Iranian Journal of Electrical and Electronic Engineering (Jun 2022)
CFL Stability Conditions for the FDTD Method in Bounded Inhomogeneous Anisotropic Media
Abstract
Anisotropic media appear regularly in electromagnetic wave engineering. The finite-difference time-domain (FDTD) method is a robust technique to model such media. However, the value of the time step in the FDTD algorithm is bounded by the Courant-Friedrichs-Lewy (CFL) condition. In this paper, a simple analytical approach is developed using the Gershgorin circle theorem to derive a point-wise closed-form relation for the CFL condition in bounded inhomogeneous anisotropic media. The proposed technique includes objects of arbitrary shapes with straight, tilted, or curved interfaces located in a computational space with uniform or adaptive gridding schemes. Both axial and non-axial anisotropies are considered in the analysis. The proposed method is able to investigate the effect of boundaries and interfaces on the stability of the algorithm. It is shown that in the presence of an interface between two anisotropic media, the von-Neumann criterion is not able to predict the stability bound for specific ranges of the permittivity tensor components and unit cell aspect ratios. Exploiting the proposed closed-form formulations, it is possible to tune the CFL time step and avoid the temporal instability by the wise selection of the gridding scheme especially in curved boundaries where subcell modelings such as Yu-Mittra formalism are applicable. Some illustrative examples are provided to verify the method by comparing the results with those of the eigenvalue analysis and time-domain simulations.
