Phase error control for FD-TD methods of second and fourth order accuracy

For FD-TD methods(used to solve Maxwell´s equations) we determine the spatial resolution of the discretized domain in terms of the total computation time and the desired phase error. It is shown that the spatial step should vary as Δx~g[e_φ/t_c] ¹s/ in order to maintain a prescribed phase error level e_φ throughout the computation time t_c, where s (=2 or 4) is the spatial order of accuracy of the scheme and g is a geometric factor. Significantly, we show that the rule of thumb of using 10-20 points per wavelength to determine the spatial cell size for the standard scheme is not optimal. Our results are verified by numerical simulations in two dimensions with the Yee (1966) scheme and a new fourth-order accurate FD-TD scheme