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

TL;DR

This paper introduces a high-order FDTD scheme utilizing the correction function method to accurately handle complex interfaces in Maxwell's equations with continuous coefficients, achieving high-order convergence and precise discontinuity capturing.

Contribution

The paper develops a novel high-order FDTD scheme with correction functions for complex interfaces, maintaining simplicity and achieving high accuracy without increasing computational complexity.

Findings

Second-order convergence for corrected second-order FDTD scheme

High-order convergence with corrected fourth-order FDTD scheme

Accurate capture of discontinuities without spurious oscillations

Abstract

We propose a high-order FDTD scheme based on the correction function method (CFM) to treat interfaces with complex geometry without 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.