Direct Simulation and the Boltzmann Equation

1970 ◽  
Vol 13 (11) ◽  
pp. 2676 ◽  
Author(s):  
G. A. Bird
Keyword(s):  
Boltzmann Equation ◽  
Direct Simulation ◽  
The Boltzmann Equation

DSMC: Fast direct simulation Monte Carlo solver for the Boltzmann equation by Multi-Chain Markov Chain and multicore programming

2016 ◽  
Vol 07 (02) ◽  
pp. 1650009
Author(s):  
Di Zhao ◽  
Haiwu He
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Boltzmann Equation ◽  
Direct Simulation Monte Carlo ◽  
Collision Term ◽  
Direct Simulation ◽  
Multicore Programming ◽  
The Boltzmann Equation ◽  
Simulation Monte Carlo

Direct Simulation Monte Carlo (DSMC) solves the Boltzmann equation with large Knudsen number. The Boltzmann equation generally consists of three terms: the force term, the diffusion term and the collision term. While the first two terms of the Boltzmann equation can be discretized by numerical methods such as the finite volume method, the third term can be approximated by DSMC, and DSMC simulates the physical behaviors of gas molecules. However, because of the low sampling efficiency of Monte Carlo Simulation in DSMC, this part usually occupies large portion of computational costs to solve the Boltzmann equation. In this paper, by Markov Chain Monte Carlo (MCMC) and multicore programming, we develop Direct Simulation Multi-Chain Markov Chain Monte Carlo (DSMC3): a fast solver to calculate the numerical solution for the Boltzmann equation. Computational results show that DSMC3 is significantly faster than the conventional method DSMC.

Convergence of A Distributional Monte Carlo Method for the Boltzmann Equation

2012 ◽  
Vol 4 (1) ◽  
pp. 102-121 ◽  
Author(s):  
Christopher R. Schrock ◽  
Aihua W. Wood
Keyword(s):  
Monte Carlo ◽  
Boltzmann Equation ◽  
Substantial Reduction ◽  
Direct Simulation Monte Carlo ◽  
Collision Operator ◽  
Direct Simulation ◽  
The Boltzmann Equation ◽  
Distributional Method ◽  
Point Measure ◽  
Simulation Monte Carlo

AbstractDirect 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.

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