Convergence of a discrete-velocity model for the Boltzmann-BGK equation

We prove the convergence of a conservative and entropic discrete-velocity model for the Bathnagar-Gross-Krook (BGK) equation. In this model, the approximation of the Maxwellian is based on a discrete entropy minimization principle. The main difficulty, due to its implicit definition, is to prove that this approximation is consistent. We also demonstrate the existence and uniqueness of a solution to the discrete-velocity model, by using a fixed-point theorem. Finally, the model is written in a continuous equation form, and we prove the convergence of its solution toward a solution of the BGK equation.