Fourth order compact implicit method for the Maxwell equations with discontinuous coefficients

Yefet, Amir; Turkel, Eli

doi:10.1016/s0168-9274(99)00075-6

Cited by 44 publications

(53 citation statements)

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“…A few attempts have been made to accelerate the convergence speed of embedding FDTD methods, giving rise to several fully fourth-order FDTD methods on a medium with material discontinuities by using a simple structured grid [4,[47][48][49], including the curvilinear grid [50]. The single structured grid is logically equivalent to a Cartesian grid.…”

Section: Introductionmentioning

confidence: 99%

“…The single structured grid is logically equivalent to a Cartesian grid. The structured grids used in these fourth-order FDTD methods [47][48][49][50] are, nevertheless, less general than the Cartesian grids of the embedding methods [4,44,45]. In particular, these fourth-order approaches typically require that the electric grid points coincide with the plane interface in 2D configuration, so that the staircasing problem remains unsolved in these high-order methods.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, without appropriate numerical interface schemes, the conventional high-order time-domain methods would be easily degraded to produce loworder accuracy [4]. Therefore, the significant accomplishment of these fourth-order schemes [47][48][49][50] is that the physical jump conditions at material interfaces are correctly enforced up to high-order, so that fourth-order convergence is uniformly assured over the entire domain.…”

Section: Introductionmentioning

confidence: 99%

“…Similar to the embedding methods [4,44,45], the modeling of interfaces results in a local modification of the differential scheme in a preprocessing stage [4,[47][48][49][50].…”

Section: Introductionmentioning

confidence: 99%

“…The studies of fourth-order embedding FDTD schemes [4,[47][48][49][50] open up the opportunity for developing other more general and robust high-order interface schemes, which deserve further exploration. There are many important questions remaining unanswered.…”

Section: Introductionmentioning

confidence: 99%

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High-order FDTD methods via derivative matching for Maxwell's equations with material interfaces

Zhao

Wang

2004

Journal of Computational Physics

199

164

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This paper introduces a series of novel hierarchical implicit derivative matching methods to restore the accuracy of high-order finite difference time-domain (FDTD) schemes of computational electromagnetics (CEM) with material interfaces in one (1D) and two spatial dimensions (2D). By making use of fictitious points, systematic approaches are proposed to locally enforce the physical jump conditions at material interfaces in a preprocessing stage, to arbitrarily high orders of accuracy in principle. While often limited by numerical instability, orders up to 16 in 1D and 2D are achieved. Detailed stability analyses are presented for the present approach to examine the up limit in constructing embedded FDTD methods. As natural generalizations of the high-order FDTD schemes, the proposed derivative matching methods automatically reduce to the standard FDTD schemes when the material interfaces are absent. An interesting feature of the present approach is that it encompasses a variety of schemes of different orders in a single code. Another feature of the present approach is that it can be robustly implemented with other high accuracy time-domain approaches, such as the multiresolution time-domain (MRTD) method and the local spectral time-domain (LSTD) method, to cope 1 with material interfaces. Numerical experiments on both 1D and 2D problems are carried out to test the convergence, examine the stability, access the efficiency, and explore the limitation of the proposed methods. It is found that operating at their best capacity, the proposed high order schemes are at least a million times more efficient than their fourth order versions in both 1D and 2D. Therefore, it is believed that the proposed hierarchical derivative matching methods are highly accurate, efficient, and robust for CEM.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Similar to the embedding methods [4,44,45], the modeling of interfaces results in a local modification of the differential scheme in a preprocessing stage [4,[47][48][49][50].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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High-order FDTD methods via derivative matching for Maxwell's equations with material interfaces

Zhao

Wang

2004

Journal of Computational Physics

199

164

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Hybrid unconditionally stable high‐order nonstandard schemes with optimal error‐controllable spectral resolution for complex microwave problems

