  A key property of the linear Boltzmann semiconductor model is that as the
collision 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-diffusion
limit, where $M$ is a Maxwellian and the physical density $\rho$ satisfies a
second-order parabolic PDE known as the drift-diffusion equation. Numerical
approximations that mirror this property are said to be asymptotic preserving.
In this paper we build two discontinuous Galerkin methods to the semiconductor
model: one with the standard upwinding flux and the other with a
$\varepsilon$-scaled Lax-Friedrichs flux, where 1/$\varepsilon$ is the scale of
the collision frequency. We show that these schemes are uniformly stable in
$\varepsilon$ and are asymptotic preserving. In particular, we discuss what
properties the discrete Maxwellian must satisfy in order for the schemes to
converge in $\varepsilon$ to an accurate $h$-approximation of the drift
diffusion limit. Discrete versions of the drift-diffusion equation and error
estimates in several norms with respect to $\varepsilon$ and the spacial
resolution are also included.
