A New Minimization Solver for Computing the Ground States of Bose–Einstein Condensates

In this paper, we present a highly accurate discretization model for directly computing the ground states of Bose–Einstein condensates (BECs). The energy functional and constraint are discretized by introducing the fourth‐order compact finite difference and the composite Simpson quadrature, and thus the original infinite‐dimensional constrained minimization problem is approximated by a finite‐dimensional minimization problem with a spherical constraint uniformly. We develop a projected Riemannian gradient (PRG) algorithm to solve the optimization problem and establish the corresponding convergence results. Extensive numerical tests of non‐rotating BECs as well as rotating BECs under different potentials and parameter regimes are provided to verify the feasibility and high accuracy of our method. In addition, we apply it to investigate the ground states of multi‐component BECs and the numerical results again validate the power of this method.