High-accuracy finite-difference equations for dielectric waveguide analysis I: uniform regions and dielectric interfaces
Citations
181 citations
Cites background from "High-accuracy finite-difference equ..."
...The importance of elliptic interface problems has been well recognized in a variety of disciplines, such as fluid dynamics [16,19,29,47], electromagnetics [23,24] and material science [26]....
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129 citations
Cites background from "High-accuracy finite-difference equ..."
...This class of problems have demonstrated a growing impact to fluid dynamics [14,18,19,30,53], electromagnetics [23,24,33], material science [25,27] and biologic systems [20,44,70,75]....
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129 citations
Cites background from "High-accuracy finite-difference equ..."
...This problem is commonly referred to as an elliptic interface problem and occurs widely in practical applications, such as fluid mechanics [23], electromagnetic wave propagation [14,18,41,40], materials science [20,22], and biological science [39,12,7]....
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91 citations
Cites background or methods from "High-accuracy finite-difference equ..."
...This relaxation of zoning symmetry will require a new derivation for uniform and interface points [ 1 ], with a concomitant loss of accuracy, but will be found to be much easier than performing a new derivation for corner points with nonuniform grid sizes....
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...(Uniform, horizontal and vertical interface points may be treated using the formalism described in the previous article [ 1 ])....
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...8‐11 indicates second-order truncation errors instead of the third-order that might be expected from the order of the finite-difference equation at the corner (first) and the “dilution” behavior reported in the previous article [ 1 ]....
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...This equation is ostensibly first-order accurate, and when combined with highly accurate difference equations in the interior regions and at simple planar boundaries [ 1 ], has resulted in a highly accurate full-vector waveguide eigenmode solver whose truncation error ranges from second to third order....
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...Second, geometrical effects dilute the effect of the corner-generated error on the total error in a global quantity such as modal index by two orders [ 1 ]....
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89 citations
Cites background from "High-accuracy finite-difference equ..."
...Elliptic interface problems with discontinuous coefficients and singular sources are found in many applications, such as fluid mechanics [31], computational electromagnetics [18,23,56,55], materials science [25,29], and biological science [54,15,8]....
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References
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"High-accuracy finite-difference equ..." refers background in this paper
...where is the Gamma function and is the hypergeometric function [ 9 ] defined by...
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...Using (A2) and the series definition for the hypergeometric function [ 9 ], we may evaluate these quantities as follows:...
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27,354 citations
"High-accuracy finite-difference equ..." refers background in this paper
...The previous four relations may be solved for the ’s and ’s in terms of the ’s and ’s by utilizing integral properties of the Bessel functions [ 8 ]....
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...The integrals in their most general form are [ 8 ]...
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"High-accuracy finite-difference equ..." refers methods in this paper
...These include, but are not limited to, the imaginary distance beam propagation method [1], the finite-element method [ 2 ], the method of lines [3], and the finite-difference method [4]....
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