Volume 4, Issue 1
Convergence of A Distributional Monte Carlo Method for the Boltzmann Equation

Adv. Appl. Math. Mech., 4 (2012), pp. 102-121.

Published online: 2012-04

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• Abstract

Direct Simulation Monte Carlo (DSMC) methods for the Boltzmann equation employ a point measure approximation to the distribution function, as simulated particles may possess only a single velocity. This representation limits the method to converge only weakly to the solution of the Boltzmann equation.  Utilizing kernel density estimation we have developed a stochastic Boltzmann solver which possesses strong convergence for bounded and $L^\infty$ solutions of the Boltzmann equation.  This is facilitated by distributing the velocity of each simulated particle instead of using the point measure approximation inherent to DSMC.  We propose that the development of a distributional method which incorporates distributed velocities in collision selection and modeling should improve convergence and potentially result in a substantial reduction of the variance in comparison to DSMC methods.  Toward this end, we also report initial findings of modeling collisions distributionally using the Bhatnagar-Gross-Krook collision operator.

• Keywords

Direct simulation monte carlo rarefied gas dynamics Boltzmann equation convergence proof

82B40 76P05 65C35 82C80

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@Article{AAMM-4-102, author = {Christopher R. Schrock and Aihua W. Wood}, title = {Convergence of A Distributional Monte Carlo Method for the Boltzmann Equation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2012}, volume = {4}, number = {1}, pages = {102--121}, abstract = {

Direct Simulation Monte Carlo (DSMC) methods for the Boltzmann equation employ a point measure approximation to the distribution function, as simulated particles may possess only a single velocity. This representation limits the method to converge only weakly to the solution of the Boltzmann equation.  Utilizing kernel density estimation we have developed a stochastic Boltzmann solver which possesses strong convergence for bounded and $L^\infty$ solutions of the Boltzmann equation.  This is facilitated by distributing the velocity of each simulated particle instead of using the point measure approximation inherent to DSMC.  We propose that the development of a distributional method which incorporates distributed velocities in collision selection and modeling should improve convergence and potentially result in a substantial reduction of the variance in comparison to DSMC methods.  Toward this end, we also report initial findings of modeling collisions distributionally using the Bhatnagar-Gross-Krook collision operator.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.10-m11113}, url = {http://global-sci.org/intro/article_detail/aamm/109.html} }
TY - JOUR T1 - Convergence of A Distributional Monte Carlo Method for the Boltzmann Equation AU - Christopher R. Schrock & Aihua W. Wood JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 102 EP - 121 PY - 2012 DA - 2012/04 SN - 4 DO - http://doi.org/10.4208/aamm.10-m11113 UR - https://global-sci.org/intro/article_detail/aamm/109.html KW - Direct simulation monte carlo KW - rarefied gas dynamics KW - Boltzmann equation KW - convergence proof AB -

Direct Simulation Monte Carlo (DSMC) methods for the Boltzmann equation employ a point measure approximation to the distribution function, as simulated particles may possess only a single velocity. This representation limits the method to converge only weakly to the solution of the Boltzmann equation.  Utilizing kernel density estimation we have developed a stochastic Boltzmann solver which possesses strong convergence for bounded and $L^\infty$ solutions of the Boltzmann equation.  This is facilitated by distributing the velocity of each simulated particle instead of using the point measure approximation inherent to DSMC.  We propose that the development of a distributional method which incorporates distributed velocities in collision selection and modeling should improve convergence and potentially result in a substantial reduction of the variance in comparison to DSMC methods.  Toward this end, we also report initial findings of modeling collisions distributionally using the Bhatnagar-Gross-Krook collision operator.

Christopher R. Schrock & Aihua W. Wood. (1970). Convergence of A Distributional Monte Carlo Method for the Boltzmann Equation. Advances in Applied Mathematics and Mechanics. 4 (1). 102-121. doi:10.4208/aamm.10-m11113
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