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Treatment of Complex Interfaces for Maxwell’s Equations with Continuous Coefficients Using the Correction Function Method

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Abstract

We propose a high-order FDTD scheme based on the correction function method (CFM) to treat interfaces with complex geometry without significantly increasing the complexity of the numerical approach for constant coefficients. Correction functions are modeled by a system of PDEs based on Maxwell’s equations with interface conditions. To be able to compute approximations of correction functions, a functional that is a square measure of the error associated with the correction functions’ system of PDEs is minimized in a divergence-free discrete functional space. Afterward, approximations of correction functions are used to correct a FDTD scheme in the vicinity of an interface where it is needed. We perform a perturbation analysis on the correction functions’ system of PDEs. The discrete divergence constraint and the consistency of resulting schemes are studied. Numerical experiments are performed for problems with different geometries of the interface. A second-order convergence is obtained for a second-order FDTD scheme corrected using the CFM. High-order convergence is obtained with a corrected fourth-order FDTD scheme. The discontinuities within solutions are accurately captured without spurious oscillations.

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Acknowledgements

The authors are grateful to Professor Charles Audet for interesting and helpful conversations. The research of Professor Jean-Christophe Nave was partially supported by the NSERC Discovery Program.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, McGill University, Montreal, QC, H3A 0B9, Canada

    Yann-Meing Law & Jean-Christophe Nave

  2. Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA, 02139-4307, USA

    Alexandre Noll Marques

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  1. Yann-Meing Law
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  2. Alexandre Noll Marques
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  3. Jean-Christophe Nave
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Law, YM., Marques, A.N. & Nave, JC. Treatment of Complex Interfaces for Maxwell’s Equations with Continuous Coefficients Using the Correction Function Method. J Sci Comput 82, 56 (2020). https://doi.org/10.1007/s10915-020-01148-6

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  • DOI: https://doi.org/10.1007/s10915-020-01148-6

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