Numerical analysis of the Gross-Pitaevskii Equation with a randomly varying potential in time

TL;DR

This paper develops and analyzes a spectral Crank-Nicolson numerical scheme for the stochastic Gross-Pitaevskii equation with time-varying potential, proving its convergence and strong order of at least one.

Contribution

It introduces a spectral Crank-Nicolson scheme for the stochastic Gross-Pitaevskii equation and proves its convergence with a strong order of at least one.

Findings

The scheme converges in the case of cubic non-linearity.

The scheme has a strong order of convergence of at least one.

Convergence is proven under the assumption of a globally defined unique solution.

Abstract

The Gross-Pitaevskii equation with white noise in time perturbations of the harmonic potential is considered. In this article we define a Crank-Nicolson scheme based on a spectral discretization and we show the convergence of this scheme in the case of cubic non-linearity and when the exact solution is uniquely defined and global in time. We prove that the strong order of convergence in probability is at least one.