# Convergence analysis of a symplectic semi-discretization for stochastic NLS equation with quadratic potential

## TL;DR

This paper analyzes the convergence of a symplectic semi-discretization method for stochastic nonlinear Schrödinger equations with quadratic potential, demonstrating order one convergence in probability and exploring long-term behavior through numerical experiments.

## Contribution

It provides a theoretical convergence analysis of a symplectic scheme for stochastic NLS with quadratic potential, including order one convergence and numerical validation.

## Key findings

- Convergence order of the scheme is one in probability.
- Numerical experiments confirm theoretical convergence and analyze long-term behavior.
- The scheme effectively simulates the influence of potential and noise on the system.

## Abstract

In this paper, we investigate the convergence in probability of a stochastic symplectic scheme for stochastic nonlinear Schr\"{o}dinger equation with quadratic potential and an additive noise. Theoretical analysis shows that our symplectic semi-discretization is of order one in probability under appropriate regularity conditions for the initial value and noise. Numerical experiments are given to simulate the long time behavior of the discrete average charge and energy as well as the influence of the external potential and noise, and to test the convergence order.