Analysis of discrete energy-decay preserving schemes for Maxwell's equations in Cole-Cole dispersive medium

This work investigates the design and analysis of energy-decay preserving numerical schemes for Maxwell’s equations in a Cole-Cole (C-C) dispersive medium. A continuous energy-decay law is first established for the C-C model through a modified energy functional. Subsequently, a novel $\theta$-scheme is proposed for temporal discretization, which is rigorously proven to preserve a discrete energy dissipation property under the condition $\theta \in [\frac{\alpha }{2},\frac{1}{2}]$. The temporal convergence rate of the scheme is shown to be first-order for $\theta \mathrm{\ne }0.5$ and second-order for $\theta =0.5$. Extensive numerical experiments validate the theoretical findings, including convergence tests and energy-decay comparisons. The proposed SFTR-$\theta$ scheme demonstrates superior performance in maintaining monotonic energy decay compared to an alternative 2nd-order fractional backward difference formula, particularly in long-time simulations, highlighting its robustness and physical fidelity.