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

Guoyu Zhang, Ziming Dong, Baoli Yin, Yang Liu, Hong Li

TL;DR

The paper addresses energy-dissipation in Maxwell's equations within a Cole-Cole dispersive medium by formulating a modified continuous energy and proving its decay. It introduces a discrete energy-dissipation preserving SFTR-$\theta$ time-stepping scheme, with convergence guarantees: first-order when $\theta \neq 1/2$ and second-order when $\theta=1/2$. Numerical experiments corroborate monotone energy decay and show superior long-time stability compared to a fractional BDF-2 approach, validating the method's physical fidelity. The work lays a foundation for adaptive time stepping in dispersive media simulations and points to extensions to more complex models such as Havriliak–Negami.

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 \(θ\)-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.

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

TL;DR

The paper addresses energy-dissipation in Maxwell's equations within a Cole-Cole dispersive medium by formulating a modified continuous energy and proving its decay. It introduces a discrete energy-dissipation preserving SFTR- time-stepping scheme, with convergence guarantees: first-order when and second-order when . Numerical experiments corroborate monotone energy decay and show superior long-time stability compared to a fractional BDF-2 approach, validating the method's physical fidelity. The work lays a foundation for adaptive time stepping in dispersive media simulations and points to extensions to more complex models such as Havriliak–Negami.

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 -scheme is proposed for temporal discretization, which is rigorously proven to preserve a discrete energy dissipation property under the condition . The temporal convergence rate of the scheme is shown to be first-order for and second-order for . Extensive numerical experiments validate the theoretical findings, including convergence tests and energy-decay comparisons. The proposed SFTR- 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.

Paper Structure

This paper contains 8 sections, 7 theorems, 56 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Continuous energy dissipation
  3. Discrete energy dissipation
  4. Error Analysis
  5. Numerical tests
  6. C-C model with source terms
  7. C-C model without source terms
  8. Conclusion

Key Result

Lemma 2.1

(Energy dissipation of Debye model of order 1) Let $\alpha=1$ in (b3). There holds where $\mathcal{E}(t):=\tau_0\int_{0}^{t}\|\partial_t \boldsymbol P(s)\|^2\mathrm{d}s +\|\boldsymbol P\|^2 +\epsilon_0(\epsilon_s-\epsilon_\infty)(\epsilon_0\epsilon_\infty\|\boldsymbol E\|^2+\mu_0\|H\|^2)$ and therefore,

Figures (3)

  • Figure 1: Values of $\min_{\theta \in [\frac{\alpha}{2},\frac{1}{2}]}\Theta(x,\alpha,\theta)$ for $(x,\alpha)\in (0,1]\times (0,1)$.
  • Figure 2: Illustration of the discrete energy-decay property of the SFTR-$\theta$ scheme with $h = \sqrt{2}/60$ and $\tau = 0.01$. (a) Energy evolution for $\alpha = 0.5$ with varying $\theta = 0.3, 0.4, 0.5$; (b) Energy evolution for $\theta = 0.5$ with varying $\alpha = 0.1, 0.3, 0.5, 0.7, 0.9$.
  • Figure 3: Comparison of the discrete energy-decay between the SFTR-$\theta$ scheme and the F-BDF-2 method, with parameters $h = \sqrt{2}/60$, $\tau = 0.01$, and $\theta = 0.5$: (a) $\alpha = 0.2$; (b) $\alpha = 0.5$; (c) $\alpha = 0.8$; (d) $\alpha = 0.99$.

Theorems & Definitions (19)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • proof
  • ...and 9 more