Multi-symplectic discontinuous Galerkin methods for the stochastic Maxwell equations with additive noise

TL;DR

This paper develops high-order discontinuous Galerkin methods for stochastic Maxwell equations with additive noise, preserving multi-symplectic structure and linear energy growth, with proven error estimates and demonstrated numerical performance.

Contribution

It introduces novel high-order DG methods that maintain the multi-symplectic structure and energy growth properties for stochastic Maxwell equations with additive noise.

Findings

Methods preserve discrete multi-symplectic structure.

Energy growth is linear and well-captured.

Numerical results confirm optimal error estimates.

Abstract

One- and multi-dimensional stochastic Maxwell equations with additive noise are considered in this paper. It is known that such system can be written in the multi-symplectic structure, and the stochastic energy increases linearly in time. High order discontinuous Galerkin methods are designed for the stochastic Maxwell equations with additive noise, and we show that the proposed methods satisfy the discrete form of the stochastic energy linear growth property and preserve the multi-symplectic structure on the discrete level. Optimal error estimate of the semi-discrete DG method is also analyzed. The fully discrete methods are obtained by coupling with symplectic temporal discretizations. One- and two-dimensional numerical results are provided to demonstrate the performance of the proposed methods, and optimal error estimates and linear growth of the discrete energy can be observed for all cases.