An entropy stable discontinuous Galerkin finite-element moment method for the Boltzmann equation

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Abstract

This paper presents a numerical approximation technique for the Boltzmann equation based on a moment-system approximation in velocity dependence and a discontinuous Galerkin finite-element approximation in position dependence. The closure relation for the moment systems derives from minimization of a suitable φ-divergence. This divergence-based closure yields a hierarchy of tractable symmetric hyperbolic moment systems that retain the fundamental structural properties of the Boltzmann equation. The approximation in position dependence is based on the discontinuous Galerkin finite-element method. The resulting combined discontinuous Galerkin moment method corresponds to a Galerkin approximation of the Boltzmann equation in renormalized form. The new moment–closure formulation engenders a new upwind numerical flux function, based on half-space integrals of the approximate distribution. We establish that the proposed upwind flux ensures entropy dissipation of the discontinuous Galerkin finite-element approximation. Numerical results are presented for a one-dimensional test case.

LanguageEnglish
Pages1988-1999
Number of pages12
JournalComputers and Mathematics with Applications
Volume72
Issue number8
DOIs
StatePublished - 1 Oct 2016

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Moment Method
Discontinuous Galerkin
Boltzmann equation
Galerkin Approximation
Method of moments
Boltzmann Equation
Entropy
Finite Element Method
Moment
Finite Element Approximation
Closure
F-divergence
Entropy Dissipation
Moment Closure
Fluxes
Discontinuous Galerkin Finite Element Method
Approximation
Numerical Approximation
Galerkin Method
Structural Properties

Keywords

  • Boltzmann equation
  • Discontinuous Galerkin finite-element methods
  • Entropy dissipation
  • Moment systems

Cite this

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abstract = "This paper presents a numerical approximation technique for the Boltzmann equation based on a moment-system approximation in velocity dependence and a discontinuous Galerkin finite-element approximation in position dependence. The closure relation for the moment systems derives from minimization of a suitable φ-divergence. This divergence-based closure yields a hierarchy of tractable symmetric hyperbolic moment systems that retain the fundamental structural properties of the Boltzmann equation. The approximation in position dependence is based on the discontinuous Galerkin finite-element method. The resulting combined discontinuous Galerkin moment method corresponds to a Galerkin approximation of the Boltzmann equation in renormalized form. The new moment–closure formulation engenders a new upwind numerical flux function, based on half-space integrals of the approximate distribution. We establish that the proposed upwind flux ensures entropy dissipation of the discontinuous Galerkin finite-element approximation. Numerical results are presented for a one-dimensional test case.",
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An entropy stable discontinuous Galerkin finite-element moment method for the Boltzmann equation. / Abdel Malik, Michael; van Brummelen, E.H.

In: Computers and Mathematics with Applications, Vol. 72, No. 8, 01.10.2016, p. 1988-1999.

Research output: Contribution to journalArticleAcademicpeer-review

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AB - This paper presents a numerical approximation technique for the Boltzmann equation based on a moment-system approximation in velocity dependence and a discontinuous Galerkin finite-element approximation in position dependence. The closure relation for the moment systems derives from minimization of a suitable φ-divergence. This divergence-based closure yields a hierarchy of tractable symmetric hyperbolic moment systems that retain the fundamental structural properties of the Boltzmann equation. The approximation in position dependence is based on the discontinuous Galerkin finite-element method. The resulting combined discontinuous Galerkin moment method corresponds to a Galerkin approximation of the Boltzmann equation in renormalized form. The new moment–closure formulation engenders a new upwind numerical flux function, based on half-space integrals of the approximate distribution. We establish that the proposed upwind flux ensures entropy dissipation of the discontinuous Galerkin finite-element approximation. Numerical results are presented for a one-dimensional test case.

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