Some fundamental characteristics of the one-dimensional alternate-direction-implicit finite-difference timedomain method

Some fundamental characteristics are investigated for the alternate-direction-implicit finite-difference time-domain (ADI-FDTD) method in the one-dimensional case, such as growth and dissipation, numerical dispersion, and a time-step size limit. It is shown that this two sub-step method alternates dissipation and growth that exactly compensate and, thus, is unconditionally stable. The numerical dispersion error is larger than for Yeeʹs method and there is an "intrinsic temporal numerical dispersion" accuracy limit at zero mesh size, which is the highest accuracy one can obtain with a meaningful time-step size. Also, it is shown that, for some combinations of time step and mesh size, the ADI-FDTD method does not propagate a wave. There is a minimum numerical velocity limited by the mesh density, and the wave attenuates for time-step sizes larger than an "ADI limit." Thus, the time-step size does have an upper bound, which is smaller than the Nyquist limit. The results of numerical experiments are shown to agree well with the theoretical prediction.