Investigation of numerical dispersion with time step of the FDTD methods: avoiding erroneous conclusions

Abstract

To obtain high accuracy and efficiency in the simulations, an optimum time step should be taken in finite-difference time-domain (FDTD) methods. The authors investigated how time steps impact on numerical dispersion of two FDTD methods including the FDTD(2,2) method and the FDTD(2,4) method. Through rigorously analytical and numerical analysis, it is found that small time steps of the FDTD methods do not always have small numerical errors. Our findings reveal that these two FDTD methods present different behaviours with respect to time steps: (1) for the FDTD(2,2) method, smaller time steps limited by the Courant-Friedrichs-Lewy condition increase numerical dispersion and lead to larger simulation errors and (2) for the FDTD(2,4) method, as the time step increases, numerical dispersion errors first decrease and then increase. Our findings are also comprehensively validated from one- to three-dimensional cases through several numerical examples including wave propagation, resonant frequencies of cavities and a practical engineering problem.