Exponential integrators for stochastic Schr\"odinger equations driven by Ito noise

TL;DR

This paper introduces an explicit exponential numerical scheme for stochastic Schrödinger equations driven by Ito noise, demonstrating convergence properties and analyzing how well the scheme preserves physical quantities like mass, energy, and momentum.

Contribution

The paper develops and analyzes a new exponential integrator for stochastic Schrödinger equations, establishing convergence orders and examining the preservation of trace formulas in numerical solutions.

Findings

Strong order 1 convergence for additive noise

Strong order 1/2 convergence for multiplicative noise

Numerical results confirm theoretical convergence and trace preservation

Abstract

We study an explicit exponential scheme for the time discretisation of stochastic Schr\"odinger equations driven by additive or multiplicative Ito noise. The numerical scheme is shown to converge with strong order $1$ if the noise is additive and with strong order $1/2$ for multiplicative noise. In addition, if the noise is additive, we show that the exact solutions of our problems satisfy trace formulas for the expected mass, energy, and momentum (i.e., linear drifts in these quantities). Furthermore, we inspect the behaviour of the numerical solutions with respect to these trace formulas. Several numerical simulations are presented and confirm our theoretical results.