Ergodic numerical approximations for stochastic Maxwell equations

TL;DR

This paper introduces a new numerical method for stochastic Maxwell equations that preserves ergodic properties, providing convergence guarantees for the invariant measure with proven regularity estimates and a convergence order of 1/2.

Contribution

It develops an ergodic-preserving numerical approximation for stochastic Maxwell equations with proven convergence order and regularity estimates.

Findings

Numerical solutions exhibit uniform regularity over time.

The mean-square convergence order is 1/2 in both space and time.

Numerical invariant measure converges to the exact measure in L2-Wasserstein distance.

Abstract

In this paper, we propose a novel kind of numerical approximations to inherit the ergodicity of stochastic Maxwell equations. The key to proving the ergodicity lies in the uniform regularity estimates of the numerical solutions with respect to time, which are established by analyzing some important physical quantities. By introducing an auxiliary process, we show that the mean-square convergence order of the ergodic discontinuous Galerkin full discretization is $\frac{1}{2}$ in the temporal direction and $\frac{1}{2}$ in the spatial direction, which provides the convergence order of the numerical invariant measure to the exact one in $L^2$-Wasserstein distance.