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 (θ)-scheme is proposed for temporal discretization, which is rigorously proven to preserve a discrete energy dissipation property under the condition (θ\in [\fracα{2}, \frac{1}{2}]). The temporal convergence rate of the scheme is shown to be first-order for (θ\neq 0.5) and second-order for (θ= 0.5). Extensive numerical experiments validate the theoretical findings, including convergence tests and energy-decay comparisons. The proposed SFTR-(θ) 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.

💡 Research Summary

This paper presents a comprehensive study on the design and analysis of numerical schemes that preserve the energy-decay property for Maxwell’s equations embedded in a Cole-Cole dispersive medium. The Cole-Cole model describes wave propagation in materials with frequency-dependent permittivity, involving a coupled system of Maxwell’s equations and a fractional differential equation governing the polarization field.

The authors first establish a continuous energy-decay law for the model. By introducing a modified energy functional, E_α(t), which incorporates a fractional integral term, they rigorously prove that the system’s energy monotonically decreases over time (dE_α/dt ≤ 0). This extends and refines previous results that only showed energy stability (non-increase).

The core contribution is the proposal and analysis of a novel temporal discretization scheme, termed the θ-scheme, based on a shifted fractional trapezoidal rule (SFTR) for approximating the Caputo fractional derivative. The fully discrete scheme is derived by applying finite element discretization in space. The key theoretical result is Theorem 3.7, which proves that this semi-discrete θ-scheme preserves a discrete counterpart of the energy-decay law under the condition θ ∈