A gradient flow model for the Gross--Pitaevskii problem: Mathematical and numerical analysis

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.