Discrete Velocity Models and Half-Space Problems

Abstract: We study some questions related to discrete velocity models (DVMs) of the Boltzmann equation. In the case of plane stationary problems the typical DVM reduces to a dynamical system (system of ODEs). Properties of such systems are studied in this paper in the most general case. In particular, a topological classification of their singular points is made and dimensions of the corresponding stable, unstable and center manifolds are computed. These results are applied to typical half-space problems of rarefied gas dynamics. A classification of well-posed half-space problems for linearized DVMs is made. Exact solutions of a (simplified) linearized kinetic model of BGK type are found as limiting case of corresponding discrete models. The main results of the paper can be also used for moment approximations and other versions of discretizised kinetic equations