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

Bessemoulin-Chatard, Marianne; Herda, Maxime; Rey, Thomas

doi:10.1090/mcom/3490

Hypocoercivity and diffusion limit of a finite volume scheme for linear kinetic equations
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Math. Comp. 89 (2020), 1093-1133 Request permission

Abstract:

In this article, we are interested in the asymptotic analysis of a finite volume scheme for one-dimensional linear kinetic equations, with either a 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, no. 6 (2015)] to prove the exponential return to equilibrium of the approximate solution. We obtain decay rates that are uniformly bounded 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.

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