Charge-conserving FEM–PIC schemes on general grids

Pinto, Martin Campos; Jund, Sébastien; Salmon, Stéphanie; Sonnendrücker, Éric

doi:10.1016/j.crme.2014.06.011

Cited by 51 publications

(51 citation statements)

References 29 publications

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“…These shapes have the advantage of being polynomial on their support unlike spline particles which are made of several polynomial pieces. Because the longterm charge conservation properties of the scheme require an accurate time-average of the particle current in the deposition method as demonstrated in [30,17], this feature actually simplifies the involved algorithms. For more details on these algorithms we refer to [18].…”

Section: Compatible Maxwell Solvers With Particles Imentioning

confidence: 99%

“…The first remedies have been to correct the electric field by solving exact [9] or approximate [51,47] Poisson equations, until Eastwood, Villasenor and Buneman [30,70] noticed that stable solvers could be obtained by an adequate computation of the current from the particles which would preserve a discrete Gauss law. In the framework of curl-conforming finite element method, we have described a generic algorithm in [17], revisited in a geometric perspective in [66,57]. More recently it was noticed that this algorithm fits in a semi-discrete Hamiltonian structure, where the divergence constraints are identified as Casimirs and thus are automatically conserved [46].…”

Section: Compatible Maxwell Solvers With Particles Imentioning

confidence: 99%

See 1 more Smart Citation

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

Pinto¹,

Sonnendrücker²

2017

The SMAI Journal of Computational Mathematics

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Abstract. This article is the first 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 chargeconserving schemes are provided for the 2D case. In this article we study two schemes which include a strong discretization of the Ampere law. The first one is based on a standard conforming mixed finite element discretization and the long-time stability is ensured by a Raviart-Thomas finite element interpolation for the current source, thanks to its commuting diagram properties. The second one is a new non-conforming variant where the numerical fields are sought in fully discontinuous spaces. Numerical experiments involving Maxwell and Maxwell-Vlasov problems are then provided to validate the stability of the proposed methods.Math. classification. 35Q61, 65M12, 65M60, 65M75.

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Section: Compatible Maxwell Solvers With Particles Imentioning

confidence: 99%

Section: Compatible Maxwell Solvers With Particles Imentioning

confidence: 99%

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

Pinto¹,

Sonnendrücker²

2017

The SMAI Journal of Computational Mathematics

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“…This space discretization is standard ans has been studied in, e.g., Ref. [20,19,25,9] where the source term J is approximated with a standard orthogonal projection on V 2 h , leading to define J h by…”

Section: Conforming Finite Elements With a Strong Faraday Lawmentioning

confidence: 99%

“…Following our stability analysis we would like that the resulting solutions satisfy the proper discrete Gauss laws which involve the structure-preserving divergence operators identified in this work, namely (5.5) and (5.7) respectively. In [9] this construction was described for the conforming FEM method, using a time averaging and an extension of the L 2 projection (5.13) for current densities carried by Dirac particles. Specifically, it was shown that the quantities To assess the basic convergence and stability properties of the proposed schemes we use the analytical current source proposed in [17,12] to study the charge conservation properties of a penalized finite volume scheme.…”

Section: Charge-conserving Coupling With Point Particlesmentioning

confidence: 99%

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

Pinto¹,

Sonnendrücker²

2017

The SMAI Journal of Computational Mathematics

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Abstract. 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 chargeconserving 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.Math. classification. 35Q61, 65M12, 65M60, 65M75.

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“…Since a discrete version of the continuity equation (6) can be obtained in particle schemes by averaging the time-dependent current density (20) over the time step, and evaluating the charge density at t n (see [2] and the reference therein), we consider the following smoothed, time-averaged current density:…”

Section: Smooth Particle Current Deposition With Numerical Quadraturesmentioning

confidence: 99%

Electromagnetic PIC simulations with smooth particles: a numerical study

Pinto

Lutz

Mounier³

2016

ESAIM: Proc.

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Abstract. In this article we study a charge-conserving finite-element particle scheme for the Maxwell-Vlasov system that is based on a div-conforming representation of the electric field and we propose a high-order deposition algorithm for smooth particles with piecewise polynomial shape. The numerical performances of the method are assessed with an academic beam test-case, and it is shown that for an appropriate choice of the particle parameters the efficiency of the resulting method overcomes that of similar finite-element schemes using point particles.Résumé. Cet article présente un schémaéléments-finis conservant la charge pour le système de Vlasov-Maxwell basé sur une représentation div-conforme du champélectrique, ainsi qu'un algorithme d'ordreélevé pour déposer le courant porté par des particules régulières polynômiales par morceaux. Les performances numériques du schéma couplé sontévaluéesà l'aide d'un cas test académique de faisceau, et nous montrons que pour un choix approprié des paramètres des particules, cette méthode se révèle plus efficace que d'autres schémas similaires couplés avec des particules ponctuelles.

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Charge-conserving FEM–PIC schemes on general grids

Cited by 51 publications

References 29 publications

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

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

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

Electromagnetic PIC simulations with smooth particles: a numerical study

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