2022/06/20 by Victor DeCaria, DeCaria, Victor, Cory D. Hauck +3
Computer Science · Engineering · #65M08 #65M12 #65M15 #65M60 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Lattice Boltzmann Simulation Studies #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2206.09805
openalex publication_date 2022/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.