Divergence Preserving Discrete Surface Integral Methods for Maxwell's Curl Equations Using Non-orthogonal Unstructured Grids

Madsen, Niel K.

doi:10.1006/jcph.1995.1114

Cited by 123 publications

(60 citation statements)

References 12 publications

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“…Problem (36), (37) is standard for the finite element method [13]. Typical results concerning the convergence estimates, the superconvergence of the finite volume method under different assumptions on the smoothness of the solution can be found in [16,35].…”

Section: Boundary Value Problem For the Stationary Diffusion Equationmentioning

confidence: 99%

“…From later works in this direction we note the paper [15], in which the accuracy estimates in different norms are given using the finite volume method considered as the Petrov-Galerkin finite element method. Here we will show the simple possibility of construction and investigation of difference schemes for problem (36), (37) on the basis of the vector analysis grid operators using the Delaunay triangulation and the Voronoi diagram.…”

Section: Boundary Value Problem For the Stationary Diffusion Equationmentioning

confidence: 99%

“…Construction of consistent approximations using the Delauney triangulation for some problems of mathematical physics was performed in [60]. For electrodynamics problems discrete approximations with the use of the Delauney triangulation were constructed in [23,[36][37][38].…”

Section: Introductionmentioning

confidence: 99%

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Finite-difference Approximation of Mathematical Physics Problems on Irregular Grids

Vabishchevich¹

2005

Comput. Methods Appl. Math.

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-Mathematical physics problems are often formulated by means of the vector analysis differential operators: divergence, gradient and rotor. For approximate solutions of such problems it is natural to use the corresponding operator statements for the grid problems, i.e., to use the so-called VAGO (Vector Analys Grid Operators) method. In this paper, we discuss the possibilities of such an approach in using general irregular grids. The vector analysis difference operators are constructed using the Delaunay triangulation and the Voronoi diagrams. The truncation error and the consistency property of the difference operators constructed on two types of grids are investigated. Construction and analysis of the difference schemes of the VAGO method for applied problems are illustrated by the examples of stationary and non-stationary convection-diffusion problems. The other examples concerned the solution of the nonstationary vector problems described by the second-order equations or the systems of first-order equations.2000 Mathematics Subject Classification: 65N06; 65M06.

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Section: Boundary Value Problem For the Stationary Diffusion Equationmentioning

confidence: 99%

Section: Boundary Value Problem For the Stationary Diffusion Equationmentioning

confidence: 99%

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Finite-difference Approximation of Mathematical Physics Problems on Irregular Grids

Vabishchevich¹

2005

Comput. Methods Appl. Math.

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“…This anomaly is due to the fact that the corresponding dependent quantity Q j (·,·,·,n) is defined on a space-time domain which spans a time step going from the time instant (n − . The FIT method [7] adopts a set of discrete constitutive operators similar to those of the FDTD method, whereas some generalization of the FDTD method, such as the the Discrete Surface Integral method (DSI) [4], adopt a more complex discrete representation of constitutive equations, while maintaining the topological time-stepping of FDTD. This follows from the fact that, contrary to the case of topological equations, many possible approaches to the discretization of constitutive equations are possible.…”

Section: The Missing Linkmentioning

confidence: 99%

The Geometry of Time-Stepping

Mattiussi¹

2001

PIER

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Abstract-The space-time geometric structure of Maxwell's equations is examined and a subset of them is found to define a pair of exact discrete time-stepping relations. The desirability of adopting an approach to the discretization of electromagnetic problems which exploits this fact is advocated, and the name topological time-stepping for numerical schemes complying with it is suggested. The analysis of the equations leading to this kind of time-stepping reveals that these equations are naturally written in terms of integrated field quantities associated with space-time domains. It is therefore suggested that these quantities be adopted as state variables within numerical methods. A list of supplementary prescriptions for a discretization of electromagnetic problems suiting this philosophy is given, with particular emphasis on the necessity to adopt a space-time approach in each discretization step. It is shown that some existing methods already comply with these tenets, but that this fact is not explicitly recognized and exploited. The role of the constitutive equations in this discretization philosophy is briefly analyzed. The extension of this approach to more general kinds of space-time meshes, to other sets of basic time-stepping equations and to other field theories is finally considered.

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“…This approximation can lead to inaccurate solutions and has prompted the development of a wide range of schemes designed to reduce or eliminate staircasing errors (e.g., [2]- [6]). Unfortunately, these alternative approaches have costs, both in terms of computational overhead and algorithm complexity, above that of the original FDTD scheme.…”

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confidence: 99%

FDTD simulations of TEM horns and the implications for staircased representations

Schneider

Shlager

1997

IEEE Trans. Antennas Propagat.

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Abstract-Two-dimensional (2-D) TEM horns are modeled using the finite-difference time-domain (FDTD) method. The boundary walls are perfect electric conductors and one wall, which does not align with the Cartesian grid, is approximated using a staircased representation. By carefully comparing the FDTD results to those of the analytic solution, one can make conclusions about the coarseness with which a boundary can be represented. It is found that staircasing errors are small when the staircase diagonal (the hypotenuse of the right triangle created by the stairstep) is smaller than half a wavelength at the highest significant frequency in the excitation. This rule-of-thumb is put forward as a necessary condition for the discretization of general problems. Results are also provided for some simple FDTD schemes that are designed to reduce staircasing errors. By using large aspect-ratio cells, a grid can be constructed that satisfies the rule-of-thumb given above. While this approach eliminates general staircasing errors, some errors persist owing to the presence of step discontinuities immediately adjacent to the horn feed. These errors can be further reduced by using a cell-splitting approach. It is shown that the contour path FDTD technique can be used to eliminate nearly all staircasing errors, while some additional improvement is shown to be provided by using a stabilized contour path FDTD approach. Finally, a recently proposed conformal technique that permits simple implementation is shown to provide results comparable with those of the stabilized contour path approach.Index Terms-FDTD methods, horn antennas.

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Divergence Preserving Discrete Surface Integral Methods for Maxwell's Curl Equations Using Non-orthogonal Unstructured Grids

Cited by 123 publications

References 12 publications

Finite-difference Approximation of Mathematical Physics Problems on Irregular Grids

Finite-difference Approximation of Mathematical Physics Problems on Irregular Grids

The Geometry of Time-Stepping

FDTD simulations of TEM horns and the implications for staircased representations

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