Energy-conserving method for Stochastic Maxwell Equations with Multiplicative Noise

TL;DR

This paper demonstrates that stochastic Maxwell equations with multiplicative noise have a geometric structure and energy conservation property, and introduces a numerical method that preserves these features with verified accuracy and efficiency.

Contribution

It proposes a novel stochastic multi-symplectic energy-conserving numerical method combining wavelet collocation and stochastic symplectic techniques for these equations.

Findings

The method accurately preserves energy in simulations.

Numerical experiments confirm the method's energy conservation and accuracy.

Comparison shows advantages over finite difference methods.

Abstract

In this paper, it is shown that three-dimensional stochastic Maxwell equations with multiplicative noise are stochastic Hamiltonian partial differential equations possessing a geometric structure (i.e. stochastic mutli-symplectic conservation law), and the energy of system is a conservative quantity almost surely. We propose a stochastic multi-symplectic energy-conserving method for the equations by using the wavelet collocation method in space and stochastic symplectic method in time. Numerical experiments are performed to verify the excellent abilities of the proposed method in providing accurate solution and preserving energy. The mean square convergence result of the method in temporal direction is tested numerically, and numerical comparisons with finite difference method are also investigated.