A Comprehensive New Methodology for Formulating FDTD Schemes With Controlled Order of Accuracy and Dispersion

Numerical dispersion errors in the wave-equation-finite-difference-time-domain (WE-FDTD) method have been treated by higher order schemes, coefficient modification schemes, dispersion relation preserving and non-standard schemes. In this work, a unified methodology is formulated for the systematic generation of WE-FDTD schemes tailored to the spectrum of the excitation. The methodology enables the scheme designer to gradually trade order of accuracy (OoA) for lower dispersion errors in a controlled manner at the cost of sacrificing low frequency behavior, that is not deemed critical for this type of excitation. The methodology is shown to encompass both existing and new schemes. Stability analysis is carried out concurrently with the generation of each scheme. Using a stencil size of 3 and 5 temporal and spatial samples, respectively, long term errors of a scheme designed for a specific pulse are compared with the standard (4,4) scheme that has the same computational complexity, via simulation of a modulated pulse that propagates over a million time steps.