Numerical Stability of FDTD Algorithms in Gyrotropic Media Analysed through Their Dispersion Relations

A rigorous and unified numerical stability analysis of FDTD algorithms in gyrotropic regions is launched. Off-diagonal terms of permittivity or permeability tensors lead to the coupling between two orthogonal E-or H-field components, respectively. To avoid the need for large matrix inversion, several explicit and implicit ways of decoupling the system have been proposed in the literature, and are investigated herein, together with a new asymmetrical semi-implicit approach. To accentuate the effect of component coupling, as opposed to the dispersion effects, a unified model of gyrotropic media is proposed reflecting the worse case dispersion asymptotic behaviour of magnetised ferrites and magnetised plasma. It is shown that explicit and, counter-intuitively, also implicit FDTD schemes are inherently unstable. It is also shown that the FDTD algorithms are stabilised via either semi-implicit or, again counter-intuitively, higher order implicit approach.