A Fourier penalty method for solving the time-dependent Maxwell's equations in domains with curved boundaries

TL;DR

This paper introduces a high-order Fourier penalty method for solving time-dependent Maxwell's equations near curved boundaries, enabling accurate enforcement of boundary conditions and high convergence rates in complex geometries.

Contribution

The paper develops a novel Fourier penalty approach that systematically enforces boundary conditions with high order accuracy, suitable for spectral methods in Maxwell's equations.

Findings

Achieves convergence orders up to 3.5 in 1D

Demonstrates no dispersion errors in the numerical method

Successfully applied to complex 2D and 3D geometries

Abstract

We present a high order, Fourier penalty method for the Maxwell's equations in the vicinity of perfect electric conductor boundary conditions. The approach relies on extending the smooth non-periodic domain of the equations to a periodic domain by removing the exact boundary conditions and introducing an analytic forcing term in the extended domain. The forcing, or penalty term is chosen to systematically enforce the boundary conditions to high order in the penalty parameter, which then allows for higher order numerical methods. We present an efficient numerical method for constructing the penalty term, and discretize the resulting equations using a Fourier spectral method. We demonstrate convergence orders of up to 3.5 for the one-dimensional Maxwell's equations, and show that the numerical method does not suffer from dispersion (or pollution) errors. We also illustrate the approach in two dimensions and demonstrate convergence orders of 2.5 for transverse magnetic modes and 1.5 for the transverse electric modes. We conclude the paper with numerous test cases in dimensions two and three including waves traveling in an irregular waveguide, and scattering off of a windmill-like geometry.