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Computer Physics Communications

Computation of the dynamics of rotating 2D/3D Gross-Pitaevskii equations based on the HPC pseudo-spectral solver BEC2HPC

Jérémie Gaidamour, Xavier Antoine

Published Jun 19, 2026• 0 citations
Publisher

Abstract

The aim of this paper is to present the extension of the HPC pseudo-spectral solver BEC2HPC to compute the dynamics of 2D/3D rotating Bose-Einstein condensates modeled by the Gross-Pitaevskii equations with a rotation term. Numerical examples are provided to show the efficiency of the solver for large-scale simulations. NEW VERSION PROGRAM SUMMARY Program Title: BEC2HPC CPC Library link to program files: https://doi.org/10.17632/mdzpw4dr4t.2 Licensing provisions: GPLv2 Programming language: C++, Python Journal reference of previous version: Comput. Phys. Commun. 265 (2021) 108007, https://doi.org/10.1016/j.cpc.2021.108007 Does the new version supersede the previous version?: Yes Reasons for the new version: The first version of BEC2HPC provided a stationary-state solver, while this new version also enables the computation of the dynamics of rotating Gross–Pitaevskii equations. Summary of revisions: We have implemented ADI time-splitting and relaxation pseudo-spectral schemes in BEC2HPC, following the same parallel design and domain decomposition strategy as for the stationary-state solver. Nature of problem: This extension of the software BEC2HPC computes the dynamics of rotating Bose–Einstein condensates (BEC) modeled by the Gross–Pitaevskii equation (GPE). The available numerical methods are accurate and efficient for BEC with large rotation speeds and high nonlinearities. The parallel implementation allows performing large-scale simulations of 2D or 3D problems on parallel computing platforms. Solution method: Four time discretization methods are implemented for the dynamics of the GPE: a first-, second- and a fourth-order splitting schemes, and a second-order relaxation scheme. The spatial discretization is based on a Fourier pseudo-spectral method. This solver is implemented in distributed memory using MPI and a decomposition of the computational domain. Additional comments including restrictions and unusual features: The parallel algorithms are implemented in C++ and MPI, with a Python interface available for defining the physics of the problem. A pure Python version of sequential algorithms is also provided for rapid prototyping using the same interface. Results can be exported and visualized with external tools (HDF5 and VTK formats for ParaView or PyVista).

DOI
10.1016/j.cpc.2026.110261

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Computation of the dynamics of rotating 2D/3D Gross-Pitaevskii equations based on the HPC pseudo-spectral solver BEC2HPC

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The aim of this paper is to present the extension of the HPC pseudo-spectral solver BEC2HPC to compute the dynamics of 2D/3D rotating Bose-Einstein condensates modeled by the Gross-Pitaevskii equations with a rotation term. Numerical examples are provided to show the efficiency of the solver for large-scale simulations.

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Weizhu Bao, Daniel Marahrens, Qinglin Tang +1 more

We propose a simple, efficient and accurate numerical method for simulating the dynamics of rotating Bose-Einstein condensates (BECs) in a rotational frame with/without a long-range dipole-dipole interaction. We begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with an angular momentum rotation term and/or long-range dipole-dipole interaction, state the two-dimensional (2D) GPE obtained from the 3D GPE via dimension reduction under anisotropic external potential and review some dynamical laws related to the 2D and 3D GPE. By introducing a rotating Lagrangian coordinate system, the original GPEs are re-formulated to GPEs without the angular momentum rotation which is replaced by a time-dependent potential in the new coordinate system. We then cast the conserved quantities and dynamical laws in the new rotating Lagrangian coordinates. Based on the new formulation of the GPE for rotating BECs in the rotating Lagrangian coordinates, a time-splitting spectral method is presented for computing the dynamics of rotating BECs. The new numerical method is explicit, simple to implement, unconditionally stable and very efficient in computation. It is spectral order accurate in space and second-order accurate in time, and conserves the mass in the discrete level. Extensive numerical results are reported to demonstrate the efficiency and accuracy of the new numerical method. Finally, the numerical method is applied to test the dynamical laws of rotating BECs such as the dynamics of condensate width, angular momentum expectation and center-of-mass, and to investigate numerically the dynamics and interaction of quantized vortex lattices in rotating BECs without/with the long-range dipole-dipole interaction.

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A Simple and Efficient Numerical Method for Computing the Dynamics of Rotating Bose--Einstein Condensates via Rotating Lagrangian Coordinates

