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A key property of the linear Boltzmann semiconductor model is that as thecollision frequency tends to infinity, the phase space density $f = f(x,v,t)$converges to an isotropic function $M(v)\rho(x,t)$, called the drift-diffusionlimit, where $M$ is a Maxwellian and the physical density $\rho$ satisfies asecond-order parabolic PDE known as the drift-diffusion equation. Numericalapproximations that mirror this property are said to be asymptotic preserving.In this paper we build two discontinuous Galerkin methods to the semiconductormodel: one with the standard upwinding flux and the other with a$\varepsilon$-scaled Lax-Friedrichs flux, where 1/$\varepsilon$ is the scale ofthe collision frequency. We show that these schemes are uniformly stable in$\varepsilon$ and are asymptotic preserving. In particular, we discuss whatproperties the discrete Maxwellian must satisfy in order for the schemes toconverge in $\varepsilon$ to an accurate $h$-approximation of the driftdiffusion limit. Discrete versions of the drift-diffusion equation and errorestimates in several norms with respect to $\varepsilon$ and the spacialresolution are also included.

Asymptotic Preserving Discontinuous Galerkin Methods for a Linear Boltzmann Semiconductor Model