Abstract
A numerical framework is proposed and analyzed for computing the ground state of Bose–Einstein condensates. A gradient flow approach is developed, incorporating both a Lagrange multiplier to enforce theL²conservation and a free energy dissipation. An explicit approximation is applied to the chemical potential, combined with an exponential time differencing (ETD) operator to the diffusion part, as well a stabilizing operator, to obtain an intermediate numerical profile. Afterward, anL²normalization is applied at the next numerical stage. A theoretical analysis reveals a free energy dissipation under a maximum norm bound assumption for the numerical solution, and such a maximum norm bound could be recovered by a careful convergence analysis and error estimate. In the authors' knowledge, the proposed method is the first numerical work that preserves the following combined theoretical properties: (1) an explicit computation at each time step, (2) unconditional free energy dissipation, (3)L²norm conservation at each time step, (4) a theoretical justification of convergence analysis and optimal rate error estimate. Comprehensive numerical experiments validate these theoretical results, demonstrating excellent agreement with established reference solutions.