# Strong Convergence Rate of Splitting Schemes for Stochastic Nonlinear Schr\"odinger Equations

## TL;DR

This paper establishes the optimal strong convergence rate of a fully discrete splitting scheme for stochastic nonlinear Schrödinger equations, combining spectral Galerkin and Crank--Nicolson methods, with theoretical proofs and numerical validation.

## Contribution

It introduces the first proven strong convergence rates for fully discrete schemes solving stochastic NLS equations, utilizing novel uniform a priori estimates and exponential integrability.

## Key findings

- Convergence rate of order 1/2 for splitting processes to the solution.
- Full discretization achieves optimal strong convergence rate of O(N^{-2} + τ^{1/2}).
- Numerical experiments confirm theoretical convergence rates.

## Abstract

We prove the optimal strong convergence rate of a fully discrete scheme, based on a splitting approach, for a stochastic nonlinear Schr\"odinger (NLS) equation. The main novelty of our method lies on the uniform a priori estimate and exponential integrability of a sequence of splitting processes which are used to approximate the solution of the stochastic NLS equation. We show that the splitting processes converge to the solution with strong order $1/2$. Then we use the Crank--Nicolson scheme to temporally discretize the splitting process and get the temporal splitting scheme which also possesses strong order $1/2$. To obtain a full discretization, we apply this splitting Crank--Nicolson scheme to the spatially discrete equation which is achieved through the spectral Galerkin approximation. Furthermore, we establish the convergence of this fully discrete scheme with optimal strong convergence rate $\mathcal{O}(N^{-2}+\tau^\frac12)$, where $N$ denotes the dimension of the approximate space and $\tau$ denotes the time step size. To the best of our knowledge, this is the first result about strong convergence rates of temporally numerical approximations and fully discrete schemes for stochastic NLS equations, or even for stochastic partial differential equations (SPDEs) with non-monotone coefficients. Numerical experiments verify our theoretical result.