Hypocoercivity and diffusion limit of a finite volume scheme for linear kinetic equations
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- by Marianne Bessemoulin-Chatard, Maxime Herda and Thomas Rey;
- Math. Comp. 89 (2020), 1093-1133
- DOI: https://doi.org/10.1090/mcom/3490
- Published electronically: November 26, 2019
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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.References
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Bibliographic Information
- Marianne Bessemoulin-Chatard
- Affiliation: Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629, Université de Nantes, F-44000 Nantes, France
- MR Author ID: 960946
- Email: marianne.bessemoulin@univ-nantes.fr
- Maxime Herda
- Affiliation: Inria, Laboratoire Paul Painlevé, CNRS, UMR 8524, Université de Lille, F-59000 Lille, France
- MR Author ID: 1154786
- Email: maxime.herda@inria.fr
- Thomas Rey
- Affiliation: Inria, Laboratoire Paul Painlevé, CNRS, UMR 8524, Université de Lille, F-59000 Lille, France
- Email: thomas.rey@univ-lille.fr
- Received by editor(s): December 14, 2018
- Received by editor(s) in revised form: July 16, 2019
- Published electronically: November 26, 2019
- Additional Notes: The first author was partially funded by the Centre Henri Lebesgue (ANR-11-LABX-0020-01) and ANR Project MoHyCon (ANR-17-CE40-0027-01)
The second and third authors were partially funded by Labex CEMPI (ANR-11-LABX-0007-01) and ANR Project MoHyCon (ANR-17-CE40-0027-01) - © Copyright 2019 American Mathematical Society
- Journal: Math. Comp. 89 (2020), 1093-1133
- MSC (2010): Primary 82B40, 65M08, 65M12
- DOI: https://doi.org/10.1090/mcom/3490
- MathSciNet review: 4063313