Preservation of Physical Properties of Stochastic Maxwell Equations with Additive Noise via Stochastic Multi-symplectic Methods

TL;DR

This paper develops and analyzes three stochastic multi-symplectic numerical methods for stochastic Maxwell equations with additive noise, demonstrating their ability to preserve key physical properties like divergence and energy evolution.

Contribution

The paper introduces three novel stochastic multi-symplectic methods that preserve divergence and energy properties of stochastic Maxwell equations numerically.

Findings

All methods preserve the discrete averaged divergence.

Each method exhibits dissipative energy behavior consistent with the continuous case.

Numerical experiments confirm theoretical preservation properties.

Abstract

Stochastic Maxwell equations with additive noise are a system of stochastic Hamiltonian partial differential equations intrinsically, possessing the stochastic multi-symplectic conservation law.It is shown that the averaged energy increases linearly with respect to the evolution of time and the flow of stochastic Maxwell equations with additive noise preserves the divergence in the sense of expectation. Moreover, we propose three novel stochastic multi-symplectic methods to discretize stochastic Maxwell equations in order to investigate the preservation of these properties numerically. We made theoretical discussions and comparisons on all of the three methods to observe that all of them preserve the corresponding discrete version of the averaged divergence. Meanwhile, we obtain the corresponding dissipative property of the discrete averaged energy satisfied by each method. Especially, the evolution rates of the averaged energies for all of the three methods are derived which are in accordance with the continuous case. Numerical experiments are performed to verify our theoretical results.