Hybrid Monte Carlo methods for the diffusion of impurities in a gas

The goal of this work is to construct suitable Monte Carlo methods for kinetic equations modelling the dynamics of fine polluting powders (for example, particulate matter) interacting inelastically with a background gas. The evolution of the particles and the background motion can be described by a system of Boltzmann equations for hard spheres. The complexity of the model can be strongly reduced by assuming that the background is in thermodynamic equilibrium and that the polluting particles are sufficiently few (in comparison to the background molecules) to admit that there are no collisions among couples of them. As a consequence, for each species of pollutant, we obtain a single dissipative linear Boltzmann equation with a collision dynamic related to the gas-impurity mass ratio. The development of hybrid Monte Carlo schemes coupling a deterministic fluid solver for the background moments and a Time Relaxed Monte Carlo (TRMC) method for the diffusion of the impurities is developed and some numerical results are presented.