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

TL;DR

This paper analyzes the mathematical properties and numerical schemes for a gradient flow model used to compute the ground state of Bose--Einstein condensates, providing theoretical validation and numerical experiments.

Contribution

It offers the first rigorous mathematical analysis of the model's well-posedness and asymptotic behavior, along with a new fully discrete numerical scheme with proven convergence.

Findings

The model is well-posed and solutions exhibit expected asymptotic behavior.

The proposed numerical scheme is well-posed and converges optimally.

Numerical experiments confirm the theoretical results.

Abstract

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.