DISPERSION AND LOCAL-ERROR ANALYSIS OF COMPACT LFE-27 FORMULA FOR OBTAINING SIXTH-ORDER ACCURATE NUMERICAL SOLUTIONS OF 3D HELMHOLTZ EQUATION

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.