Efficient and accurate gradient flow methods for computing ground states of spinor Bose-Einstein condensates

Abstract

An accurate and efficient normalized gradient flow (NGF) method is developed to compute the ground state of a spinor Bose-Einstein condensate (BEC) with an arbitrary positive integer spin, which is described as a vector wave function that minimizes the Gross-Pitaevskii energy functional under the two constraints of the total mass and magnetization. Similar to the NGF for computing the ground states of single component BEC, the proposed method consists of two steps, i.e. the gradient flow part to decrease the energy and the projection step for the mass and magnetization constraints. By introducing the Lagrange multipliers in the gradient flow part, the constructed method allows various possible temporal discretizations and inexact projections could be applied in the projection steps. Specifically, we adopt a semi-implicit Euler time discretization scheme to discretize the gradient flow with Lagrange multipliers, thus only a completely decoupled linear elliptic system with constant coefficients needs to be solved at each time step. Moreover, two simple inexact projection methods are proposed and analyzed. Finally, the numerical results for spin-1, spin-2 and spin-3 cases are presented to show the accuracy and effectiveness of our algorithm, and the numerical comparisons for different choices of projections are reported.