On deterministic approximation of the Boltzmann equation in a bounded domain

In this paper we present a fully deterministic method for the numerical solution to the Boltzmann equation of rarefied gas dynamics in a bounded domain for multi-scale problems. Periodic, specular reflection and diffusive boundary conditions are discussed and investigated numerically. The collision operator is treated by a Fourier approximation of the collision integral, which guarantees spectral accuracy in velocity with a computational cost of $N\log (N)$, where $N$ is the number of degree of freedom in velocity space. This algorithm is coupled with a second order finite volume scheme in space and a time discretization allowing to deal for rarefied regimes as well as their hydrodynamic limit. Finally, several numerical tests illustrate the efficiency and accuracy of the method for unsteady flows (Poiseuille flows, ghost effects, trend to equilibrium).