An Updated Review of General Dispersion Relation for Conditionally and Unconditionally Stable FDTD Algorithms

Abstract

We present a general numerical dispersion equation which is applicable to all known conditionally and unconditionally stable finite-difference time-domain (FDTD) algorithms on staggered rectangular grid, including Yee's FDTD, wavelet-based FDTD, extended curl FDTD, alternating direction implicit (ADI)-FDTD, Crank-Nicolson (CN)-FDTD, Crank-Nicolson split-step (CNSS)-FDTD, and their modifications of higher order spatial stencils. The real part of the complex eigenvalue of the total amplification matrix defines and distinguishes the dispersion relation for each individual scheme. Easy-to-check conditions are provided, under which the numerical dispersion of a particular time-domain scheme is governed by the proposed dispersion equation. These conditions are on the amplification matrix eigenvalues. The proposed dispersion equation includes each considered dispersion relation as a special case, and presents itself a general governing equation to estimate 3-D numerical dispersion of the aforementioned schemes in the frame of plane waves.