Hypocoercivity and diffusion limit of a finite volume scheme for linear kinetic equations

Abstract

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.

arXiv comment39 pages, 10 figures, 2 tables

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Cited by in corpus (5)

• Structure preserving schemes for Fokker-Planck equations with nonconstant diffusion matrices
• A uniform-in-time nonlocal approximation of the standard Fokker-Planck equation
• Study of an entropy dissipating finite volume scheme for a nonlocal cross-diffusion system
• A hypocoercivity-exploiting stabilised finite element method for Kolmogorov equation
• Large time behavior of nonlinear finite volume schemes for convection-diffusion equations