Invariant measures of stochastic Maxwell equations and ergodic numerical approximations

Abstract

This paper studies the existence and uniqueness of the invariant measure for a class of stochastic Maxwell equations and proposes a novel kind of ergodic numerical approximations to inherit the intrinsic properties. The key to proving the ergodicity lies in the uniform regularity estimates of the exact and 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 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.