Convergent Cartesian Grid Methods for Maxwell's Equations in Complex Geometries

Ditkowski, Adi; Dridi, Kim; Hesthaven, Jan S.

doi:10.1006/jcph.2001.6719

Cited by 94 publications

(103 citation statements)

References 19 publications

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“…The method has been applied successfully to IBVPs on constant domains by embedding the boundary operator into the numerical scheme in a penalty-based approach, see for example [26], [27], [28], [29].…”

Section: A Brief Introduction To Embedded Finite-differencementioning

confidence: 99%

High-Order Embedded Finite Difference Schemes for Initial Boundary Value Problems on Time Dependent Irregular Domains

Ditkowski

Harness

2009

J Sci Comput

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This paper considers a family of spatially discrete approximations, including boundary treatment, to initial boundary value problems in evolving bounded domains. The presented method is based on the Cartesian grid embedded Finite-Difference method, which was initially introduced by Abarbanel and Ditkowski [24] [25], for initial boundary value problems on constant irregular domains.We perform a comprehensive theoretical analysis of the numerical issues, which arise when dealing with domains, whose boundaries evolve smoothly in the spatial domain as a function of time. In this class of problems the moving boundaries are impenetrable with either Dirichlet or Neumann boundary conditions, and should not be confused with the class of moving interface problems such as multiple phase flow, solidification, and the stefan problem.Unlike other similar works on this class of problems, the resulting method is not restricted to domains of up to 3-D, can achieve higher then 2 nd -order accuracy both in time and space, and is strictly stable in semi-discrete settings. The strict stability property of the method also implies that the numerical solution remains consistent and valid for a long integration time.A complete convergence analysis is carried in semi-discrete settings, including a detailed analysis for the implementation of the diffusion equation. Numerical solutions of the diffusion equation, using the method for a 2 nd and a 4 th -order of accuracy are carried out in one dimension and two dimensions respectively, which demonstrates the efficacy of the method.

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Section: A Brief Introduction To Embedded Finite-differencementioning

confidence: 99%

High-Order Embedded Finite Difference Schemes for Initial Boundary Value Problems on Time Dependent Irregular Domains

Ditkowski

Harness

2009

J Sci Comput

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“…Alternatively, some embedding FDTD methods which make use of simple Cartesian grids, have been introduced to defer the problem of complex geometries to be solely dealt with by the discretizing schemes [4,44,45]. These methods are also termed embedded interface methods [46].…”

Section: Introductionmentioning

confidence: 99%

“…To overcome the staircasing problems and to impose the physically correct jump conditions, which are often encountered in the modeling of complex geometries, appropriate local modifications of the differentiation scheme close to boundaries and interfaces in a preprocessing stage are essential. While maintaining the simplicity and computational efficiency of the Yee scheme [1,2], these modified FDTD methods fully restore second-order accuracy, even in case of curved boundaries and interfaces, by using a simple Cartesian grid [4,44,45]. However, the extension of embedding FDTD methods to higher-order accuracy remains a significant challenge [4].…”

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%

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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

197

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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“…As known, a second-order accuracy can be obtained for the spatially-extended space derivative operators in the regular Yee scheme, which assume the materials are homogeneous, at least within the extent of their stencil. However, due to the inhomogeneity of the coefficients in Maxwell's equations, this full accuracy can not be realized across discontinuous material interfaces and will be degraded to first order [103]. In this section, the error analysis is presented for the regular Yee scheme at dielectric interfaces in 2D case.…”

Section: Error Analysis At Materials Interfacesmentioning

confidence: 99%

Reducing the losses of optical metamaterials

Fang

2010

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Convergent Cartesian Grid Methods for Maxwell's Equations in Complex Geometries

Cited by 94 publications

References 19 publications

High-Order Embedded Finite Difference Schemes for Initial Boundary Value Problems on Time Dependent Irregular Domains

High-Order Embedded Finite Difference Schemes for Initial Boundary Value Problems on Time Dependent Irregular Domains

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

Reducing the losses of optical metamaterials

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