Stability and numerical dispersion of symplectic fourth-order time-domain schemes for optical field simulation

Hirono, T.; Lui, W.W.; Yokoyama, K.; Seki, S.

doi:10.1109/50.721080

Cited by 32 publications

(18 citation statements)

References 20 publications

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“…The electromagnetics (or acoustics) equations are first discretized, then the finite-dimensional system of ODEs obtained is considered as an input for symplectic methods. However, in some cases only, the discretization of Maxwell's equations actually leads to a Hamiltonian system of ODEs: it is indeed the case for some FDTD methods [13], more generally for FETD methods [24], and, also for the case considered here: DGTD methods with totally centered numerical fluxes. One particular feature of symplectic schemes is their ability to reach high accuracy and to deal with local time-stepping.…”

Section: Symplectic Schemes For Hamiltonian Systemsmentioning

confidence: 99%

“…These integrators are well established for finite-dimensional Hamiltonian systems (see [19] for several references), most applications being devoted to N-body mechanical systems. However, the number of applications of symplectic schemes in the context of computational electromagnetics is currently growing [13,24]. The electromagnetics (or acoustics) equations are first discretized, then the finite-dimensional system of ODEs obtained is considered as an input for symplectic methods.…”

Section: Symplectic Schemes For Hamiltonian Systemsmentioning

confidence: 99%

“…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%

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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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Section: Symplectic Schemes For Hamiltonian Systemsmentioning

confidence: 99%

Section: Symplectic Schemes For Hamiltonian Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Piperno

2006

European Journal of Computational Mechanics

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“…On the contrary, the preservation of the symplectic structure is known to lead to improved conservation of energy in long-term simulations. As the Maxwell's equations can be written as an infinite-dimensional Hamiltonian system of PDEs, people are now considering the use of symplectic schemes for the time discretization in time-domain simulations [15,16,28].…”

Section: Introductionmentioning

confidence: 99%

Symplectic local time-stepping in non-dissipative DGTD methods applied to wave propagation problems

Piperno

2006

ESAIM: M2AN

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Abstract. The Discontinuous Galerkin Time Domain (DGTD) methods are now popular for the solution of wave propagation problems. Able to deal with unstructured, possibly locally-refined meshes, they handle easily complex geometries and remain fully explicit with easy parallelization and extension to high orders of accuracy. Non-dissipative versions exist, where some discrete electromagnetic energy is exactly conserved. However, the stability limit of the methods, related to the smallest elements in the mesh, calls for the construction of local-time stepping algorithms. These schemes have already been developed for N -body mechanical problems and are known as symplectic schemes. They are applied here to DGTD methods on wave propagation problems.Mathematics Subject Classification. 65L05, 65L20, 65M12, 65M60, 78M10.

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“…For the FDTD, the fourth-order accurate FD scheme is used to reduce the phase error of a propagating wave [43], [44]. However, as far as we know, there is no application of the higher order FD scheme considering the boundary condition at a dielectric interface to the FDTD (although the second-order FDTD has been developed for an arbitrary dielectric interface [37]- [39], the corresponding fourth-order FDTD has not yet been developed).…”

Section: B Application Of Fourth-order Accurate Fd Formula To Adim-bmentioning

confidence: 99%