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

TL;DR

This paper develops and analyzes energy-decay preserving numerical schemes for Maxwell's equations in Cole-Cole dispersive media, ensuring physical energy decay and demonstrating superior long-term stability through rigorous proofs and numerical validation.

Contribution

It introduces a novel heme that preserves discrete energy decay for Maxwell's equations in Cole-Cole media, with proven stability and convergence properties.

Findings

The heme maintains energy decay in simulations.

Convergence rates are first-order for  and second-order at .

Numerical results confirm theoretical energy decay and stability advantages.

Abstract

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 \(\theta \neq 0.5\) 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.