Efficient time integration methods for Gross--Pitaevskii equations with rotation term

TL;DR

This paper introduces and investigates efficient exponential time integration methods for the Gross--Pitaevskii equation with rotation, utilizing a reformulation in rotating coordinates and tailored schemes, confirmed by numerical experiments.

Contribution

It presents a novel combination of commutator-free quasi-Magnus integrators with operator splitting and spectral discretisation for this specific equation.

Findings

Numerical experiments demonstrate the good performance of the proposed integrators.

The tailored schemes effectively handle the structure of the Hamilton operator.

The methods show improved efficiency and accuracy for rotating Gross--Pitaevskii equations.

Abstract

The objective of this work is the introduction and investigation of favourable time integration methods for the Gross--Pitaevskii equation with rotation term. Employing a reformulation in rotating Lagrangian coordinates, the equation takes the form of a nonlinear Schr{\"o}dinger equation involving a space-time-dependent potential. A natural approach that combines commutator-free quasi-Magnus exponential integrators with operator splitting methods and Fourier spectral space discretisations is proposed. Furthermore, the special structure of the Hamilton operator permits the design of specifically tailored schemes. Numerical experiments confirm the good performance of the resulting exponential integrators.