Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

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


Authors: Marianne Bessemoulin-Chatard, Maxime Herda and Thomas Rey
Journal: Math. Comp. 89 (2020), 1093-1133
MSC (2010): Primary 82B40, 65M08, 65M12
DOI: https://doi.org/10.1090/mcom/3490
Published electronically: November 26, 2019
Full-text PDF
View in AMS MathViewer New

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 82B40, 65M08, 65M12

Retrieve articles in all journals with MSC (2010): 82B40, 65M08, 65M12


Additional Information

Marianne Bessemoulin-Chatard
Affiliation: Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629, Université de Nantes, F-44000 Nantes, France
Email: marianne.bessemoulin@univ-nantes.fr

Maxime Herda
Affiliation: Inria, Laboratoire Paul Painlevé, CNRS, UMR 8524, Université de Lille, F-59000 Lille, France
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

DOI: https://doi.org/10.1090/mcom/3490
Keywords: Kinetic equations, finite volume methods, hypocoercivity, diffusion limit, asymptotic-preserving schemes
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)
Article copyright: © Copyright 2019 American Mathematical Society