Crank-Nicolson FDTD Method in Media Described by Time-Fractional Constitutive Relations

In this contribution, we present the Crank-Nicolson finite-difference time-domain (CN-FDTD) method, implemented for simulations of wave propagation in media described by time-fractional (TF) constitutive relations. That is, the considered constitutive relations involve fractional-order (FO) derivatives based on the Grunwald-Letnikov definition, allowing for description of hereditary properties and memory effects of media and processes. Therefore, the TF constitutive relations make it possible to include, in a dielectric response, diffusion processes which are modelled mathematically by the diffusion-wave equation. We formulate fundamental equations of the proposed CN-FDTD method, and then we execute simulations which confirm its accuracy and applicability. Additionally, we perform numerical tests of stability, which confirm unconditional stability of the method. The proposed method is useful for researchers investigating numerical techniques in media described by FO derivatives.