Semi-analytical Solutions of the 2-d Homo- Geneous 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 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.