Approximation of Invariant Measure for Damped Stochastic Nonlinear Schr\"{o}dinger Equation via an Ergodic Numerical Scheme

TL;DR

This paper introduces a fully discrete numerical scheme combining spectral Galerkin and modified implicit Euler methods to approximate the invariant measure of the damped stochastic nonlinear Schrödinger equation, ensuring ergodicity and providing error estimates.

Contribution

The paper develops and analyzes a novel fully discrete scheme that preserves ergodicity and quantifies the approximation error of the invariant measure for the stochastic Schrödinger equation.

Findings

Proves unique ergodicity of the numerical scheme.

Establishes second-order spatial and half-order temporal error estimates.

Demonstrates the scheme's effectiveness in approximating invariant measures.

Abstract

In order to inherit numerically the ergodicity of the damped stochastic nonlinear Schr\"odinger equation with additive noise, we propose a fully discrete scheme, whose spatial direction is based on spectral Galerkin method and temporal direction is based on a modification of the implicit Euler scheme. We not only prove the unique ergodicity of the numerical solutions of both spatial semi-discretization and full discretization, but also present error estimations on invariant measures, which gives order $2$ in spatial direction and order ${\frac12}$ in temporal direction.