Explicit local time-stepping methods for Maxwell’s equations

Grote, Marcus J.; Mitkova, Teodora

doi:10.1016/j.cam.2010.04.028

Cited by 60 publications

(51 citation statements)

References 27 publications

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“…Assuming (1.3), one of the results states that if τ ∼ h its second ODE order is maintained for h → 0 (no order reduction with stiff source terms). Another result is that, with zero source terms, 4) showing that the conservation behavior is actually very good. Herewith it is of course tacitly assumed that τ is limited such that the method integrates in a stable way, something which cannot be concluded from this result due to the minus sign in front of the third term.…”

Section: The Composition Scheme and The Trapezoidal Rulementioning

confidence: 97%

“…1, the method from [2] and [4] also serves to overcome step size limitations by grid-induced stiffness. That method is designed for second-order wave equations and the work reported focuses also on the Maxwell equations.…”

Section: Future Workmentioning

confidence: 99%

“…The new scheme also bears a relationship to a component splitting scheme discussed in [3] which is especially designed for a discontinuous Galerkin discretization. A novel local time-stepping technique for second-order wave equations discretized in space by a continuous or discontinuous finite element method has been proposed in [2] and is further discussed in [4]. This local time-stepping technique also serves to overcome step size limitations by grid-induced stiffness.…”

Section: Introductionmentioning

confidence: 99%

“…5 on some plans for future work. Also a few remarks on the approach from [2] and [4] are included in this final section.…”

Section: Introductionmentioning

confidence: 99%

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Component splitting for semi-discrete Maxwell equations

Verwer

2010

Bit Numer Math

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A time-integration scheme for semi-discrete linear Maxwell equations is proposed. Special for this scheme is that it employs component splitting. The idea of component splitting is to advance the greater part of the components of the semidiscrete system explicitly in time and the remaining part implicitly. The aim is to avoid severe step size restrictions caused by grid-induced stiffness emanating from locally refined space grids. The proposed scheme is a blend of an existing secondorder composition scheme which treats wave terms explicitly and the second-order implicit trapezoidal rule. The new blended scheme retains the composition property enabling higher-order composition.

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Section: The Composition Scheme and The Trapezoidal Rulementioning

confidence: 97%

Section: Future Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…5 on some plans for future work. Also a few remarks on the approach from [2] and [4] are included in this final section.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Component splitting for semi-discrete Maxwell equations

Verwer

2010

Bit Numer Math

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“…By combining a symplectic integrator with a DG discretization of Maxwell's equations in first-order form, Piperno [18] proposed a second-order explicit local time-stepping scheme, which also conserves a discrete energy. Starting from the standard LF method, the authors proposed energy conserving fully explicit LTS integrators of arbitrarily high accuracy for the wave equation [1]; that approach was extended to Maxwell's equations in [19]. An hp-version, where not only the time-step but also the order of approximation is adapted within di↵erent regions of the mesh, was proposed in [20] and later applied to a realistic geological model [21].…”

Section: Introductionmentioning

confidence: 99%

Multi-level explicit local time-stepping methods for second-order wave equations

Diaz

Grote

2015

Computer Methods in Applied Mechanics and Engineering

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To cite this version:Julien Diaz, Marcus Grote. Multi-level explicit local time-stepping methods for second-order wave equations. Computer Methods in Applied Mechanics and Engineering, Elsevier, 2015, 291, pp.240-265. 10 Abstract Local mesh refinement severly impedes the e ciency of explicit time-stepping methods for numerical wave propagation. Local time-stepping (LTS) methods overcome the bottleneck due to a few small elements by allowing smaller time-steps precisely where those elements are located.Yet when the region of local mesh refinement itself contains a sub-region of even smaller elements, any local time-step again will be overly restricted. To remedy the repeated bottleneck

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High‐order discontinuous Galerkin method for time‐domain electromagnetics on geometry‐independent Cartesian meshes

Navarro-García

Sevilla

Nadal

et al. 2021

Numerical Meth Engineering

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In this work we present the Cartesian grid discontinuous Galerkin (cgDG) finite element method, a novel numerical technique that combines the high accuracy and efficiency of a high‐order discontinuous Galerkin discretization with the simplicity and hierarchical structure of a geometry‐independent Cartesian mesh. The elements that intersect the boundary of the physical domain require special treatment in order to minimize their effect on the performance of the algorithm. We considered the exact representation of the geometry for the boundary of the domain avoiding any nonphysical artifacts. We also define a stabilization procedure that eliminates the step size restriction of the time marching scheme due to extreme cut patterns. The unstable degrees of freedom are eliminated and the supporting regions of their shape functions are reassigned to neighboring elements. A subdomain matching algorithm and an a posterior enrichment strategy are presented. Combining these techniques we obtain a final spatial discretization that preserves stability and accuracy of the standard body‐fitted discretization. The method is validated through a series of numerical tests and it is successfully applied to the solution of problems of interest in the context of electromagnetic scattering with increasing complexity.

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Explicit local time-stepping methods for Maxwell’s equations

Cited by 60 publications

References 27 publications

Component splitting for semi-discrete Maxwell equations

Component splitting for semi-discrete Maxwell equations

Multi-level explicit local time-stepping methods for second-order wave equations

High‐order discontinuous Galerkin method for time‐domain electromagnetics on geometry‐independent Cartesian meshes

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