A Pick function approach for designing energy-decay preserving schemes of the Maxwell equations in Havriliak-Negami dispersive media

TL;DR

This paper introduces a novel high-order energy-decaying scheme for Maxwell's equations in dispersive media using Pick functions, overcoming limitations of traditional convolution quadrature methods and ensuring unconditional stability.

Contribution

It develops a new approach based on Pick functions to construct energy-decay preserving schemes of higher order for Maxwell's equations in complex media.

Findings

Successfully constructs a second-order completely monotonic sequence.

The proposed scheme guarantees unconditional energy decay and stability.

Numerical experiments confirm convergence and energy dissipation properties.

Abstract

This work proposes a novel approach for designing high-order energy-decaying schemes for Maxwell's equations in Havriliak-Negami dispersive media. It is shown that conventional convolution quadrature (CQ) methods, which rely directly on the generating function of linear multistep methods, cannot generate completely monotonic sequences beyond first-order accuracy. We rigorously prove that for any linear multistep method of second-or higher-order, the associated generating function $\delta(\zeta)$ cannot satisfy both that \(-\delta(\zeta)\) is a Pick function and that it is analytic on \((-\infty,1)\) - a key requirement for constructing completely monotonic sequences. To overcome this fundamental limitation, we introduce a reconstruction of the generating function's structure. By strategically incorporating the theory of Pick functions, we successfully construct a second-order completely monotonic sequence. This theoretical advance leads to a discrete scheme that inherits the continuous model's energy decay property, guaranteeing unconditional stability. Numerical experiments confirm the convergence rates and energy dissipation behavior of the proposed method.