Kantartzis

2012

Int J Numerical Modelling

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A hybrid unconditionally stable time-domain technique for the precise analysis and wideband performance characterization of 3D microwave systems is developed in this paper. Founded on a nontraditional differential discretization basis, the proposed technique launches a class of robust operators via an error-controllable procedure that offers enhanced spectral resolution and optimal spatial stencils. The key asset of the frequency-dependent algorithm is the novel high-order nonstandard approximators, whose tensorial properties preserve the hyperbolic character of Maxwell's equations. In this manner, the resulting formulation remains completely explicit and generates effective dual meshes free of artificial vector parasites and spurious modes. Moreover, the preceding schemes are fruitfully hybridized, in the context of nonoverlapping subspaces, with an alternating-direction implicit finite-element time-domain method in an effort to handle abruptly varying media boundaries and intricate geometries. Hence, an extensive decrease of dispersion errors is achieved, even when time-steps are chosen appreciably beyond stability limits. These advanced simulation competences are successfully applied to diverse real-world setups and composite configurations, thus validating the efficiency and universality of the proposed methodology.Amid the broad assortment of numerical tools, the contribution of the finite-difference time-domain (FDTD) algorithm [8] to the investigation of waveguide and antenna devices has been widely acknowledged, furnishing beneficial outcomes, along with its various powerful high-order renditions [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Nevertheless, its applicability to contemporary scenarios is proven cumbersome, especially on a wideband basis, as most of the devices have many geometric details, arbitrary discontinuities, or involve dispersive materials, which call for prolonged simulations because of the Courant stability condition. Furthermore, the large electrical size of some cases dictates fine lattice resolutions together with the enforcement of nonphysical assumptions. Toward the circumvention of the stability limit, the alternating-direction implicit (ADI) concept has, lately, offered a promising means, which can allow the use of large time-steps during the update of field components [28,29]. In fact, since its initial advent, the ADI-FDTD algorithm became the topic of a systematic examination [30][31][32], whose principal deductions unveiled a serious shortcoming in practical implementations; the appearance of gradually increasing dispersion errors as time-steps depart from the Courant limit. For the mitigation of this problem, various efficient techniques have been, hitherto, presented for Cartesian [33][34][35][36][37][38][39][40][41] and nonorthogonal lattices [42,43].It is the objective of this paper to introduce a 3D frequency-dependent unconditionally stable timedomain methodology, based on a family of high-order nonstandard schemes, for the accurate modeling ...

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A new parallelization strategy for solving time-dependent 3D Maxwell equations using a high-order accurate compact implicit scheme

Kashdan

Galanti

2006

Int. J. Numer. Model.

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SUMMARYWith progress in computer technology there has been renewed interest in a time-dependent approach to solving Maxwell equations. The commonly used Yee algorithm (an explicit central difference scheme for approximation of spatial derivatives coupled with the Leapfrog scheme for approximation of temporal derivatives) yields only a second-order of accuracy. On the other hand, an increasing number of industrial applications, especially in optic and microwave technology, demands high-order accurate numerical modelling. The standard way to increase accuracy of the finite difference scheme without increasing the differential stencil is to replace a 2nd-order accurate explicit scheme for approximation of spatial derivatives with the 4th-order accurate compact implicit scheme. In general, such a replacement requires additional memory resources and slows the computations. However, the curl-based form of Maxwell equations allows us to construct an effective parallel algorithm with the alternating domain decomposition (ADD) minimizing the communication time. We present a new parallel approach to the solution of threedimensional time-dependent Maxwell equations and provide a theoretical and experimental analysis of its performance.

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Fourth order compact implicit method for the Maxwell equations with discontinuous coefficients

Cited by 44 publications

References 0 publications

High-order FDTD methods via derivative matching for Maxwell's equations with material interfaces

High-order FDTD methods via derivative matching for Maxwell's equations with material interfaces

Hybrid unconditionally stable high‐order nonstandard schemes with optimal error‐controllable spectral resolution for complex microwave problems

A new parallelization strategy for solving time-dependent 3D Maxwell equations using a high-order accurate compact implicit scheme

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