Testing Stability of FDTD in Media Described by Time-Fractional Constitutive Relations

We present the procedure that allows one to establish the maximum time-step size guaranteeing stable finite-difference time-domain (FDTD) simulations in media described by time-fractional constitutive relations. The considered relations involve fractional-order (FO) derivatives based on the GrünwaldLetnikov definition, which describes hereditary properties and memory effects of media and processes. Therefore, the current values of the electromagnetic field in the FDTD iterative procedure depend on all the previous ones, which is a new type of computational scheme for this method. In this contribution, we focus on the derivation of a new stability condition for such an FDTD computational scheme.We formulate fundamental equations of the proposed FDTD method and, then, derive the stability condition. In the next step, we analyse the properties of this condition on the complex plane based on the characteristic equation of the method. Our results are useful for researchers investigating numerical techniques for modeling electromagnetic processes described by the diffusion-wave equation and FO derivatives.