A second-order space-time accurate scheme for Maxwell’s equations in a Cole–Cole dispersive medium

A fully implicit finite-difference time-domain (FDTD) scheme with second-order space-time accuracy for Maxwell’s equations in a Cole–Cole dispersive medium is proposed and analyzed. The temporal discretization is built upon Crank-Nicolson method, and the Caputo derivative term is based on the recently established L2-1σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}2\text {-}1_{\sigma }$$\end{document} formula and a weighted approach. A rigorous analysis is carried out to show that the proposed scheme is unconditionally stable and has second-order accuracy in both time and space. 2D and 3D numerical examples are presented to validate our theoretical findings.