Efficient and stable diffusion generated methods for ground state computation in Bose--Einstein condensates

TL;DR

This paper introduces relaxed formulations of the Gross--Pitaevskii energy functional for Bose--Einstein condensates, enabling stable, accurate, and efficient numerical computation of ground states through energy-dissipative algorithms and adaptive strategies.

Contribution

It proposes novel relaxed formulations with proven convergence and stability, along with adaptive algorithms that improve computational efficiency for ground state calculations in BECs.

Findings

Methods are stable and converge to the ground state.

Adaptive strategy enhances computational efficiency.

Numerical experiments confirm energy dissipation and stability.

Abstract

This paper investigates numerical methods for approximating the ground state of Bose--Einstein condensates (BECs) by introducing two relaxed formulations of the Gross--Pitaevskii energy functional. These formulations achieve first- and second-order accuracy with respect to the relaxation parameter \( \tau \), and are shown to converge to the original energy functional as \( \tau \to 0 \). A key feature of the relaxed functionals is their concavity, which ensures that local minima lie on the boundary of the concave hull. This property prevents energy increases during constraint normalization and enables the development of energy-dissipative algorithms. Numerical methods based on sequential linear programming are proposed, accompanied by rigorous analysis of their stability with respect to the relaxed energy. To enhance computational efficiency, an adaptive strategy is introduced, dynamically refining solutions obtained with larger relaxation parameters to achieve higher accuracy with smaller ones. Numerical experiments demonstrate the stability, convergence, and energy dissipation of the proposed methods, while showcasing the adaptive strategy's effectiveness in improving computational performance.