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We propose a simple, efficient, and accurate numerical method for simulating the dynamics of rotating Bose--Einstein condensates (BECs) in a rotational frame with or without long-range dipole-dipole interaction (DDI). We begin with the three-dimensional (3D) Gross--Pitaevskii equation (GPE) with an angular momentum rotation term and/or long-range DDI, state the two-dimensional (2D) GPE obtained from the 3D GPE via dimension reduction under anisotropic external potential, and review some dynamical laws related to the 2D and 3D GPEs. By introducing a rotating Lagrangian coordinate system, the original GPEs are reformulated to GPEs without the angular momentum rotation, which is replaced by a time-dependent potential in the new coordinate system. We then cast the conserved quantities and dynamical laws in the new rotating Lagrangian coordinates. Based on the new formulation of the GPE for rotating BECs in the rotating Lagrangian coordinates, a time-splitting spectral method is presented for computing the dynamics of rotating BECs. The new numerical method is explicit, simple to implement, unconditionally stable, and very efficient in computation. It is spectral-order accurate in space and second-order accurate in time and conserves the mass on the discrete level. We compare our method with some representative methods in the literature to demonstrate its efficiency and accuracy. In addition, the numerical method is applied to test the dynamical laws of rotating BECs such as the dynamics of condensate width, angular momentum expectation, and center of mass, and to investigate numerically the dynamics and interaction of quantized vortex lattices in rotating BECs without or with the long-range DDI.

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Advances in Mathematical Physics•Aug 4, 2022•2 citations

Computation of Stationary States for Rotating Bose-Einstein Condensates using Spectral Methods

Adam Erlandsson, Paul Hedvall

The Bose-Einstein condensate is a phase of matter that arises when cooling gases of bosons to extremely low temperatures. When studying these condensates one may use the Gross-Pitaevskii equation, which is a non-linear variant of the Schrödinger equation. An interesting phenomenon that arises when rotating a Bose-Einstein condensate is the appearance of vortices. We implement a semi-implicit Euler scheme using spectral methods proposed in [1] to numerically calculate the ground state of a rotating Bose-Einstein condensate. We start with implementing a simpler iterative fixed-point method to solve the Euler scheme but show that this method fails to converge for large rotations. Because of this we implement multiple Krylov subspace solvers that in fact do converge for large rotations and show that the Preconditioned Conjugate Gradient method has better performance than the BiConjugate Gradient Stabilized method. After the implementation we briefly look at the performance of the method and improve it with simple tricks that do not compromise the accuracy or robustness and which reduce the computation time slightly. Lastly we look at the formation of vortices in 2-dimensional and 3-dimensional Bose-Einstein condensates. We show that the number of vortices increases exponentially for increasing angular velocity in 2D until the condensate breaks apart, but in 3D we ultimately find that the required computation time and RAM storage is too large to be able to analyze the vortices in a similar way on our personal computers.

