Convergence in probability of an ergodic and conformal multi-symplectic numerical scheme for a damped stochastic NLS equation

TL;DR

This paper introduces a new numerical scheme for the damped stochastic nonlinear Schrödinger equation that converges in probability with order one, preserves key physical properties, and is validated through numerical experiments.

Contribution

The paper presents a novel ergodic, conformal multi-symplectic numerical scheme with proven convergence order and ergodicity for the stochastic NLS equation, along with numerical validation.

Findings

Scheme has convergence order one in probability.

Scheme preserves discrete conformal multi-symplectic law.

Numerical experiments confirm theoretical convergence and long-term behavior.

Abstract

In this paper, we investigate the convergence order in probability of a novel ergodic numerical scheme for damped stochastic nonlinear Schr\"{o}dinger equation with an additive noise. Theoretical analysis shows that our scheme is of order one in probability under appropriate assumptions for the initial value and noise. Meanwhile, we show that our scheme possesses the unique ergodicity and preserves the discrete conformal multi-symplectic conservation law. Numerical experiments are given to show the longtime behavior of the discrete charge and the time average of the numerical solution, and to test the convergence order, which verify our theoretical results.