Comptes Rendus
Theoretical and numerical approaches for Vlasov–Maxwell equations
Charge-conserving FEM–PIC schemes on general grids
Comptes Rendus. Mécanique, Volume 342 (2014) no. 10-11, pp. 570-582.

In this article, we aim at proposing a general mathematical formulation for charge-conserving finite-element Maxwell solvers coupled with particle schemes. In particular, we identify the finite-element continuity equations that must be satisfied by the discrete current sources for several classes of time-domain Vlasov–Maxwell simulations to preserve the Gauss law at each time step, and propose a generic algorithm for computing such consistent sources. Since our results cover a wide range of schemes (namely curl-conforming finite element methods of arbitrary degree, general meshes in two or three dimensions, several classes of time discretization schemes, particles with arbitrary shape factors and piecewise polynomial trajectories of arbitrary degree), we believe that they provide a useful roadmap in the design of high-order charge-conserving FEM–PIC numerical schemes.

Published online:
DOI: 10.1016/j.crme.2014.06.011
Keywords: Maxwell–Vlasov system, Conservation of charge, Continuity equation, Finite element method, Particle-In-Cell, Unstructured grids

Martin Campos Pinto 1; Sébastien Jund 2; Stéphanie Salmon 3; Éric Sonnendrücker 4, 5

1 UPMC Université Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 4, place Jussieu, 75005, Paris, France
2 IRMA – Université de Strasbourg and CNRS, CALVI – INRIA Nancy-Grand Est, France
3 Laboratoire de mathématiques, EA 4535, Fédération ARC CNRS FR 3399, Université de Reims–Champagne-Ardenne, UFR Sciences Exactes et Naturelles, Moulin de la Housse, BP 1039, 51687 Reims cedex 2, France
4 Max-Planck Institute for Plasma Physics, Boltzmannstr. 2, 85748 Garching, Germany
5 Mathematics Center, TU Munich, Boltzmannstr. 3, 85748 Garching, Germany
     author = {Martin Campos Pinto and S\'ebastien Jund and St\'ephanie Salmon and \'Eric Sonnendr\"ucker},
     title = {Charge-conserving {FEM{\textendash}PIC} schemes on general grids},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {570--582},
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     volume = {342},
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     year = {2014},
     doi = {10.1016/j.crme.2014.06.011},
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TI  - Charge-conserving FEM–PIC schemes on general grids
JO  - Comptes Rendus. Mécanique
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DO  - 10.1016/j.crme.2014.06.011
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%A Sébastien Jund
%A Stéphanie Salmon
%A Éric Sonnendrücker
%T Charge-conserving FEM–PIC schemes on general grids
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Martin Campos Pinto; Sébastien Jund; Stéphanie Salmon; Éric Sonnendrücker. Charge-conserving FEM–PIC schemes on general grids. Comptes Rendus. Mécanique, Volume 342 (2014) no. 10-11, pp. 570-582. doi : 10.1016/j.crme.2014.06.011.

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This work has been partially supported by Agence Nationale de la Recherche under the grant ANR-06-CIS6-0013.

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