Runge-Kutta semidiscretizations for stochastic Maxwell equations with additive noise

TL;DR

This paper develops and analyzes stochastic Runge-Kutta methods for time discretization of stochastic Maxwell equations with additive noise, preserving key physical and mathematical properties and proving convergence.

Contribution

It introduces symplecticity-preserving stochastic Runge-Kutta methods and proves their convergence for stochastic Maxwell equations, addressing an open problem in the field.

Findings

Methods are well-posed and convergent with order one in mean-square sense.

The schemes preserve physical properties like energy and divergence evolution.

The approach solves an open problem for stochastic Maxwell equations with additive noise.

Abstract

The paper concerns semidiscretizations in time of stochastic Maxwell equations driven by additive noise. We show that the equations admit physical properties and mathematical structures, including regularity, energy and divergence evolution laws, and stochastic symplecticity, etc. In order to inherit the intrinsic properties of the original system, we introduce a general class of stochastic Runge-Kutta methods, and deduce the condition of symplecticity-preserving. By utilizing a priori estimates on numerical approximations and semigroup approach, we show that the methods, which are algebraically stable and coercive, are well-posed and convergent with order one in mean-square sense, which answers an open problem in [Chen and Hong, SIAM J. Numer. Anal., 2016] for stochastic Maxwell equations driven by additive noise.