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.

开发了一种准确高效的归一化梯度流（NGF）方法，以计算具有任意正整数自旋的自旋玻色-爱因斯坦凝聚物（BEC）的基态，该基波被描述为使Gross-Pitaevskii能量最小的矢量波函数在总质量和磁化强度的两个约束下起作用。类似于用于计算单组分BEC的基态的NGF，该方法包括两个步骤，即减少能量的梯度流部分和针对质量和磁化约束的投影步骤。通过在梯度流动部分引入拉格朗日乘数，所构造的方法可以进行各种可能的时间离散化，并且不精确的投影可以应用于投影步骤。具体来说，我们采用半隐式Euler时间离散化方案，通过Lagrange乘法器离散化梯度流，因此在每个时间步仅需要求解一个具有常数系数的完全解耦的线性椭圆系统。此外，提出并分析了两种简单的不精确投影方法。最后，给出了spin-1，spin-2和spin-3案例的数值结果，以证明我们算法的准确性和有效性，并报告了不同投影选择的数值比较。