On a discrete framework of hypocoercivity for kinetic equations

Blaustein, Alain; Filbet, Francis

doi:10.1090/mcom/3862

On a discrete framework of hypocoercivity for kinetic equations
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by Alain Blaustein and Francis Filbet

Math. Comp. 93 (2024), 163-202

DOI: https://doi.org/10.1090/mcom/3862

Published electronically: July 19, 2023

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Abstract:

We propose and study a fully discrete finite volume scheme for the linear Vlasov-Fokker-Planck equation written as an hyperbolic system using Hermite polynomials in velocity. This approach naturally preserves the stationary solution and the weighted $L^2$ relative entropy. Then, we adapt the arguments developed in Dolbeault, Mouhot, and Schmeiser [Trans. Amer. Math. Soc. 367 (2015), pp. 3807–3828] based on hypocoercivity methods to get quantitative estimates on the convergence to equilibrium of the discrete solution. Finally, we prove that in the diffusive limit, the scheme is asymptotic preserving with respect to both the time variable and the scaling parameter at play.

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