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.
Abstract-We advance the theory of the two-dimensional method of connected local fields (CLF) to the three-dimensional cases. CLF is suitable for obtaining semi-analytical solutions of Helmholtz equation. The fundamental building block (cell) of the 3-D CLF is a cube consisting of a central point and twenty six points on the cube's surface. These surface points form three symmetry groups: six on the planar faces, twelve on the edges and eight on the vertices (corners). The local field within the unit cell is expanded in a truncated spherical FourierBessel series. From this representation we develop a closed-form, 3D local field expansion (LFE) coefficients that relate the central point to its immediate neighbors. We also compute the CLF-based FD-FD numerical solutions of the 3D Green's function in free space. Compared with the analytic solution, we found that even at a low three points per wavelength spatial sampling, the accumulated phase errors of the CLF 3D Green's function after propagating a distance of ten wavelengths are well under ten percent.
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.
Abstract-We present the dispersion and local-error analysis of the twenty-seven point local field expansion (LFE-27) formula for obtaining highly accurate semi-analytical solutions of the Helmholtz equation in a 3D homogeneous medium. Compact finite-difference (FD) stencils are the cornerstones in frequency-domain FD methods. They produce block tri-diagonal matrices which require much less computing resources compared to other non-compact stencils. LFE-27 is a 3D compact FD-like stencil used in the method of connected local fields (CLF) [1]. In this paper, we show that LFE-27 possesses such good numerical quality that it is accurate to the sixth order. Our analyses are based on the relative error studies of numerical phase and group velocities. The classical second-order FD formula requires more than twenty sampling points per wavelength to achieve less than 1% relative error in both phase and group velocities whereas LFE-27 needs only three points per wavelength to match the same performance.
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