Energy dissipation and global convergence of a discrete normalized gradient flow for computing ground states of two-component Bose-Einstein condensates

TL;DR

This paper rigorously proves energy dissipation and global convergence of a discrete normalized gradient flow method for computing ground states in two-component Bose-Einstein condensates, supported by numerical validation.

Contribution

It introduces a reformulation with a Lagrange multiplier to prove energy dissipation and convergence for the GFSI method in complex MBEC models.

Findings

Energy dissipation is rigorously proven for GFSI in two-component MBECs.

Numerical experiments confirm the theoretical energy dissipation and convergence.

An upper bound on time step size related to particle interactions is verified.

Abstract

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.