  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.
