A new 9-point sixth-order accurate compact finite-difference method for the Helmholtz equation

Nabavi, Majid; Siddiqui, Muhammad Haroon; Dargahi, Javad

doi:10.1016/j.jsv.2007.06.070

Cited by 87 publications

(70 citation statements)

References 7 publications

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“…This novel compact 9-point FD-FD stencil [17] provides fairly accurate results with low sampling densities of three to four points per wavelength.…”

Section: Normalized Discrete Fd Helmholtz Operatorsmentioning

confidence: 99%

Theoretical Foundation for the Method of Connected Local Fields

Mu¹,

Chang²

2011

PIER

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Abstract-The method of connected local fields (CLF), developed for computing numerical solutions of the two-dimensional (2-D) Helmholtz equation, is capable of advancing existing frequency-domain finitedifference (FD-FD) methods by reducing the spatial sampling density nearly to the theoretical limit of two points per wavelength. In this paper, we show that the core theory of CLF is the result of applying the uniqueness theorem to local EM waves. Furthermore, the mathematical process for computing the local field expansion (LFE) coefficients from eight adjacent points on a square is similar to that in the theory of discrete Fourier transform. We also present a theoretical analysis of both the local and global errors in the theory of connected local fields and provide closed-form expressions for these errors.

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“…This novel compact 9-point FD-FD stencil [17] provides fairly accurate results with low sampling densities of three to four points per wavelength.…”

Section: Normalized Discrete Fd Helmholtz Operatorsmentioning

confidence: 99%

Theoretical Foundation for the Method of Connected Local Fields

Mu¹,

Chang²

2011

PIER

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“…As a result, most existing FD-FD applications adopt the FD2-5 formulation. Currently, sixth-order accurate compact 9-point FD methods [15][16][17] are the most advanced FD formulations for the 2-D inhomogeneous Helmholtz equation given below:…”

Section: Classical Second-order Accurate 2d Fd-fd Formulationmentioning

confidence: 99%

“…Under normal circumstances, the standard second-order FD-FD method requires at least 10 points per wavelength to discretize the Helmholtz equation in order to minimize unwanted numerical dispersion. Currently, high-order finite difference formulae are being developed [13][14][15][16][17] to reduce FD discretization density. First, a new approach based on 2-D scalar wave extrapolation leads to ninepoint coefficients [13] for the FD-FD method.…”

Section: Introductionmentioning

confidence: 99%

“…Immediately, we see that numerical anisotropy will be reduced with this new nine-point scheme covering eight propagating directions instead of the previous four. Furthermore, instead of using fixed coefficients for the Laplace operator, we can adjust and fine-tune the nine-point stencil for each given frequency to reduce temporal dispersion To lower the sampling density at high frequencies, there are also approaches based on fourth and sixth-order approximation of the 2D Helmholtz with an inhomogeneous source [14][15][16][17]. The advantages of these methods are that the frequencydependent FD stencils are compact i.e., they do not involve points other than four side points and four corner points.…”

Section: Introductionmentioning

confidence: 99%

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Semi-Analytical Solutions of 2-D Homogeneous Helmholtz Equation by the Method of Connected Local Fields

Chang¹,

Mu²

2010

PIER

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Abstract-The frequency-domain finite-difference (FD-FD) methods have been successfully used to obtain numerical solutions of the twodimensional (2-D) Helmholtz equation. The standard second-order accurate FD-FD scheme is known to produce unwanted numerical spatial and temporal dispersions when the sampling is inadequate. Recently compact higher-order accurate FD-FD methods have been proposed to reduce the spatial sampling density. We present a semianalytical solution of the 2-D homogeneous Helmholtz equation by connecting overlapping square patches of local fields where each patch is expanded in a set of Fourier-Bessel (FB) series. These local FB coefficients correspond to a total of eight points, four on the sides and four on the corners of the square patch. The local field expansion (LFE) analysis leads to an improved compact nine-point FD-FD stencil for the 2-D homogeneous Helmholtz equation. We show that LFE formulation possesses superior numerical properties of being less dispersive and nearly isotropic because this method of connecting local fields merely ties these overlapping EM field patches which already satisfy the Helmholtz equation.

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“…Equation (1) has been solved by different techniques such as finitedifference method (FDM) [10], fast Fourier transform-based (FFT) methods [11], finite element method (FEM) [12], the spectral-element method [13], compact finite-difference method [14] and multigrid methods [15]. Multigrid method based on HOC schemes is among the most efficient iterative technique for solving PDEs [16,17].…”

Section: Introductionmentioning

confidence: 99%