AN ASYMPTOTIC PRESERVING DISCONTINUOUS GALERKIN METHOD FOR A LINEAR BOLTZMANN SEMICONDUCTOR MODEL

Victor P. Decaria; Cory D. Hauck; Stefan R. Schnake

doi:10.1137/22M1485784

AN ASYMPTOTIC PRESERVING DISCONTINUOUS GALERKIN METHOD FOR A LINEAR BOLTZMANN SEMICONDUCTOR MODEL

Victor P. Decaria
, Cory D. Hauck
, Stefan R. Schnake

Multiscale Methods

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

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 a discontinuous Galerkin method to the semiconductor model, and we show this scheme is both uniformly stable in \varepsilon, where 1/\varepsilon is the scale of the collision frequency, and 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.

Original language	English
Pages (from-to)	1067-1097
Number of pages	31
Journal	SIAM Journal on Numerical Analysis
Volume	62
Issue number	3
DOIs	https://doi.org/10.1137/22M1485784
State	Published - 2024

Funding

\ast Received by the editors April 4, 2022; accepted for publication (in revised form) January 30, 2024; published electronically May 6, 2024. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, as part of their Applied Mathematics Research Program. The work was performed at the Oak Ridge National Laboratory, which is managed by UT-Battelle, LLC under contract DE-AC05-00OR22725. The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for U.S. Government purposes. Copyright is owned by SIAM to the extent not limited by these rights. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan). https://doi.org/10.1137/22M1485784 \dagger Mathematics in Computation Section, Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 USA ([email protected], [email protected]).

Keywords

asymptotic preserving
discontinuous Galerkin
drift-diffusion
semiconductor models

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10.1137/22M1485784

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title = "AN ASYMPTOTIC PRESERVING DISCONTINUOUS GALERKIN METHOD FOR A LINEAR BOLTZMANN SEMICONDUCTOR MODEL",

abstract = "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)\textbackslash{}rho(x,t), called the drift-diffusion limit, where M is a Maxwellian and the physical density \textbackslash{}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 a discontinuous Galerkin method to the semiconductor model, and we show this scheme is both uniformly stable in \textbackslash{}varepsilon, where 1/\textbackslash{}varepsilon is the scale of the collision frequency, and asymptotic preserving. In particular, we discuss what properties the discrete Maxwellian must satisfy in order for the schemes to converge in \textbackslash{}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 \textbackslash{}varepsilon and the spacial resolution are also included.",

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year = "2024",

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PY - 2024

Y1 - 2024

N2 - 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 a discontinuous Galerkin method to the semiconductor model, and we show this scheme is both uniformly stable in \varepsilon, where 1/\varepsilon is the scale of the collision frequency, and 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.

AB - 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 a discontinuous Galerkin method to the semiconductor model, and we show this scheme is both uniformly stable in \varepsilon, where 1/\varepsilon is the scale of the collision frequency, and 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.

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AN ASYMPTOTIC PRESERVING DISCONTINUOUS GALERKIN METHOD FOR A LINEAR BOLTZMANN SEMICONDUCTOR MODEL

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