Higher Stability Limits for the Symplectic FDTD Method by Making Use of Chebyshev Polynomials

A new scheme is introduced for obtaining higher stability performance for the symplectic finite-difference time-domain (FDTD) method. Both the stability limit and the numerical dispersion of the symplectic FDTD are determined by a function zeta. It is shown that when the zeta function is a Chebyshev polynomial the stability limit is linearly proportional to the number of the exponential operators. Thus, the stability limit can be increased as much as possible at the cost of increased number of operators. For example, the stability limit of the four-exponential operator scheme is 0.989 and of the eight-exponential operator scheme it is 1.979 for fourth-order space discretization in three dimensions, which is almost three times the stability limit of previously published symplectic FDTD schemes with a similar number of operators. This study also shows that the numerical dispersion errors for this new scheme are less than those of the previously reported symplectic FDTD schemes     

An Unconditionally Stable Split-Step FDTD Method for Low Anisotropy

Split-step unconditionally stable finite-difference time-domain (FDTD) methods have higher dispersion and anisotropic errors for large stability factors. A new split-step method with four sub-steps is introduced and shown to have much lower anisotropy compared with the well known alternating direction implicit finite-difference time-domain (ADI-FDTD) and other known split step methods. Another important aspect of the new method is that for each space step value there is a stability factor value that the numerical propagation phase velocity is isotropic.     

Optimized exponential operator coefficients for symplectic FDTD method

A new symplectic finite difference time domain scheme is introduced. The scheme uses fourth-order finite differencing for space and a symplectic scheme with a propagator of exponential differential operators for time. The coefficients of the exponential operators are obtained by optimizing the higher order terms of the growth factor for high Courant stability limit as well as by using the Taylor's series expansion of the exponential operator for up to the second-order term. When the Taylor's series expansion of the exponential operator is considered the new scheme is second-order in time, but the dispersion performance of the scheme is similar to the performance of the fourth-order symplectic schemes previously reported. The stability performance is shown to be better, and as the new scheme uses smaller number of exponential operators it also reduces the computational time. One other advantage of this scheme is that it is flexible in the choice of the coefficients, which allows the coefficients to be chosen according to performance requirements.