On a time and space discretized approximation of the Boltzmann equation in the whole space

In this paper, convergence results on the solutions of a time and space discrete model approximation of the Boltzmann equation for a gas of Maxwellian particles in a bounded domain, obtained by Babovsky and Illner (1989), are extended to approximate the solutions of the Boltzmann equation in the whole physical space. This is done for a class of particle interactions including Maxwell and soft cut-off potentials in the sense of Grad.The main result shows that the solutions of the discrete model converge in L1L1 to the solutions of the Boltzmann equation, when the discretization parameters go simultaneously to zero. The convergence is uniform with respect to the discretization parameters.In addition, a sufficient condition for the implementation of the main result is provided.The techniques detailed in this paper may be also applied in other contexts.