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

Abstract

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.

Theoretical Foundation for the Method of Connected Local Fields

“…[1] that this LFE-9 equation leads to an improved compact nine-point FD-FD stencil for the 2-D homogeneous Helmholtz equation. It would help advance all existing FD-FD methods by reducing much needed computational resources without sacrificing accuracy.…”

“…[1] upon which we shall investigate their error characteristics. The theory of CLF for 1-D Helmholtz equation is stated as followed: A point u 0 is related to its two neighbors u −1 , u 1 , by:…”

“…This parameter behaves much like the FD parameter N λ = λ/∆, the sampling density of Equation (52) of Ref. [1]. We can say that when N psd > 2, all the traveling waves within the surface S are captured by this set of discrete points on the perimeter of the area S. When all sampled points on contours C1 or C2 are applied we can compute N psd without any ambiguity.…”

“…[1] we present numerical dispersion analysis for LFE-5 and LFE-9 formulation. In this section, we provide a closed-form, first-order error analysis for the method of connected local fields.…”

“…Recently we presented a method called connected local fields (CLF) [1][2][3][4] which was developed for obtaining numerical solutions of the Helmholtz equation in a homogeneous, and ultimately, an inhomogeneous medium. The method started with a Fourier-Bessel series expansion of a local field defined over a square patch consisting of nine points arranged as shown in Figure 1.…”