Convergence of A Distributional Monte Carlo Method for the Boltzmann Equation

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.

Stochastic Boltzmann equation for magnetic relaxation in high-spin molecules

“…The idea of solving the regular Boltzmann equation for rarified gas dynamics and heat transport in solids by stochastic trajectory simulation of gas particles is an active field with a long history 43 . In this research, it has been pointed out that the solutions obtained by such simulations do not converge in a mathematically strong sense ('convergence in probability') to solutions to the regular Boltzmann equation as the number of simulated particles tends to infinity, but rather in the mathematically weaker sense of 'convergence in distribution', and so special techniques have been developed to deal with this problem 42 . Based on comparisons between these Monte Carlo approaches to solving the regular Boltzmann equation for rarified gas dynamics, and our boson gas simulation method for solving the SBE for molecules adsorbed to a surface, we might expect that the solutions presented in section 4 are only such 'weak' solutions to the SBE.…”

“…Even if we are satisfied working with the relaxation-time approximation, integrating the SBE directly appears to be very difficult due to the need to average over all realisations of the paths of the frequency processes. To proceed, we consider a Monte Carlo approach that is used to solve the regular Boltzmann equation for the case of rarefied gas dynamics, in which a sample of gas particles undergoing a stochastic motion are simulated, and the the laws describing this stochastic motion are chosen such that the solution to the Boltzmann equation is satisfied 42,43 . A similar approach has also been used in the field of heat transport in solids 44 .…”

“…To proceed, we consider a Monte Carlo 16 approach that is used to solve the regular Boltzmann equation for the case of rarefied gas dynamics, in which a sample of gas particles undergoing a stochastic motion are simulated, and the the laws describing this stochastic motion are chosen such that the solution to the Boltzmann equation is satisfied 42,43 . A similar approach has also been used in the field of heat transport in solids 44 .…”

A deterministic-stochastic method for computing the Boltzmann collision integral in $\mathcal{O}(MN)$ operations

“…A possible alternative to DSMC is to employ distributed velocities in collision selection and modeling as opposed to the point measure approximation to the distribution function inherent to DSMC. To this end, distributional Monte Carlo methods have been proposed using kernel density estimation for space homogeneous Boltzmann equation [42,41].…”