The gradient flow with semi-implicit discretization (GFSI) is the most widely used algorithm for computing the ground state of Gross-Pitaevskii energy functional. Numerous numerical experiments have shown that the energy dissipation holds when calculating the ground states of multicomponent Bose-Einstein condensates (MBECs) with GFSI, while rigorous proof remains an open challenge. By introducing a Lagrange multiplier, we reformulate the GFSI into an equivalent form and thereby prove the energy dissipation for GFSI in two-component scenario with Josephson junction and rotating term, which is one of the most important and topical model in MBECs. Based on this, we further establish the global convergence to stationary states. Also, the numerical results of energy dissipation in practical experiments corroborate our rigorous mathematical proof, and we numerically verified the upper bound of time step that guarantees energy dissipation is indeed related to the strength of particle interactions.

翻译：梯度流半隐式离散化（GFSI）是计算Gross-Pitaevskii能量泛函基态最广泛使用的算法。大量数值实验表明，在使用GFSI计算多组分玻色-爱因斯坦凝聚体（MBECs）基态时能量耗散成立，但其严格证明仍是一个公开的挑战。通过引入拉格朗日乘子，我们将GFSI重构为等价形式，从而证明了GFSI在含约瑟夫森结与旋转项的双组分情形下的能量耗散性——该模型是MBECs中最重要且前沿的模型之一。在此基础上，我们进一步建立了向稳态的全局收敛性。同时，实际实验中的能量耗散数值结果验证了我们严格的数学证明，并且我们通过数值实验证实了保证能量耗散的时间步长上界确实与粒子间相互作用强度相关。