Bird's direct simulation Monte Carlo method for the Boltzmann equation is considered. The limit (as the number of particles tends to infinity) of the random empirical measures associated with the Bird algorithm is shown to be a deterministic measure-valued function satisfying an equation close (in a certain sense) to the Boltzmann equation. A Markov jump process is introduced, which is related to Bird's collision simulation procedure via a random time transformation. Convergence is established for the Markov process and the random time transformation. These results, together with some general properties concerning the convergence of random measures, make it possible to characterize the limiting behavior of the Bird algorithm.

A convergence proof for Bird's direct simulation Monte Carlo method for the Boltzmann equation

▪ Abstract The principles of and procedures for implementing direct simulation Monte Carlo (DSMC) are described. Guidelines to inherent and external errors common in DSMC applications are provided. Three applications of DSMC to transitional and nonequilibrium flows are considered: rarefied atmospheric flows, growth of thin films, and microsystems. Selected new, potentially important advances in DSMC capabilities are described: Lagrangian DSMC, optimization on parallel computers, and hybrid algorithms for computations in mixed flow regimes. Finally, the limitations of current computer technology for using DSMC to compute low-speed, high–Knudsen-number flows are outlined as future challenges.

Direct Simulation Monte Carlo: Recent Advances and Applications

Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models

The use of high plasma density and low gas density, a recent trend in plasma-assisted materials processing, requires a particle simulation method for plasmas and gas flows. The kinetic theory basis of the particle simulation method is first described. Based on this theoretical viewpoint, state-of-the-art probabilistic treatments of collisions are described for electron-molecule, ion-molecule, molecule-molecule, and Coulomb collisions.

Probability theory of electron-molecule, ion-molecule, molecule-molecule, and Coulomb collisions for particle modeling of materials processing plasmas and cases

We specify the Markov process corresponding to a generalized mollified Boltzmann equation with general motion between collisions and nonlinear bounded jump (collision) operator, and give the nonlinear martingale problem it solves. We consider various linear interacting particle systems in order to approximate this nonlinear process. We prove propagation of chaos, in variation norm on path space with a precise rate of convergence, using coupling and interaction graph techniques and a representation of the nonlinear process on a Boltzmann tree. No regularity nor uniqueness assumption is needed. We then consider a nonlinear equation with both Vlasov and Boltzmann terms and give a weak pathwise propagation of chaos result using a compactness-uniqueness method which necessitates some regularity. These results imply functional laws of large numbers and extend to multitype models. We give algorithms simulating or approximating the particle systems.

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Stochastic particle approximations for generalized Boltzmann models and convergence estimates

It is shown that the discrete measures given by the Nanbu simulation method converge with respect to the weak topology of measures to solutions of the Boltzmann equation. The main conditions for this result are that the Cauchy problem for the Boltzmann equation has a sufficiently smooth solution and that the discretization parameters (cell size, timestep and test particle number) satisfy suitable constraints.

Hans Babovsky

Papers

A convergence proof for Nanbu's simulation method for the full Boltzmann equation