheme.