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High-Order FDTD Schemes for Maxwell’s Interface Problems with Discontinuous Coefficients and Complex Interfaces Based on the Correction Function Method

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Abstract

We propose high-order FDTD schemes based on the Correction Function Method (CFM) (Marques et al. in J Comput Phys 230:7567–7597, 2011) for Maxwell’s interface problems with discontinuous coefficients and complex interfaces. The key idea of the CFM is to model the correction function near an interface to retain the order of a finite difference approximation. To do so, we solve a system of PDEs based on the original problem by minimizing an energy functional. The CFM is applied to the standard Yee scheme and a fourth-order FDTD scheme. The proposed CFM-FDTD schemes are verified in 2-D using the transverse magnetic (\(\hbox {TM}_z\)) mode. Numerical examples include scattering of magnetic and non-magnetic dielectrics, and problems with manufactured solutions using various complex interfaces and discontinuous piecewise varying coefficients. Long-time simulations are also performed to investigate the stability of CFM-FDTD schemes. The proposed CFM-FDTD schemes achieve up to fourth-order convergence in \(L^2\)-norm and provide approximations devoid of spurious oscillations.

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Acknowledgements

The authors are grateful to Alexis Montoison of Polytechnique Montréal for his help on the Julia programming language. The authors also thank Dr. Jessica Lin and Dr. Gantumur Tsogtgerel of McGill University for their support.

Funding

The research of JCN was partially supported by the NSERC Discovery Program.

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

    Yann-Meing Law & Jean-Christophe Nave

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  1. Yann-Meing Law
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  2. Jean-Christophe Nave
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Correspondence to Yann-Meing Law.

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Law, YM., Nave, JC. High-Order FDTD Schemes for Maxwell’s Interface Problems with Discontinuous Coefficients and Complex Interfaces Based on the Correction Function Method. J Sci Comput 91, 26 (2022). https://doi.org/10.1007/s10915-022-01797-9

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  • DOI: https://doi.org/10.1007/s10915-022-01797-9

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