This paper concerns the mathematical and numerical analysis of the $L^2$ normalized gradient flow model for the Gross--Pitaevskii eigenvalue problem, which has been widely used to design the numerical schemes for the computation of the ground state of the Bose--Einstein condensate. We first provide the mathematical analysis for the model, including the well-posedness and the asymptotic behavior of the solution. Then we propose a normalized implicit-explicit fully discrete numerical scheme for the gradient flow model, and give some numerical analysis for the scheme, including the well-posedness and optimal convergence of the approximation. Some numerical experiments are provided to validate the theory.

翻译：本文研究用于Gross-Pitaevskii特征值问题的$L^2$归一化梯度流模型的数学与数值分析，该模型已被广泛用于设计计算玻色-爱因斯坦凝聚基态的数值格式。我们首先对该模型进行数学分析，包括解的存在唯一性及渐近行为。随后针对该梯度流模型提出一种归一化隐式-显式全离散数值格式，并对该格式进行数值分析，包括近似解的存在唯一性及最优收敛性。最后通过数值实验验证理论结果。