Compatible Maxwell solvers with particles II: conforming and non-conforming 2D schemes with a strong Faraday law

Campos Pinto, Martin¹ ; Sonnendrücker, Eric²

¹ CNRS, Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 4, place Jussieu 75005, Paris, France
² Max-Planck Institute for plasma physics, Boltzmannstr. 2, D-85748 Garching, Germany, Mathematics Center, TU Munich, Boltzmannstr. 3, D-85747 Garching, Germany

The SMAI Journal of computational mathematics, Tome 3 (2017), pp. 91-116

Résumé

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 $L^2$ 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 $L^2$ projections in discontinuous spaces. Numerical experiments involving Maxwell and Maxwell-Vlasov problems are then provided to validate the stability of the proposed methods.

Détail

Publié le : 2017-09-15
DOI : 10.5802/smai-jcm.21

Classification : 35Q61, 65M12, 65M60, 65M75
Keywords: Maxwell equations, Gauss laws, structure-preserving, PIC, charge-conserving current deposition, conforming finite elements, discontinuous Galerkin, Conga method.