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Uniformly accurate methods for Vlasov equations with non-homogeneous strong magnetic field


Authors: Philippe Chartier, Nicolas Crouseilles, Mohammed Lemou, Florian Méhats and Xiaofei Zhao
Journal: Math. Comp. 88 (2019), 2697-2736
MSC (2010): Primary 65L05, 65L20, 65L70
DOI: https://doi.org/10.1090/mcom/3436
Published electronically: April 29, 2019
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Abstract: In this paper, we consider uniformly accurate methods for solving the highly-oscillatory Vlasov equations with non-homogeneous magnetic field. The specific difficulty (and the resulting novelty of our approach) stems from the presence of a non-periodic oscillation, which necessitates a careful ad-hoc reformulation of the equations by using a confinement property of the solution. A two-scale numerical scheme is proposed where the error estimates show that our scheme not only remains insensitive to the stiffness of the problem in terms of both accuracy and computational cost, but also preserves the confinement property at the discrete level. Our results are illustrated numerically on several examples.


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Additional Information

Philippe Chartier
Affiliation: Université de Rennes, INRIA-MINGuS, CNRS, IRMAR-UMR 6625, F-35000 Rennes, France
Email: Philippe.Chartier@inria.fr

Nicolas Crouseilles
Affiliation: Université de Rennes, INRIA-MINGuS, CNRS, IRMAR-UMR 6625, F-35000 Rennes, France
Email: nicolas.crouseilles@inria.fr

Mohammed Lemou
Affiliation: Université de Rennes, CNRS, INRIA-MINGuS, IRMAR-UMR 6625, F-35000 Rennes, France
Email: mohammed.lemou@univ-rennes1.fr

Florian Méhats
Affiliation: Université de Rennes, INRIA-MINGuS, CNRS, IRMAR-UMR 6625, F-35000 Rennes, France
Email: florian.mehats@univ-rennes1.fr

Xiaofei Zhao
Affiliation: School of Mathematics and Statistics, Wuhan University, 430072 Wuhan, People’s Republic of China; and Université de Rennes, INRIA-MINGuS, CNRS, IRMAR-UMR 6625, F-35000 Rennes, France
Email: matzhxf@whu.edu.cn

DOI: https://doi.org/10.1090/mcom/3436
Keywords: Vlasov and Vlasov-Poisson equations, non-homogeneous strong magnetic field, high oscillations, uniform accuracy, two-scale methods
Received by editor(s): February 19, 2018
Received by editor(s) in revised form: December 5, 2018
Published electronically: April 29, 2019
Additional Notes: This work was supported by the French ANR project MOONRISE ANR-14-CE23-0007-01.
The second and third authors were supported by the Enabling Research EUROFusion project CfP-WP14-ER-01/IPP-03.
The fifth author is the corresponding author
The fifth author was supported by the IPL FRATRES
This work has been carried out within the framework of the French Federation for Magnetic Fusion Studies (FR-FCM) and of the Eurofusion consortium, and has received funding from the Euratom research and training programme 2014-2018 and 2019-2020 under grant agreement No. 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission.
Article copyright: © Copyright 2019 American Mathematical Society