  We present a mathematical analysis of the asymptotic preserving scheme
proposed in [M. Lemou and L. Mieussens, SIAM J. Sci. Comput., 31, pp. 334-368,
2008] for linear transport equations in kinetic and diffusive regimes. We prove
that the scheme is uniformly stable and accurate with respect to the mean free
path of the particles. This property is satisfied under an explicitly given CFL
condition. This condition tends to a parabolic CFL condition for small mean
free paths, and is close to a convection CFL condition for large mean free
paths. Ou r analysis is based on very simple energy estimates.
