On a structure-preserving numerical method for fractional Fokker-Planck equations
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- by Nathalie Ayi, Maxime Herda, Hélène Hivert and Isabelle Tristani HTML | PDF
- Math. Comp. 92 (2023), 635-693 Request permission
Abstract:
In this paper, we introduce and analyse numerical schemes for the homogeneous and the kinetic Lévy-Fokker-Planck equation. The discretizations are designed to preserve the main features of the continuous model such as conservation of mass, heavy-tailed equilibrium and (hypo)coercivity properties. We perform a thorough analysis of the numerical scheme and show exponential stability and convergence of the scheme. Along the way, we introduce new tools of discrete functional analysis, such as discrete non-local Poincaré and interpolation inequalities adapted to fractional diffusion. Our theoretical findings are illustrated and complemented with numerical simulations.References
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Additional Information
- Nathalie Ayi
- Affiliation: Sorbonne Université, Université de Paris, CNRS, Laboratoire Jacques-Louis Lions, 4 place Jussieu, 75005 Paris, France
- MR Author ID: 1198167
- Email: nathalie.ayi@sorbonne-universite.fr
- Maxime Herda
- Affiliation: Inria, Univ. Lille, CNRS UMR 8524, Laboratoire Paul Painlevé, F-59000 Lille, France
- MR Author ID: 1154786
- ORCID: 0000-0002-0590-5779
- Email: maxime.herda@inria.fr
- Hélène Hivert
- Affiliation: Univ. Lyon, École centrale de Lyon, CNRS UMR 5208, Institut Camille Jordan, F-69134 Écully, France
- ORCID: 0000-0003-0834-6156
- Email: helene.hivert@ec-lyon.fr
- Isabelle Tristani
- Affiliation: Département de mathématiques et applications, École normale supérieure, CNRS, PSL Research University, 45 rue d’Ulm, 75005 Paris, France
- MR Author ID: 1080256
- Email: isabelle.tristani@ens.fr
- Received by editor(s): July 28, 2021
- Received by editor(s) in revised form: June 3, 2022
- Published electronically: November 10, 2022
- Additional Notes: The authors were financially supported by the Hausdorff Research Institute for Mathematics in Bonn. The second author was supported by the LabEx CEMPI (ANR-11-LABX-0007-01). The fourth author was supported by the ANR EFI: ANR-17-CE40-0030 and the ANR SALVE: ANR-19-CE40-0004.
- © Copyright 2022 American Mathematical Society
- Journal: Math. Comp. 92 (2023), 635-693
- MSC (2020): Primary 82B40, 35R11, 65M06, 65M12
- DOI: https://doi.org/10.1090/mcom/3789
- MathSciNet review: 4524105