FDTD coeffcient modification schemes of the wave and Maxwell’s equations for controlling order of accuracy and dispersion errors

Abstract

Recently, a general methodology for generating FDTD dispersion relation preserving (DRP) schemes of the wave equation (WE) was introduced. These schemes can be designed to accommodate arbitrary requirements for phase and group velocity dispersion characteristics over the operating frequency range. In this work, the methodology is further generalized to include specification of both dispersion behavior and the orders of accuracy (OoAs) by minimizing the local truncation error [4, Ch. 5.1] for both the WE and Maxwell's equations (MEs). While the increase in OoA leads, in general, to a lower overall dispersion error, additional improvements of dispersion characteristics at certain frequencies can be accomplished at the expense of reduced OoA. This tradeoff affords flexibility in designing an FDTD algorithm for a wide range of applications. The OoA for a given stencil size is analyzed in this work by a Taylor expansion of the dispersion relationship in powers of the normalized frequency. This equation serves as a starting point for the methodology for the analysis and control of the OoA and dispersion errors, that encompasses both the customary analysis with Taylor expansion of the field in terms of At and Ax and the derivative-swapping approach.