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

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.