  In this article, we are interested in the asymptotic analysis of a finite
volume scheme for one dimensional linear kinetic equations, with either
Fokker-Planck or linearized BGK collision operator. Thanks to appropriate
uniform estimates, we establish that the proposed scheme is
Asymptotic-Preserving in the diffusive limit. Moreover, we adapt to the
discrete framework the hypocoercivity method proposed by [J. Dolbeault, C.
Mouhot and C. Schmeiser, Trans. Amer. Math. Soc., 367, 6 (2015)] to prove the
exponential return to equilibrium of the approximate solution. We obtain decay
rates that are bounded uniformly in the diffusive limit.
  Finally, we present an efficient implementation of the proposed numerical
schemes, and perform numerous numerical simulations assessing their accuracy
and efficiency in capturing the correct asymptotic behaviors of the models.
