# American Institute of Mathematical Sciences

December  2015, 8(4): 707-723. doi: 10.3934/krm.2015.8.707

## An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: A splitting approach

 1 Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN 47907, United States 2 Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706 3 Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, CA 90095, United States

Received  August 2014 Revised  April 2015 Published  July 2015

We present a new asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions --- a leading-order elastic collision together with a lower-order interparticle collision. When the mean free path is small, numerically solving this equation is prohibitively expensive due to the stiff collision terms. Furthermore, since the equilibrium solution is a (zero-momentum) Fermi-Dirac distribution resulting from joint action of both collisions, the simple BGK penalization designed for the one-scale collision [10] cannot capture the correct energy-transport limit. This problem was addressed in [13], where a thresholded BGK penalization was introduced. Here we propose an alternative based on a splitting approach. It has the advantage of treating the collisions at different scales separately, hence is free of choosing threshold and easier to implement. Formal asymptotic analysis and numerical results validate the efficiency and accuracy of the proposed scheme.
Citation: Jingwei Hu, Shi Jin, Li Wang. An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: A splitting approach. Kinetic & Related Models, 2015, 8 (4) : 707-723. doi: 10.3934/krm.2015.8.707
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