An accurate spectral method for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media

In this paper, we propose an accurate numerical means built upon a spectral-Galerkin method in spatial discretization and an enriched multi-step spectral-collocation approach in temporal direction, for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media in two-dimensional...

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Main Authors: Huang, Can, Wang, Li-Lian
Other Authors: School of Physical and Mathematical Sciences
Format: Article
Language:English
Published: 2021
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Online Access:https://hdl.handle.net/10356/150429
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Institution: Nanyang Technological University
Language: English
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spelling sg-ntu-dr.10356-1504292021-06-04T08:20:11Z An accurate spectral method for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media Huang, Can Wang, Li-Lian School of Physical and Mathematical Sciences Science::Mathematics Cole-Cole Media Dispersive In this paper, we propose an accurate numerical means built upon a spectral-Galerkin method in spatial discretization and an enriched multi-step spectral-collocation approach in temporal direction, for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media in two-dimensional setting. Our starting point is to derive a new model involving only one unknown field from the original model with three unknown fields: electric, magnetic fields, and the induced electric polarization (described by a global temporal convolution of the electric field). This results in a second-order integral-differential equation with a weakly singular integral kernel expressed by the Mittag-Lefler (ML) function. The most interesting but challenging issue resides in how to efficiently deal with the singularity in time induced by the ML function which is an infinite series of singular power functions with different nature. With this in mind, we introduce a spectral-Galerkin method using Fourier-like basis functions for spatial discretization, leading to a sequence of decoupled temporal integral-differential equations (IDE) with the same weakly singular kernel involving the ML function as the original two-dimensional problem. With a careful study of the regularity of IDE, we incorporate several leading singular terms into the numerical scheme and approximate much regular part of the solution. Then, we solve the IDE by a multi-step well-conditioned collocation scheme together with mapping technique to increase the accuracy and enhance the resolution. We show that such an enriched collocation method is convergent and accurate. Ministry of Education (MOE) The research of author Can Huang is supported by the National Natural Science Foundation of China (no. 11401500, 91630204, 11771363). The research of author Li-Lian Wang is partially supported by Singapore MOE AcRF Tier 1 Grant (RG 15/12) and Singapore MOE AcRF Tier 2 Grants (MOE2017-T2-2-014 and MOE2018-T2-1-059). 2021-06-04T08:20:11Z 2021-06-04T08:20:11Z 2019 Journal Article Huang, C. & Wang, L. (2019). An accurate spectral method for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media. Advances in Computational Mathematics, 45(2), 707-734. https://dx.doi.org/10.1007/s10444-018-9636-2 1019-7168 https://hdl.handle.net/10356/150429 10.1007/s10444-018-9636-2 2-s2.0-85053799357 2 45 707 734 en RG 15/12 MOE2017-T2-2-014 MOE2018-T2-1-059 Advances in Computational Mathematics © 2018 Springer Science Business Media, LLC, part of Springer Nature. All rights reserved.
institution Nanyang Technological University
building NTU Library
continent Asia
country Singapore
Singapore
content_provider NTU Library
collection DR-NTU
language English
topic Science::Mathematics
Cole-Cole Media
Dispersive
spellingShingle Science::Mathematics
Cole-Cole Media
Dispersive
Huang, Can
Wang, Li-Lian
An accurate spectral method for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media
description In this paper, we propose an accurate numerical means built upon a spectral-Galerkin method in spatial discretization and an enriched multi-step spectral-collocation approach in temporal direction, for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media in two-dimensional setting. Our starting point is to derive a new model involving only one unknown field from the original model with three unknown fields: electric, magnetic fields, and the induced electric polarization (described by a global temporal convolution of the electric field). This results in a second-order integral-differential equation with a weakly singular integral kernel expressed by the Mittag-Lefler (ML) function. The most interesting but challenging issue resides in how to efficiently deal with the singularity in time induced by the ML function which is an infinite series of singular power functions with different nature. With this in mind, we introduce a spectral-Galerkin method using Fourier-like basis functions for spatial discretization, leading to a sequence of decoupled temporal integral-differential equations (IDE) with the same weakly singular kernel involving the ML function as the original two-dimensional problem. With a careful study of the regularity of IDE, we incorporate several leading singular terms into the numerical scheme and approximate much regular part of the solution. Then, we solve the IDE by a multi-step well-conditioned collocation scheme together with mapping technique to increase the accuracy and enhance the resolution. We show that such an enriched collocation method is convergent and accurate.
author2 School of Physical and Mathematical Sciences
author_facet School of Physical and Mathematical Sciences
Huang, Can
Wang, Li-Lian
format Article
author Huang, Can
Wang, Li-Lian
author_sort Huang, Can
title An accurate spectral method for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media
title_short An accurate spectral method for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media
title_full An accurate spectral method for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media
title_fullStr An accurate spectral method for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media
title_full_unstemmed An accurate spectral method for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media
title_sort accurate spectral method for the transverse magnetic mode of maxwell equations in cole-cole dispersive media
publishDate 2021
url https://hdl.handle.net/10356/150429
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