Regularity of velocity averages for transport equations on random discrete velocity grids

Abstract

We go back to the question of the regularity of the velocity average \int f (x, v)ψ(v) dµ(v) when f and v · \nabla_x f both belong to L^2 , and the variable v lies in a discrete subset of R^D. First of all, we provide a rate, depending on the number of velocities, to the defect of H^{1/2} regularity. Second of all, we show that the H^{1/2} regularity holds in expectation when the set of velocities is chosen randomly. We apply this statement to investigate the consistency with the diffusion asymptotics of a Monte–Carlo–like discrete velocity model.