Marianne Bessemoulin-Chatard, Maxime Herda and myself just submitted a new preprint, entitled Hypocoercivity and diffusion limit of a finite volume scheme for linear kinetic equations :

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, \textit{Trans. Amed. Math. Soc.}, 367, 6 (2015)] to prove the exponential return to equilibrium of the approximate solution. We obtain decay estimates that are uniform 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.

The numerical method developed in this preprint, as well as all the numerical tests, are available on the following jupyter notebook: