Discrete velocity models for the BGK-Boltzmann equation based on a discrete equilibrium defined by using a discrete minimum entropy principle are proposed. Existence, uniqueness, positivity and local entropy dissipation are proved.
Numerical approximation is obtained by both explicit and implicit schemes.
Numerical simulations of several 1 D and 2D test examples demonstrate the performance of the method. The extension of this approach to the radiative transfer equation is addressed. In a second part moment models are considered for the same applications. Those models are based on the closure approach introduced by D. Levermore applied either to the underlying continuous velocity kinetic equation or to a discrete velocity model of this equation.