Accelerated Stabilizer-Based Normalized Gradient Flow Algorithms for Stationary States of Bose–Einstein Condensates

We present an accelerated, stabilizer-based normalized gradient flow (ASNGF) framework with a nonlocal inertial (or momentum) mechanism for computing stationary states of Bose–Einstein condensates. Two schemes, ASNGF-I and ASNGF-II, are developed, and their discrete Lyapunov energy dissipation properties are established under the stated conditions. Linearized stability analyses and numerical parameter studies provide practical guidance for selecting the tunable parameters and demonstrate the important role of the nonlocal Laplacian term in the inertial dynamics. For suitable parameter scalings, the observed iteration counts are essentially insensitive to the time step over the tested range, including τ=10±60. Numerical experiments on several Gross–Pitaevskii models show that ASNGF can substantially reduce the iteration count compared with conventional normalized gradient-flow schemes for the tested problems and parameter settings. The results also reveal that its convergence behavior is problem- and parameter-dependent. The inertial mechanism can enhance exploration of non-convex energy landscapes but does not guarantee convergence to a global minimizer. These results demonstrate the potential of ASNGF as an efficient framework for stationary-state computations in BECs.