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

Mechthild Thalhammer, Gregor Thalhammer-Thurner

TL;DR

This work tackles stable, efficient simulation of multi-species Bose–Einstein condensates described by coupled Gross–Pitaevskii equations. It introduces adaptive, fourth-order modified operator splitting based on commutator corrections to achieve positive-coefficient schemes for both dynamical evolution and imaginary-time ground-state computations, aided by an inexpensive local error control. Numerical experiments in 1D–3D show superior energy and mass conservation and highly accurate ground-state computations compared to standard splits, particularly in higher dimensions. The approach is extensible to more complex nonlinearities and rotating condensates, with potential integration of Magnus-type methods for non-autonomous problems, enhancing practical applicability to real-world BEC simulations.

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.

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

TL;DR

This work tackles stable, efficient simulation of multi-species Bose–Einstein condensates described by coupled Gross–Pitaevskii equations. It introduces adaptive, fourth-order modified operator splitting based on commutator corrections to achieve positive-coefficient schemes for both dynamical evolution and imaginary-time ground-state computations, aided by an inexpensive local error control. Numerical experiments in 1D–3D show superior energy and mass conservation and highly accurate ground-state computations compared to standard splits, particularly in higher dimensions. The approach is extensible to more complex nonlinearities and rotating condensates, with potential integration of Magnus-type methods for non-autonomous problems, enhancing practical applicability to real-world BEC simulations.

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.
Paper Structure (6 sections, 59 equations, 13 figures)

This paper contains 6 sections, 59 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Gross--Pitaevskii equations
  3. Gross--Pitaevskii systems
  4. Operator splitting methods
  5. Numerical experiments
  6. Conclusions

Figures (13)

  • Figure 1: Time evolution of linear (first column) versus nonlinear (second column) reversible-in-time model problems in one (first row) and two (second row) space dimensions. Application of standard and modified operator splitting methods, specifically the first-order Lie--Trotter splitting \ref{['eq:SchemeOrder1']}, the second-order Strang splitting \ref{['eq:SchemeOrder2']}, the optimised fourth-order splitting by Blanes, MoanBlanesMoan2002, and the fourth-order modified splitting based on \ref{['eq:SchemeOrder4']}. The slopes of the lines reflect the temporal orders of convergence and confirm the preservation of the nonstiff orders for problems with sufficiently regular solutions. Modified splitting methods remain stable and competitive in accuracy and efficiency in all test cases.
  • Figure 2: Comparisons of the results displayed in Figure \ref{['fig:FigureTO1']} with related two-component systems (dotted lines). The good agreement of the global errors for the cases $(J, d) \in \{(1,1), (1,2), (2,1)\}$ suggests a similar behaviour regarding stability and accuracy for more complex settings in three space dimensions, see Figure \ref{['fig:FigureTO5']}.
  • Figure 3: Time evolution of linear (first column) versus nonlinear (second column) non-reversible model problems in one (first row) and two (second row) space dimensions. Application of standard and modified operator splitting methods, see Figure \ref{['fig:FigureTO1']}. The slopes of the lines reflect the temporal orders of convergence and confirm the preservation of the nonstiff orders for problems with sufficiently regular solutions. For standard fourth-order splitting methods applied to non-reversible systems, severe stability issues due to the occurence of negative method coefficients and hence even failures for larger time stepsizes are observed. On the contrary, modified splitting methods remain stable and yield highly accurate results for one space dimension (first row). For problems in two space dimensions with increased stiffness (second row), the time integration of the subproblems comprising the potentials and the nonlinearites with refined time increments (factor $\tfrac{1}{4}$ for Strang splitting, factor $\tfrac{1}{8}$ for modified splitting) leads to significant improvements.
  • Figure 4: Comparisons of the results displayed in Figure \ref{['fig:FigureTO3']} with related two-component systems (dotted lines). The good agreement of the global errors for the cases $(J, d) \in \{(1,1), (1,2), (2,1)\}$ suggests a similar behaviour regarding stability and accuracy for more complex settings in three space dimensions, see Figure \ref{['fig:FigureTO5']}. Due to severe instabililies for non-reversible systems, standard higher-order operator splitting methods fail for larger time increments, whereas modified splitting methods with an appropriate resolution of the nonlinear subproblems remain stable and favourable in accuracy.
  • Figure 5: Confirmation of the favourable stability and accuracy behaviour of fourth-order modified splitting methods applied to linear (left column) versus nonlinear (right column) and reversible-in-time (first row) versus non-reversible (second row) model problems in three space dimensions. In order to ensure a reliable numerical approximation of the subproblems comprising the potentials and the nonlinearites for larger time stepsizes, these stepsizes are refined by a factor $\tfrac{1}{16}$.
  • ...and 8 more figures