Reconfigured continuous normalized accelerated gradient flow for computing ground states of spin-1 Bose–Einstein condensates

Abstract

Based on the mean field Gross–Pitaevskii theory, the ground state of a spin-1 Bose–Einstein condensate (BEC) can be modeled as a vector wave function that minimizes the energy functional of the coupled Gross–Pitaevskii equations, subject to conservation constraints on total mass and magnetization. The normalized gradient flow methods are widely used for computing the ground states of spin-1 BECs due to their simplicity in implementation and efficiency in each time step. However, their slow convergence speed requires many iterations to reach the ground state, which results in excessive computational costs. In this paper, we propose a highly efficient method for computing the ground states of spin-1 BECs, based on the rapid convergence properties of accelerated gradient flow. We establish a reconfigured continuous normalized accelerated gradient flow (RCNAGF) to simulate the spin-1 BEC ground state. One explicit and two semi-implicit schemes are presented for the numerical solution of the RCNAGF, incorporating a Fourier pseudospectral method for spatial discretization. Our proposed schemes are efficient at each time step because the explicit scheme can be directly updated, while the semi-implicit schemes only need to solve several linear systems with constant coefficients. All schemes require solving an additional nonlinear algebraic system, but this computational cost is considered negligible. The numerical experimental results indicate that our schemes are more efficient than those based on normalized gradient flow schemes for computing the ground states of spin-1 BECs. Specifically, our explicit scheme allows for larger stable time step sizes, while the semi-implicit schemes significantly reduce the number of iterations required to meet the stopping criterion at the same step size.