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We design an asymptotic-preserving scheme for the semiconductor Boltzmannequation which leads to an energy-transport system for electron mass andinternal energy as mean free path goes to zero. To overcome the stiffnessinduced by the convection terms, we adopt an even-odd decomposition toformulate the equation into a diffusive relaxation system. New difficultyarises in the two-scale stiff collision terms, whereas the simple BGKpenalization does not work well to drive the solution to the correct limit. Wepropose a clever variant of it by introducing a threshold on the stiffercollision term such that the evolution of the solution resembles a Hilbertexpansion at the continuous level. Formal asymptotic analysis and numericalresults are presented to illustrate the efficiency and accuracy of the newscheme.

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