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Convergence of A Distributional Monte Carlo Method for the Boltzmann Equation

Published online by Cambridge University Press:  03 June 2015

Christopher R. Schrock  and
Aihua W. Wood
Christopher R. Schrock*
Affiliation:
Department of Mathematics and Statistics, Air Force Institute of Technology, Wright-Patterson AFB, OH 45433, USA
Aihua W. Wood*
Affiliation:
Department of Mathematics and Statistics, Air Force Institute of Technology, Wright-Patterson AFB, OH 45433, USA
*
Email: christopher.schrock@wpafb.af.mil
Corresponding author. Email: aihua.wood@afit.edu
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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∞ 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

82B4076P0565C3582C80Direct simulation monte carlorarefied gas dynamicsBoltzmann equationconvergence proof
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 1 , February 2012 , pp. 102 - 121
Copyright
Copyright © Global-Science Press 2012

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