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

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 $δ(ζ)$ cannot satisfy both that \(-δ(ζ)\) 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.