### 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.

Language | English |
---|---|

Pages | 1988-1999 |

Number of pages | 12 |

Journal | Computers and Mathematics with Applications |

Volume | 72 |

Issue number | 8 |

DOIs | |

State | Published - 1 Oct 2016 |

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### Keywords

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

### Cite this

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**An entropy stable discontinuous Galerkin finite-element moment method for the Boltzmann equation.** / Abdel Malik, Michael; van Brummelen, E.H.

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

TY - JOUR

T1 - An entropy stable discontinuous Galerkin finite-element moment method for the Boltzmann equation

AU - Abdel Malik,Michael

AU - van Brummelen,E.H.

PY - 2016/10/1

Y1 - 2016/10/1

N2 - 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.

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.

KW - Boltzmann equation

KW - Discontinuous Galerkin finite-element methods

KW - Entropy dissipation

KW - Moment systems

UR - http://www.scopus.com/inward/record.url?scp=84979704163&partnerID=8YFLogxK

U2 - 10.1016/j.camwa.2016.05.021

DO - 10.1016/j.camwa.2016.05.021

M3 - Article

VL - 72

SP - 1988

EP - 1999

JO - Computers and Mathematics with Applications

T2 - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 8

ER -