Analysis of a splitting scheme for a class of nonlinear stochastic Schr\"odinger equations

TL;DR

This paper investigates a splitting numerical scheme for nonlinear stochastic Schr"odinger equations, demonstrating its symplectic nature, mass preservation, and convergence properties through theoretical analysis and numerical experiments.

Contribution

It provides a detailed analysis of the scheme's qualitative properties, convergence order, and applicability to equations with nonlocal nonlinearities, which is novel in this context.

Findings

The scheme is symplectic and preserves mass.

Strong and probabilistic convergence orders are established.

Numerical experiments confirm theoretical results.

Abstract

We analyze the qualitative properties and the order of convergence of a splitting scheme for a class of nonlinear stochastic Schr\"odinger equations driven by additive It\^o noise. The class of nonlinearities of interest includes nonlocal interaction cubic nonlinearities. We show that the numerical solution is symplectic and preserves the expected mass for all times. On top of that, for the convergence analysis, some exponential moment bounds for the exact and numerical solutions are proved. This enables us to provide strong orders of convergence as well as orders of convergence in probability and almost surely. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme.