FDTD Method for Electromagnetic Simulations in Media Described by Time-Fractional Constitutive Relations

In this paper, the finite-difference time-domain (FDTD) method is derived for electromagnetic simulations in media described by the time-fractional (TF) constitutive relations. TF Maxwell's equations are derived based on these constitutive relations and the Grunwald-Letnikov definition of a fractional derivative. Then the FDTD algorithm, which includes memory effects and energy dissipation of the considered media, is introduced. Finally, one-dimensional signal propagation in such electromagnetic media is considered. The proposed FDTD method is derived based on a discrete approximation of the Grunwald-Letnikov definition of the fractional derivative and evaluated in a code. The stability condition is derived for the proposed FDTD method based on a numerical-dispersion relation. The obtained numerical results are compared with the outcomes of reference frequency-domain simulations, proving the accuracy of the proposed approach. However, high spatial resolution is required in order to obtain accurate results. The developed FDTD method is, unfortunately, computation and memory demanding when compared to the ordinary FDTD algorithm.

Automated summary

This paper investigates the electromagnetic simulation method in media described by time-fractional constitutive relations. A discrete approximation based on the finite-difference time-domain (FDTD) method is proposed and its accuracy is validated through comparison with frequency-domain simulations. However, high spatial resolution and computationally demanding memory are required for accurate results.