Modified splitting methods for Gross-Pitaevskii systems modelling Bose-Einstein condensates: Time evolution and ground state computation

TL;DR

This paper introduces modified high-order operator splitting methods for simulating Bose-Einstein condensates via Gross-Pitaevskii equations, improving stability, accuracy, and efficiency in time evolution and ground state calculations.

Contribution

It develops novel modified splitting methods that remain stable under moderate time step sizes, incorporating commutators and adaptive error control for better numerical performance.

Findings

The fourth-order modified splitting method shows excellent energy and mass conservation.

Adaptive time stepping enhances convergence and accuracy.

Numerical experiments confirm the method's efficiency and stability.

Abstract

The year 2025 marks the 100 and 30 years anniversaries of the discovery of Bose--Einstein condensation and its successful experimental realisation. Inspired by these important research achievements, a conceptually simple approach is proposed to facilitate reliable and efficient numerical simulations. The structure of the underlying systems of coupled Gross--Pitaevskii equations suggests the use of optimised high-order operator splitting methods for dynamical evolution and ground state computation. A second-order barrier, however, prevents the applicability of standard operator splitting methods for both, time evolution as well as imaginary time propagation. An innovative alternative approach accomplishes the design of novel modified operator splitting methods that remain stable under moderate smallness assumptions on the time increments. The core idea is to incorporate commutators of the defining differential and nonlinear multiplication operators, since this permits to fulfill the basic stability requirement of positive method coefficients. Further improvements with respect to convergence at the targeted precision arise from automatic adjustments of the time stepsizes by an inexpensive local error control. The presented numerical experiments confirm the favourable performance of a specific fourth-order modified operator splitting method. Amongst others, it is demonstrated that the excellent mass and energy conservation in long-term evolutions, intrinsic attributes of geometric numerical integrators for Hamiltonian systems, is maintained for a sensible variation of the time stepsizes. Moreover, the benefits of adaptive higher-order approximations in ground state computations are illustrated.