Numerical Analysis on Ergodic Limit of Approximations for Stochastic NLS Equation via Multi-symplectic Scheme

TL;DR

This paper develops a fully discrete symplectic scheme for a finite dimensional approximation of the stochastic nonlinear Schrödinger equation, proving ergodic limit convergence and verifying charge conservation and ergodicity through numerical experiments.

Contribution

It introduces a novel fully discrete scheme that preserves key properties and proves convergence to the ergodic limit for stochastic NLS approximations.

Findings

The scheme preserves charge conservation and ergodicity.

Temporal averages converge to the ergodic limit with order one.

Numerical experiments confirm theoretical properties.

Abstract

We consider a finite dimensional approximation of the stochastic nonlinear Schr\"odinger equation driven by multiplicative noise, which is derived by applying a symplectic method to the original equation in spatial direction. Both the unique ergodicity and the charge conservation law for this finite dimensional approximation are obtained on the unit sphere. To simulate the ergodic limit over long time for the finite dimensional approximation, we discretize it further in temporal direction to obtain a fully discrete scheme, which inherits not only the stochastic multi-symplecticity and charge conservation law of the original equation but also the unique ergodicity of the finite dimensional approximation. The temporal average of the fully discrete numerical solution is proved to converge to the ergodic limit with order one with respect to the time step for a fixed spatial step. Numerical experiments verify our theoretical results on charge conservation, ergodicity and weak convergence.