•Jan 1, 2020•0 citations
The aim of this paper is to present the extension of the HPC pseudo-spectral solver BEC2HPC to compute the dynamics of 2D/3D rotating Bose-Einstein condensates modeled by the Gross-Pitaevskii equations with a rotation term. Numerical examples are provided to show the efficiency of the solver for large-scale simulations. NEW VERSION PROGRAM SUMMARY Program Title: BEC2HPC CPC Library link to program files: https://doi.org/10.17632/mdzpw4dr4t.2 Licensing provisions: GPLv2 Programming language: C++, Python Journal reference of previous version: Comput. Phys. Commun. 265 (2021) 108007, https://doi.org/10.1016/j.cpc.2021.108007 Does the new version supersede the previous version?: Yes Reasons for the new version: The first version of BEC2HPC provided a stationary-state solver, while this new version also enables the computation of the dynamics of rotating Gross–Pitaevskii equations. Summary of revisions: We have implemented ADI time-splitting and relaxation pseudo-spectral schemes in BEC2HPC, following the same parallel design and domain decomposition strategy as for the stationary-state solver. Nature of problem: This extension of the software BEC2HPC computes the dynamics of rotating Bose–Einstein condensates (BEC) modeled by the Gross–Pitaevskii equation (GPE). The available numerical methods are accurate and efficient for BEC with large rotation speeds and high nonlinearities. The parallel implementation allows performing large-scale simulations of 2D or 3D problems on parallel computing platforms. Solution method: Four time discretization methods are implemented for the dynamics of the GPE: a first-, second- and a fourth-order splitting schemes, and a second-order relaxation scheme. The spatial discretization is based on a Fourier pseudo-spectral method. This solver is implemented in distributed memory using MPI and a decomposition of the computational domain. Additional comments including restrictions and unusual features: The parallel algorithms are implemented in C++ and MPI, with a Python interface available for defining the physics of the problem. A pure Python version of sequential algorithms is also provided for rapid prototyping using the same interface. Results can be exported and visualized with external tools (HDF5 and VTK formats for ParaView or PyVista).
We propose a simple, efficient and accurate numerical method for simulating the dynamics of rotating Bose-Einstein condensates (BECs) in a rotational frame with/without a long-range dipole-dipole interaction. We begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with an angular momentum rotation term and/or long-range dipole-dipole interaction, state the two-dimensional (2D) GPE obtained from the 3D GPE via dimension reduction under anisotropic external potential and review some dynamical laws related to the 2D and 3D GPE. By introducing a rotating Lagrangian coordinate system, the original GPEs are re-formulated to GPEs without the angular momentum rotation which is replaced by a time-dependent potential in the new coordinate system. We then cast the conserved quantities and dynamical laws in the new rotating Lagrangian coordinates. Based on the new formulation of the GPE for rotating BECs in the rotating Lagrangian coordinates, a time-splitting spectral method is presented for computing the dynamics of rotating BECs. The new numerical method is explicit, simple to implement, unconditionally stable and very efficient in computation. It is spectral order accurate in space and second-order accurate in time, and conserves the mass in the discrete level. Extensive numerical results are reported to demonstrate the efficiency and accuracy of the new numerical method. Finally, the numerical method is applied to test the dynamical laws of rotating BECs such as the dynamics of condensate width, angular momentum expectation and center-of-mass, and to investigate numerically the dynamics and interaction of quantized vortex lattices in rotating BECs without/with the long-range dipole-dipole interaction.
We propose a simple, efficient, and accurate numerical method for simulating the dynamics of rotating Bose--Einstein condensates (BECs) in a rotational frame with or without long-range dipole-dipole interaction (DDI). We begin with the three-dimensional (3D) Gross--Pitaevskii equation (GPE) with an angular momentum rotation term and/or long-range DDI, state the two-dimensional (2D) GPE obtained from the 3D GPE via dimension reduction under anisotropic external potential, and review some dynamical laws related to the 2D and 3D GPEs. By introducing a rotating Lagrangian coordinate system, the original GPEs are reformulated to GPEs without the angular momentum rotation, which is replaced by a time-dependent potential in the new coordinate system. We then cast the conserved quantities and dynamical laws in the new rotating Lagrangian coordinates. Based on the new formulation of the GPE for rotating BECs in the rotating Lagrangian coordinates, a time-splitting spectral method is presented for computing the dynamics of rotating BECs. The new numerical method is explicit, simple to implement, unconditionally stable, and very efficient in computation. It is spectral-order accurate in space and second-order accurate in time and conserves the mass on the discrete level. We compare our method with some representative methods in the literature to demonstrate its efficiency and accuracy. In addition, the numerical method is applied to test the dynamical laws of rotating BECs such as the dynamics of condensate width, angular momentum expectation, and center of mass, and to investigate numerically the dynamics and interaction of quantized vortex lattices in rotating BECs without or with the long-range DDI.
A time-splitting generalized-Laguerre–Fourier–Hermite pseudospectral method is proposed for computing the dynamics of rotating Bose–Einstein condensates (BECs) in two and three dimensions. The new numerical method is based on the following: (i) the use of a time-splitting technique for decoupling the nonlinearity; (ii) the adoption of polar coordinates in two dimensions and cylindrical coordinates in three dimensions such that the angular rotation term becomes constant coefficient; and (iii) the construction of eigenfunctions for the linear operator by properly scaling the generalized-Laguerre, Fourier, and Hermite functions. The new method enjoys the following properties: (i) it is explicit, time reversible, and time transverse invariant; (ii) it conserves the position density and is spectrally accurate in space and second-order or fourth-order accurate in time; and (iii) it solves the problem in the original whole space instead of in a truncated artificial computational domain. The method is also extended to solve the coupled Gross–Pitaevskii equations for the dynamics of rotating two-component and spin-1 BECs. Extensive numerical results for the dynamics of BECs are reported to demonstrate the accuracy and efficiency of the new method for rotating BECs.
We study efficient continuation methods for computing the ground state solution of quasi-2D rotating dipolar Bose-Einstein condensates (BECs). First, the highly accurate spectral collocation method is used to discretize the governing Gross-Pitaevskii equation (GPE). Then, we modify the two-level continuation scheme for 3D dipolar BECs described in Jeng et al. (2014) to develop a single-parameter continuation method for quasi-2D rotating dipolar BECs, where the chemical potential is treated as the continuation parameter. Further, by adding the ratio of dipolar interaction strength to contact interaction strength as the second continuation parameter, we propose an efficient two-parameter continuation method which can effectively show the change of the ground-state vortex structures as the dipolar interaction strength gradually increases. Moreover, we also study linear stability analysis for the GPE. Sample numerical results on quasi-2D rotating dipolar BECs are reported.
The Bose-Einstein condensate is a phase of matter that arises when cooling gases of bosons to extremely low temperatures. When studying these condensates one may use the Gross-Pitaevskii equation, which is a non-linear variant of the Schrödinger equation. An interesting phenomenon that arises when rotating a Bose-Einstein condensate is the appearance of vortices. We implement a semi-implicit Euler scheme using spectral methods proposed in [1] to numerically calculate the ground state of a rotating Bose-Einstein condensate. We start with implementing a simpler iterative fixed-point method to solve the Euler scheme but show that this method fails to converge for large rotations. Because of this we implement multiple Krylov subspace solvers that in fact do converge for large rotations and show that the Preconditioned Conjugate Gradient method has better performance than the BiConjugate Gradient Stabilized method. After the implementation we briefly look at the performance of the method and improve it with simple tricks that do not compromise the accuracy or robustness and which reduce the computation time slightly. Lastly we look at the formation of vortices in 2-dimensional and 3-dimensional Bose-Einstein condensates. We show that the number of vortices increases exponentially for increasing angular velocity in 2D until the condensate breaks apart, but in 3D we ultimately find that the required computation time and RAM storage is too large to be able to analyze the vortices in a similar way on our personal computers.