High-Order Symplectic Integration Methods for Finite Element Solutions to Time Dependent Maxwell Equations

Rieben, Robert N.; White, Dan; Rodrigue, Garry

doi:10.1109/tap.2004.832356

Cited by 81 publications

(62 citation statements)

References 18 publications

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“…Systems of the form (3.9) arise, for example, after spacial discretization of the Maxwell equations, of relevant interest in physics and engineering [32,27,13].…”

Section: Application Of Splitting Methods To Linear Systemsmentioning

confidence: 99%

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On the Linear Stability of Splitting Methods

Blanes

Casas

Murua³

2007

Found Comput Math

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A comprehensive linear stability analysis of splitting methods is carried out by means of a 2 × 2 matrix K(x) with polynomial entries (the stability matrix) and the stability polynomial p(x) (the trace of K(x) divided by two). An algorithm is provided for determining the coefficients of all possible timereversible splitting schemes for a prescribed stability polynomial. It is shown that p(x) carries essentially all the information needed to construct processed splitting methods for numerically approximating the evolution of linear systems. By selecting conveniently the stability polynomial, new integrators with processing for linear equations are built which are orders of magnitude more efficient than other algorithms previously available.

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“…Systems of the form (3.9) arise, for example, after spacial discretization of the Maxwell equations, of relevant interest in physics and engineering [32,27,13].…”

Section: Application Of Splitting Methods To Linear Systemsmentioning

confidence: 99%

“…According to Proposition 2.4, if x ≡ hλ ∈ (−x * , x * ), one step of the method is conjugate to the exact h-flow of the modified harmonic oscillator 27) withλ(h) = 1 h Φ(hλ). In other words, there exists a well defined matrix…”

Section: Geometric Propertiesmentioning

confidence: 99%

On the Linear Stability of Splitting Methods

Blanes

Casas

Murua³

2007

Found Comput Math

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“…. 4, that generalize the leap-frog method and are known to be stable and nondissipative [12], [23], [24]. Applied to the time-dependent system (13)-(14), they compute auxiliary solutions by first letting E n,0 := E n , B n,0 := B n , then for j = 0, .…”

Section: Consistency Criteria For Time-domain Fem Schemesmentioning

confidence: 99%

“…In this section we establish that a generic algorithm which generalizes the early virtual particle method of Eastwood [9] allows to compute charge conserving currents, i.e., discrete currents satisfying the consistency criterion (23). Despite its simplicity, the algorithm covers a large class of particle schemes.…”

Section: Consistent Coupling With Particlesmentioning

confidence: 99%

Charge-conserving FEM–PIC schemes on general grids

Pinto

Jund

Salmon

et al. 2014

Comptes Rendus. Mécanique

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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 2 or 3 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.

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“…It has been shown [20] that symplectic time-schemes, originally developed for the numerical time integration of dynamical Hamiltonian systems -astronomy, molecular dynamics, etc - [25] and currently being used for the time-integration of spatially-discretized wave propagation problems [12,13,24] can overcome this problem, via locally implicit time-integration or explicit local time-stepping.…”

Section: Introductionmentioning

confidence: 99%

DGTD methods using modal basis functions and symplectic local time-stepping

Piperno

2006

European Journal of Computational Mechanics

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The Discontinuous Galerkin Time Domain (DGTD) methods are now widely used for the solution of wave propagation problems. Able to deal with unstructured meshes past complex geometries, they remain fully explicit with easy parallelization and extension to high orders of accuracy. Still, modal or nodal local basis functions have to be chosen carefully to obtain actual numerical accuracy. Concerning time discretization, explicit nondissipative energy-preserving time-schemes exist, but their stability limit remains linked to the smallest element size in the mesh. Symplectic algorithms, based on local-time stepping or local implicit scheme formulations, can lead to dramatic reductions of computational time, which is shown here on two-dimensional acoustics problems.Key-words: waves, acoustics, Maxwell's system, Discontinuous Galerkin methods, mass matrix condition number, symplectic schemes, energy conservation, local time-stepping.Méthodes DGTD avec fonctions de base modales et schémas en temps symplectiques : applications en propagation d'ondes.Résumé : Les méthodes de Galerkine discontinu sont maintenant largement utilisées pour la résolution numérique de problèmes de propagation d'ondes. S'appuyant sur des maillages non-structurés autour des géométries les plus générales, elles restent quasiment complète-ment explicites, facilement parallélisables et d'ordre élevé. Il convient néanmoins d'optimiser le choix des fonctions de base discontinues (modales ou nodales). Pour ce qui est de la discrét-isation en temps, des schémas explicites non-dissipatifs existent, mais leur limite de stabilité reste liée aux plus petits éléments du maillage. Des algorithmes symplectiques, avec pas de temps local ou schéma localement implicite, conduisent à des diminutions considérables du temps de calcul.

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High-Order Symplectic Integration Methods for Finite Element Solutions to Time Dependent Maxwell Equations

Cited by 81 publications

References 18 publications

On the Linear Stability of Splitting Methods

On the Linear Stability of Splitting Methods

Charge-conserving FEM–PIC schemes on general grids

DGTD methods using modal basis functions and symplectic local time-stepping

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