Time exponential integrator Fourier pseudospectral methods with high accuracy and multiple conservation laws for three-dimensional Maxwell's equations

TL;DR

This paper introduces a high-accuracy, fully discrete numerical scheme for 3D Maxwell's equations that combines exponential integrators with Fourier pseudospectral methods, ensuring spectral accuracy and multiple conservation laws.

Contribution

It develops a novel scheme integrating time exponential integrators and Fourier pseudospectral methods, achieving spectral accuracy and multiple conservation laws without CFL restrictions.

Findings

Spectral accuracy in space and infinite-order in time.

Conservation of energy, helicity, momentum, and divergence-free fields.

Numerical tests confirm theoretical accuracy and conservation properties.

Abstract

Maxwell equations describe the propagation of electromagnetic waves and are therefore fundamental to understanding many problems encountered in the study of antennas and electromagnetics. The aim of this paper is to propose and analyse an efficient fully discrete scheme for solving three-dimensional Maxwell's equations. This is accomplished by combining time exponential integrator and Fourier pseudospectral methods. Fast computation is implemented in the scheme by using the Fast Fourier Transform algorithm which is well known in scientific computations. An optimal error estimate which is not encumbered by the CFL condition is established and the resulting scheme is proved to be of spectral accuracy in space and infinite-order accuracy in time. Furthermore, the scheme is shown to have multiple conservation laws including discrete energy, helicity, momentum, symplecticity, and divergence-free field conservations. All the theoretical results of the accuracy and conservations are numerically illustrated by two numerical tests.