High-order FDTD schemes for Maxwell's interface problems with discontinuous coefficients and complex interfaces based on the Correction Function Method

TL;DR

This paper introduces high-order finite-difference time-domain schemes based on the Correction Function Method for Maxwell's interface problems with complex interfaces and discontinuous coefficients, achieving high accuracy and stability.

Contribution

It develops a novel CFM-based high-order FDTD approach for Maxwell's equations with complex interfaces, improving accuracy and stability over existing methods.

Findings

Achieves up to fourth-order convergence in $L^2$-norm.

Provides stable long-time simulations.

Eliminates spurious oscillations in numerical solutions.

Abstract

We propose high-order FDTD schemes based on the Correction Function Method (CFM) 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. For this, 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 mode (TM$_z$). Numerical examples include scattering of magnetic and non-magnetic dielectric cylinders, and problems with manufactured solutions using various complex interfaces and discontinuous piecewise varying coefficients. Long-time simulations are also performed to provide numerical evidences of the stability of the proposed numerical approach. The proposed CFM-FDTD schemes achieve up to fourth-order convergence in $L^2$-norm and provide approximations devoid of spurious oscillations.