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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.20231 |
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| _version_ | 1866917166236827648 |
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| author | Yin, Baoli Zhang, Guoyu Liu, Yang Li, Hong |
| author_facet | Yin, Baoli Zhang, Guoyu Liu, Yang Li, Hong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_20231 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Pick function approach for designing energy-decay preserving schemes of the Maxwell equations in Havriliak-Negami dispersive media Yin, Baoli Zhang, Guoyu Liu, Yang Li, Hong Numerical Analysis 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. |
| title | A Pick function approach for designing energy-decay preserving schemes of the Maxwell equations in Havriliak-Negami dispersive media |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2512.20231 |