Moment closure approximations of the Boltzmann equation based on φ-divergences

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Abstract

This paper is concerned with approximations of the Boltzmann equation based on the method of moments. We propose a generalization of the setting of the moment-closure problem from relative entropy to (Formula presented.)-divergences and a corresponding closure procedure based on minimization of (Formula presented.)-divergences. The proposed description encapsulates as special cases Grad’s classical closure based on expansion in Hermite polynomials and Levermore’s entropy-based closure. We establish that the generalization to divergence-based closures enables the construction of extended thermodynamic theories that avoid essential limitations of the standard moment-closure formulations such as inadmissibility of the approximate phase-space distribution, potential loss of hyperbolicity and singularity of flux functions at local equilibrium. The divergence-based closure leads to a hierarchy of tractable symmetric hyperbolic systems that retain the fundamental structural properties of the Boltzmann equation.

LanguageEnglish
Pages77-104
Number of pages28
JournalJournal of Statistical Physics
Volume164
Issue number1
DOIs
StatePublished - Jul 2016

Fingerprint

F-divergence
Moment Closure
Boltzmann Equation
closures
divergence
Closure
Divergence
moments
Approximation
approximation
Inadmissibility
Symmetric Hyperbolic Systems
Local Equilibrium
Relative Entropy
Hermite Polynomials
Hyperbolicity
Method of Moments
entropy
hyperbolic systems
Structural Properties

Keywords

  • Boltzmann equation
  • Divergence
  • Entropy
  • Hyperbolic systems
  • Kinetic theory
  • Moment closure

Cite this

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abstract = "This paper is concerned with approximations of the Boltzmann equation based on the method of moments. We propose a generalization of the setting of the moment-closure problem from relative entropy to (Formula presented.)-divergences and a corresponding closure procedure based on minimization of (Formula presented.)-divergences. The proposed description encapsulates as special cases Grad’s classical closure based on expansion in Hermite polynomials and Levermore’s entropy-based closure. We establish that the generalization to divergence-based closures enables the construction of extended thermodynamic theories that avoid essential limitations of the standard moment-closure formulations such as inadmissibility of the approximate phase-space distribution, potential loss of hyperbolicity and singularity of flux functions at local equilibrium. The divergence-based closure leads to a hierarchy of tractable symmetric hyperbolic systems that retain the fundamental structural properties of the Boltzmann equation.",
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Moment closure approximations of the Boltzmann equation based on φ-divergences. / Abdel Malik, M.; van Brummelen, E.H.

In: Journal of Statistical Physics, Vol. 164, No. 1, 07.2016, p. 77-104.

Research output: Contribution to journalArticleAcademicpeer-review

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