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We propose a high-order FDTD scheme based on the correction function method(CFM) to treat interfaces with complex geometry without increasing thecomplexity of the numerical approach for constant coefficients. Correctionfunctions are modeled by a system of PDEs based on Maxwell's equations withinterface conditions. To be able to compute approximations of correctionfunctions, a functional that is a square measure of the error associated withthe correction functions' system of PDEs is minimized in a divergence-freediscrete functional space. Afterward, approximations of correction functionsare used to correct a FDTD scheme in the vicinity of an interface where it isneeded. We perform a perturbation analysis on the correction functions' systemof PDEs. The discrete divergence constraint and the consistency of resultingschemes are studied. Numerical experiments are performed for problems withdifferent geometries of the interface. A second-order convergence is obtainedfor a second-order FDTD scheme corrected using the CFM. High-order convergenceis obtained with a corrected fourth-order FDTD scheme. The discontinuitieswithin solutions are accurately captured without spurious oscillations.

Treatment of complex interfaces for Maxwell's equations with continuous coefficients using the correction function method