Three-dimensional dispersion-relation-preserving finite-difference time-domain schemes

We describe a methodology to develop dispersion-relation-preserving (DRP) schemes for large-scale 3D FDTD simulations. By first expanding the numerical dispersion error in spherical harmonics and properly choosing the FDTD coefficients to cancel the leading terms of the series, the dispersion error can be minimized in a minimax sense for all propagation angles. Time-domain DRP schemes are obtained by expanding the resulting frequency dependent coefficients in a Taylor series. By further employing filtering schemes, the dispersion characteristics can be adjusted to yield a smaller local dispersion error at high frequencies than at low frequencies so that a small accumulated phase error (up to a pre-assigned frequency) is produced.