Asymptotic Preserving Discontinuous Galerkin Methods for a Linear Boltzmann Semiconductor Model
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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)ρ(x,t), called the drift-diffusion limit, where M is a Maxwellian and the physical density ρ 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 ε-scaled Lax-Friedrichs flux, where 1/ε is the scale of the collision frequency. We show that these schemes are uniformly stable in ε and are asymptotic preserving. In particular, we discuss what properties the discrete Maxwellian must satisfy in order for the schemes to converge in ε 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 ε and the spacial resolution are also included.
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