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

Tianyang Chu, Xiaoying Dai, Jing Wu, Aihui Zhou

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.

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

Abstract

This paper concerns the mathematical and numerical analysis of the 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.

Paper Structure

This paper contains 24 sections, 22 theorems, 169 equations, 8 tables.

Table of Contents

  1. Introduction
  2. The gradient flow model
  3. The Gross--Pitaevskii problem
  4. The L2 gradient flow model
  5. Mathematical analysis
  6. Well-posedness and global convergence
  7. Convergence rate
  8. Numerical analysis
  9. Numerical scheme
  10. Convergence
  11. Technical lemmas
  12. Consistency
  13. Error equation
  14. Proof of Lemma \ref{['lem: induction']}
  15. Error estimate
  16. ...and 9 more sections

Key Result

Proposition 3.1

The solution of equ: PDEofL2GF satisfies normalization conservation and energy diminishment, i.e.,

Theorems & Definitions (42)

  • Remark 2.1: Projected gradient flow approach
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Theorem 3.4
  • Proof 1
  • Remark 3.5
  • Remark 3.6
  • Lemma 3.7
  • Proof 2
  • ...and 32 more