Convergence of Discrete-Velocity Schemes for the Boltzmann Equation

In this paper we prove the convergence of two discrete-velocity deterministic schemes for the Boltzmann equation, namely, Buet's scheme and a new finite-volume scheme that we introduce here. We write the discretized equation in the form of a Boltzmann continuous equation in order to be in the framework of the DiPerna-Lions theory of renormalized solutions. In order to prove convergence we have to overcome two difficulties: the convergence of the discretized collision kernel is very weak and the lemma on the compactness of velocity averages can be recovered only asymptotically when the parameter of discretization tends to zero.