Theoretical Foundation for the Method of Connected Local Fields

Abstract

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.

Dispersion and Local-Error Analysis of Compact Lfe-27 Formula for Obtaining Sixth-Order Accurate Numerical Solutions of 3d Helmholtz Equation

“…In Section 4.2 we will apply the first-order analysis of the B-V relation for the LFE-27 stencil as we did previously for the 2D LFE-9 case [12]. We can only perform this type of analysis with highly accurate compact stencils or when B and V are both small.…”

“…To analyze the numerical dispersion characteristics of Equation (1) as we have done in previous 2D LFE-9 stencil [12], we consider the plane wave solution defined as:…”

“…Numerical dispersion analyses of compact formulae on discretizing Helmholtz equation in uniform regions can be found in references [12,[18][19][20][21][22][23]. In particular, thorough dispersion research on classi-cal FD formulae for 1-D Helmholtz equation were given in Harari and Turkel's work in 1995 [18].…”

“…The works of Singer-Turkel and Sutmann are based on Taylor series expansion. Both Hadley [9,10] and Chang-Mu [11,12] used Fourier-Bessel Series as the basis function for the local field in deriving a 2D compact FD-like stencil which we refer to as LFE2D-9. Along these lines, Tsukerman [13] proposed other choices of basis functions for 2D Helmholtz equation including plane waves and harmonic polynomials.…”

Semi-Analytical Solutions of the 3-D Homogeneous Helmholtz Equation by the Method of Connected Local Fields

“…Recent advancements in highly accurate 2D FDFD algorithms [5][6][7][8][9] have overcome dispersion problems of the classical methods. Along these lines in 2010 Chang and Mu published work on the method of connected local fields (CLF, [10,11]). The twodimensional CLF method provides alternative semi-analytical solutions to the 2D Helmholtz equation.…”