This article is the second of a series where we develop and analyze structure-preserving finite element discretizations for the time-dependent 2D Maxwell system with long-time stability properties, and propose a charge-conserving deposition scheme to extend the stability properties in the case where the current source is provided by a particle method. The schemes proposed here derive from a previous study where a generalized commuting diagram was identified as an abstract compatibility criterion in the design of stable schemes for the Maxwell system alone, and applied to build a series of conforming and non-conforming schemes in the 3D case. Here the theory is extended to account for approximate sources, and specific charge-conserving schemes are provided for the 2D case. In this second article we study two schemes which include a strong discretization of the Faraday law. The first one is based on a standard conforming mixed finite element discretization and the long-time stability is ensured by the natural projection for the current, also standard. The second one is a new non-conforming variant where the numerical fields are sought in fully discontinuous spaces. In this 2D setting it is shown that the associated discrete curl operator coincides with that of a classical DG formulation with centered fluxes, and our analysis shows that a non-standard current approximation operator must be used to yield a charge-conserving scheme with long-time stability properties, while retaining the local nature of projections in discontinuous spaces. Numerical experiments involving Maxwell and Maxwell-Vlasov problems are then provided to validate the stability of the proposed methods.
DOI: 10.5802/smai-jcm.21
Keywords: Maxwell equations, Gauss laws, structure-preserving, PIC, charge-conserving current deposition, conforming finite elements, discontinuous Galerkin, Conga method.
CC-BY-NC-ND 4.0
@article{SMAI-JCM_2017__3__91_0,
author = {Martin Campos Pinto and Eric Sonnendr\"ucker},
title = {Compatible {Maxwell} solvers with particles {II:} conforming and non-conforming {2D} schemes with a strong {Faraday} law},
journal = {The SMAI Journal of computational mathematics},
pages = {91--116},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {3},
year = {2017},
doi = {10.5802/smai-jcm.21},
zbl = {1416.78029},
mrnumber = {3695789},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.21/}
}
TY - JOUR AU - Martin Campos Pinto AU - Eric Sonnendrücker TI - Compatible Maxwell solvers with particles II: conforming and non-conforming 2D schemes with a strong Faraday law JO - The SMAI Journal of computational mathematics PY - 2017 SP - 91 EP - 116 VL - 3 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.21/ DO - 10.5802/smai-jcm.21 LA - en ID - SMAI-JCM_2017__3__91_0 ER -
%0 Journal Article %A Martin Campos Pinto %A Eric Sonnendrücker %T Compatible Maxwell solvers with particles II: conforming and non-conforming 2D schemes with a strong Faraday law %J The SMAI Journal of computational mathematics %D 2017 %P 91-116 %V 3 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.21/ %R 10.5802/smai-jcm.21 %G en %F SMAI-JCM_2017__3__91_0
Martin Campos Pinto; Eric Sonnendrücker. Compatible Maxwell solvers with particles II: conforming and non-conforming 2D schemes with a strong Faraday law. The SMAI Journal of computational mathematics, Volume 3 (2017), pp. 91-116. doi : 10.5802/smai-jcm.21. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.21/
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