Modelling and numerical methods for the dynamics of impurities in a gas

The dynamics of a mixture of impurities in a gas can be represented by a system of linear Boltzmann equations for hard spheres. We assume that the background is in thermodynamic equilibrium and that the polluting particles are sufficiently few (in comparison with the background molecules) to admit that there are no collisions among couples of them. In order to derive non-trivial hydrodynamic models, we close the Euler system around local Maxwellian's which are not equilibrium states. The kinetic model is solved by using a Monte Carlo method, the hydrodynamic one by implicit-explicit Runge-Kutta schemes with weighted essentially non-oscillatory reconstruction (J. Sci. Comput. 2005; 25(1-2):129-155). Several numerical tests are then computed in order to compare the results obtained with the kinetic and the hydrodynamic models.