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C++ CUDA programs for solving the time-dependent dipolar Gross-Pitaevskii equation

Denis Mujo, D. Vudragović, Paulsamy Muruganandam, Sadhan K. Adhikari

Published May 16, 2016• 0 citations
Publisher

Abstract

In this paper we present new versions of previously published numerical programs for solving the dipolar Gross–Pitaevskii (GP) equation including the contact interaction in two and three spatial dimensions in imaginary and in real time, yielding both stationary and non-stationary solutions. New versions of programs were developed using CUDA toolkit and can make use of Nvidia GPU devices. The algorithm used is the same split-step semi-implicit Crank–Nicolson method as in the previous version (Kishor Kumar et al., 2015), which is here implemented as a series of CUDA kernels that compute the solution on the GPU. In addition, the Fast Fourier Transform (FFT) library used in the previous version is replaced by cuFFT library, which works on CUDA-enabled GPUs. We present speedup test results obtained using new versions of programs and demonstrate an average speedup of 12–25, depending on the program and input size. The original version of this program (AEWL_v1_0) may be found at dx.doi.org/10.1016/j.cpc.2015.03.024

DOI
10.17632/s858p8zsgp

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In chapter 3, the numerical techniques which have been exploited for solving Gross–Pitaevskii (GP)-type equations have been explored in detail. In particular, the algorithmic approaches employed in split-step Crank–Nicolson method, split-step Fourier transform method and Newton conjugate gradient method have been outlined. A glimpse of generating ground states of Bose–Einstein condensates using data driven approaches employing Gaussian processes as an alternative to numerical methods is shown.

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In this paper we present new versions of previously published numerical programs for solving the dipolar Gross–Pitaevskii (GP) equation including the contact interaction in two and three spatial dimensions in imaginary and in real time, yielding both stationary and non-stationary solutions. New versions of programs were developed using CUDA toolkit and can make use of Nvidia GPU devices. The algorithm used is the same split-step semi-implicit Crank–Nicolson method as in the previous version (Kishor Kumar et al., 2015), which is here implemented as a series of CUDA kernels that compute the solution on the GPU. In addition, the Fast Fourier Transform (FFT) library used in the previous version is replaced by cuFFT library, which works on CUDA-enabled GPUs. We present speedup test results obtained using new versions of programs and demonstrate an average speedup of 12–25, depending on the program and input size. The original version of this program (AEWL_v1_0) may be found at dx.doi.org/10.1016/j.cpc.2015.03.024
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 efficient and accurate numerical methods for computing the ground state and dynamics of the dipolar Bose-Einstein condensates utilising a newly developed dipole-dipole interaction (DDI) solver that is implemented with the non-uniform fast Fourier transform (NUFFT) algorithm. We begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with a DDI term and present the corresponding two-dimensional (2D) model under a strongly anisotropic confining potential. Different from existing methods, the NUFFT based DDI solver removes the singularity by adopting the spherical/polar coordinates in the Fourier space in 3D/2D, respectively, thus it can achieve spectral accuracy in space and simultaneously maintain high efficiency by making full use of FFT and NUFFT whenever it is necessary and/or needed. Then, we incorporate this solver into existing successful methods for computing the ground state and dynamics of GPE with a DDI for dipolar BEC. Extensive numerical comparisons with existing methods are carried out for computing the DDI, ground states and dynamics of the dipolar BEC. Numerical results show that our new methods outperform existing methods in terms of both accuracy and efficiency.
In chapter 3, the numerical techniques which have been exploited for solving Gross–Pitaevskii (GP)-type equations have been explored in detail. In particular, the algorithmic approaches employed in split-step Crank–Nicolson method, split-step Fourier transform method and Newton conjugate gradient method have been outlined. A glimpse of generating ground states of Bose–Einstein condensates using data driven approaches employing Gaussian processes as an alternative to numerical methods is shown.