Convergence of a Distributional Monte Carlo Method for the Boltzmann Equation (Advances in Applied Mathematics and Mechanics, 2012)
Direct Simulation Monte Carlo (DSMC) methods for the Boltzmannequation employ a point measure approximation to the distributionfunction, as simulated particles may possess only a single velocity.This representation limits the method to converge only weakly tothe solution of the Boltzmann equation. Utilizing kernel densityestimation we have developed a stochastic Boltzmann solver whichpossesses strong convergence for bounded and $L^∞$ solutionsof the Boltzmann equation. This is facilitated by distributingthe velocity of each simulated particle instead of using thepoint measure approximation inherent to DSMC. We propose that thedevelopment of a distributional method which incorporates distributedvelocities in collision selection and modeling should improve convergenceand potentially result in a substantial reduction of the variance incomparison to DSMC methods. Toward this end, we also report initialfindings of modeling collisions distributionally using theBhatnagar-Gross-Krook collision operator.