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

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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.

Originele taal-2Engels
Pagina's (van-tot)1988-1999
Aantal pagina's12
TijdschriftComputers and Mathematics with Applications
Volume72
Nummer van het tijdschrift8
DOI's
StatusGepubliceerd - 1 okt 2016

Vingerafdruk

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

Citeer dit

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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, Nr. 8, 01.10.2016, blz. 1988-1999.

Onderzoeksoutput: Bijdrage aan tijdschriftTijdschriftartikelAcademicpeer review

TY - JOUR

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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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KW - Discontinuous Galerkin finite-element methods

KW - Entropy dissipation

KW - Moment systems

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