SEMI-ANALYTICAL SOLUTIONS OF 2-D HOMOGENEOUS HELMHOLTZ EQUATION BY THE METHOD OF CONNECTED LOCAL FIELDS

The frequency-domain finite-difference (FD-FD) methods have been successfully used to obtain numerical solutions of two-dimensional (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 semi-analytical solution of 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 are related to total eight points, four on the sides and four on the corners, on the square patch. The local field expansion (LFE) analysis leads to an improved, compact FD-like, nine-point stencil for the 2-D homogeneous Helmholtz equation. We show that LFE formulation possesses superior numerical properties of being low dispersive and nearly isotropic because this method of connecting local fields merely ties these overlapping EM field patches already satisfy the Helmholtz equation.