Dispersion characteristics of compact 3-D formulae based on local spherical fourier-bessel series

Abstract

Most compact FD-FD stencils for discretizing 3-D homogeneous Helmholtz equation suffers from so-called numerical dispersion due to inadequate spatial sampling of the EM field. To verify the effectiveness of newly-derived compact local spherical Fourier-Bessel series (LSFBS) -based formulae, including local field expansion (LFE) face-centered LFE-FC-7, edge-centered LFE-EC-13, corner-point LFE-CR-9, and LFE3D-27, we investigate dispersion characteristics of those formulae. The classical FD formula (FD2-7) requires that the number of sampling points per wavelength is fifteen or more to reduce the relative error of phase velocity to less than 1% and to reduce the relative error of group velocity to less than 2.5%. However, with the LSFBS-based formula (LFE3D-27) it takes only three points per wavelength to achieve less than 1% the relative phase and group velocity errors along various chosen directions.