On a simulation scheme for the Boltzmann equation

Babovsky, Hans; Neunzert, Helmut

doi:10.1002/mma.1670080114

Cited by 114 publications

(81 citation statements)

References 5 publications

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“…As a consequence most numerical computations are based on probabilistic Monte-Carlo techniques at different levels. Examples are the direct simulation Monte-Carlo method (DSMC) by Bird [2] and the modified Monte Carlo method by Babovsky and Nanbu [1,13]. For a detailed description of such methods we refer to [2,6].…”

Section: Introductionmentioning

confidence: 99%

High order numerical methods for the space non-homogeneous Boltzmann equation

Filbet

Russo

2003

Journal of Computational Physics

101

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Section: Introductionmentioning

confidence: 99%

High order numerical methods for the space non-homogeneous Boltzmann equation

Filbet

Russo

2003

Journal of Computational Physics

101

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“…cit.). Babovsky [41] introduced a modification of N a [42][43][44]. In order to deal with chemical reactions and relaxation of energy we introduce small changes (possible only for small time increments).…”

Section: Numerical Simulationsmentioning

confidence: 99%

“…We introduce as in Refs. [38][39][40][41][42][43] a randomly chosen gij . If the collision is a reactive one the molecular velocities remain unchanged, for we neglect its influence on the velocity distribution.…”

Section: N B U ' S M E T H O D W H I C H M a D E I T S E F F I C I E mentioning

confidence: 99%

Theory of Translational Energy Relaxation in Binary Mixtures of Dilute Gases with Chemical Reaction

Cukrowski

Fritzsche²,

Popielawski

1992

Acta Phys. Pol. A

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A simple approach to the process of translational energy relaxation in dilute gases due to Dahler, Malkin, Shizgal and others is extended to the case of systems with chemical reaction. Fundamental quantities characterizing the relaxation processes such as the relaxation time and collision numbers during this time are computed for a number of molecular models of the chemical reaction (the Prigogine-Xhrouet modeł, the line-of-centers model, a modified line-of-centers model, and reverse versions of these models). Results of this analytical theory are compared with the results of numerical simulations of solutions of the appropriate Boltzmann equation with the use of the modified Nanbu-Babovsky method. This comparison leads to very good agreement between the analytical theory and numerical calculations. A marked influence of the chemical reaction on the translational relaxation in a dilute gas is

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“…The most classical example is perhaps the case of the Boltzmann equation which, due to the high dimensionality and to the necessity to avoid artificial truncation in the velocity field, is solved by Monte Carlo schemes [2,3,48,55]. However there are important flow regimes where Monte Carlo schemes loose his effectiveness: flow for which the Knudsen number is small enough that the collision rate is large and the flow is well described by fluid mechanics.…”

Section: Introductionmentioning

confidence: 99%

Hybrid Multiscale Methods for Hyperbolic and Kinetic Problems

Pareschi

2005

ESAIM: Proc.

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Abstract. In these notes we present some recent results on the development of hybrid methods for hyperbolic and kinetic equations with multiple scales. The main ingredients in the schemes are a suitable merging of particle methods in non stiff regimes with high resolution shock capturing techniques in stiff ones. The key aspect in the development of the algorithms is the choice of a suitable hybrid representation of the solution.First, after a brief review on sampling methods and Monte Carlo techniques, we introduce the hybrid schemes for hyperbolic systems with relaxation and present numerical applications to the simple JinXin relaxation model both in one and two space dimensions. Next, we show how to extend the above methodology to the case of kinetic models of BGK type and discuss the challenging case of the full Boltzmann equation. Some numerical results are also presented.Résumé. Nous présentons dans ces notes quelques résultats récents concernant le développement de méthodes hybrides pour leséquations hyperboliques et cinétiques. L'ingrédient principal de ces schémas réside dans un mélange pertinent de méthodes particulaires dans les régimes réguliers et de techniques haute résolution de capture de choc dans les zones raides. La clef du développement de ces algorithmes est donnée par le choix d'une bonne représentation hybride de la solution.Après une brève revue des méthodes d'échantillonnage et des techniques Monte-Carlo, nous introduisons des schémas hybrides pour des systèmes hyperboliques avec relaxation et présentons des applications numériques pour le modèle de Jin-Xin, en dimension un et deux. Ensuite, nous montrons comment cette méthodologie s'étendà des modèles cinétiques de type BGK et discutons le cas délicat de l'équation de Boltzmann. Des résultats numériques sont aussi présentés pour ces situations.

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On a simulation scheme for the Boltzmann equation

Abstract: A scheme for the sirnulation of solutions of the Boltzmann equation derived by Nanbu is investigated. Rigorous results concerning questions of justification, the computation effort and the energy fluctuations are presented.

Cited by 114 publications

References 5 publications

High order numerical methods for the space non-homogeneous Boltzmann equation

High order numerical methods for the space non-homogeneous Boltzmann equation

Theory of Translational Energy Relaxation in Binary Mixtures of Dilute Gases with Chemical Reaction

Hybrid Multiscale Methods for Hyperbolic and Kinetic Problems

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