On the discrete normalized gradient flow for computing ground states of rotating Bose–Einstein condensates: energy dissipation and global convergence

The discrete normalized gradient flow with semi-implicit time discretization (GFSI), introduced in [W. Bao and Q. Du, SIAM J. Sci. Comput., 25 (2004), pp. 1674–1697], has become one of the most widely used numerical method for computing the ground states of Bose–Einstein condensates (BECs). In spite of strong numerical evidence of its energy dissipation and global convergence a rigorous proof of these two properties for the GFSI remains an open challenge. In this work we comprehensively study these two properties of the GFSI in the context of computing ground states of BECs with and without angular momentum rotation. By reformulating the GFSI into an equivalent form in terms of the normalized numerical solution we rigorously establish the energy dissipation property of the algorithm. Building on this result we further establish its global convergence to a stationary state. To the best of our knowledge this study represents the first rigorous derivation of both energy dissipation and global convergence for the GFSI.