Space-time discretization of series expansion methods for the Boltzmann transport equation

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26 Citations (Scopus)

Abstract

The approximate solution of the Boltzmann transport equation via Galerkin-type series expansion methods leads to a system of first order differential equations in space and time for the expansion coefficients. This system is extremely stiff close to the fluid dynamical regime (for small Knudsen numbers), and exhibits a mildly dispersive behavior, due to the acceleration of waves by the external force (the electric field). In this paper a class of difference methods is presented and analyzed which represent a generalization of the well-known Scharfetter-Gummel exponential fitting approach for the drift-diffusion equations. It is shown that, by using appropriate operator splitting methods for the time discretization, one obtains stability properties which are only mildly dependent on the Knudsen number and essentially independent of the size of the electric field.

Original languageEnglish (US)
Pages (from-to)442-465
Number of pages24
JournalSIAM Journal on Numerical Analysis
Volume38
Issue number2
DOIs
StatePublished - 2001

Fingerprint

Boltzmann Transport Equation
Knudsen number
Series Expansion
Electric Field
Discretization
Space-time
Electric fields
Exponential Fitting
Telescoping a series
Operator Splitting Method
Drift-diffusion Equations
Time Discretization
First order differential equation
Galerkin
Approximate Solution
Differential equations
Fluid
Fluids
Dependent
Coefficient

Keywords

  • Boltzmann equation
  • Finite differences
  • Galerkin methods

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics
  • Computational Mathematics

Cite this

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abstract = "The approximate solution of the Boltzmann transport equation via Galerkin-type series expansion methods leads to a system of first order differential equations in space and time for the expansion coefficients. This system is extremely stiff close to the fluid dynamical regime (for small Knudsen numbers), and exhibits a mildly dispersive behavior, due to the acceleration of waves by the external force (the electric field). In this paper a class of difference methods is presented and analyzed which represent a generalization of the well-known Scharfetter-Gummel exponential fitting approach for the drift-diffusion equations. It is shown that, by using appropriate operator splitting methods for the time discretization, one obtains stability properties which are only mildly dependent on the Knudsen number and essentially independent of the size of the electric field.",
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AB - The approximate solution of the Boltzmann transport equation via Galerkin-type series expansion methods leads to a system of first order differential equations in space and time for the expansion coefficients. This system is extremely stiff close to the fluid dynamical regime (for small Knudsen numbers), and exhibits a mildly dispersive behavior, due to the acceleration of waves by the external force (the electric field). In this paper a class of difference methods is presented and analyzed which represent a generalization of the well-known Scharfetter-Gummel exponential fitting approach for the drift-diffusion equations. It is shown that, by using appropriate operator splitting methods for the time discretization, one obtains stability properties which are only mildly dependent on the Knudsen number and essentially independent of the size of the electric field.

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KW - Finite differences

KW - Galerkin methods

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