Uniform stability of a finite difference scheme for transport equations in diffusive regimes

An asymptotic preserving numerical scheme (with respect to diffusion scalings) for a linear transport equation is investigated. The scheme is adopted from a class of schemes developed in [S. Jin, L. Pareschi, and G. Toscani, SIAM J. Numer. Anal., 38 (2000), pp. 913-936] and [ A. Klar, SIAM J. Numer. Anal., 35 (1998), pp. 1073-1094]. Stability is proven uniformly in the mean free path under a CFL-type condition turning into a parabolic CFL condition in the diffusion limit.