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An Efficient Unconditionally Energy-Stable Numerical Scheme for Bose--Einstein Condensate

Jing Guo, Cheng Wang, Dong Wang

TL;DR

This work develops an explicit, norm-preserving numerical scheme for computing the ground state of Bose–Einstein condensates described by the Gross–Pitaevskii energy. The scheme uses an exponential time differencing (ETD) operator to discretize the diffusion term, a stabilization parameter to ensure energy dissipation, and a final $L^2$ normalization to enforce mass conservation, with a Lagrange multiplier enforcing the constraint in the continuous problem. The authors prove unconditional energy dissipation under a suitable bound on the stabilization, establish rigorous convergence and optimal error rates, and validate the theory through 1D and 2D numerical experiments showing the expected temporal and spatial rates and energy stability. The results provide a computationally efficient, provably stable framework for accurate ground-state computations in BEC simulations on periodic domains.

Abstract

A numerical framework is proposed and analyzed for computing the ground state of Bose--Einstein condensates. A gradient flow approach is developed, incorporating both a Lagrange multiplier to enforce the $L^2$ conservation and a free energy dissipation. An explicit approximation is applied to the chemical potential, combined with an exponential time differencing (ETD) operator to the diffusion part, as well a stabilizing operator, to obtain an intermediate numerical profile. Afterward, an $L^2$ normalization is applied at the next numerical stage. A theoretical analysis reveals a free energy dissipation under a maximum norm bound assumption for the numerical solution, and such a maximum norm bound could be recovered by a careful convergence analysis and error estimate. In the authors' knowledge, the proposed method is the first numerical work that preserves the following combined theoretical properties: (1) an explicit computation at each time step, (2) unconditional free energy dissipation, (3) $L^2$ norm conservation at each time step, (4) a theoretical justification of convergence analysis and optimal rate error estimate. Comprehensive numerical experiments validate these theoretical results, demonstrating excellent agreement with established reference solutions.

An Efficient Unconditionally Energy-Stable Numerical Scheme for Bose--Einstein Condensate

TL;DR

This work develops an explicit, norm-preserving numerical scheme for computing the ground state of Bose–Einstein condensates described by the Gross–Pitaevskii energy. The scheme uses an exponential time differencing (ETD) operator to discretize the diffusion term, a stabilization parameter to ensure energy dissipation, and a final normalization to enforce mass conservation, with a Lagrange multiplier enforcing the constraint in the continuous problem. The authors prove unconditional energy dissipation under a suitable bound on the stabilization, establish rigorous convergence and optimal error rates, and validate the theory through 1D and 2D numerical experiments showing the expected temporal and spatial rates and energy stability. The results provide a computationally efficient, provably stable framework for accurate ground-state computations in BEC simulations on periodic domains.

Abstract

A numerical framework is proposed and analyzed for computing the ground state of Bose--Einstein condensates. A gradient flow approach is developed, incorporating both a Lagrange multiplier to enforce the conservation and a free energy dissipation. An explicit approximation is applied to the chemical potential, combined with an exponential time differencing (ETD) operator to the diffusion part, as well a stabilizing operator, to obtain an intermediate numerical profile. Afterward, an normalization is applied at the next numerical stage. A theoretical analysis reveals a free energy dissipation under a maximum norm bound assumption for the numerical solution, and such a maximum norm bound could be recovered by a careful convergence analysis and error estimate. In the authors' knowledge, the proposed method is the first numerical work that preserves the following combined theoretical properties: (1) an explicit computation at each time step, (2) unconditional free energy dissipation, (3) norm conservation at each time step, (4) a theoretical justification of convergence analysis and optimal rate error estimate. Comprehensive numerical experiments validate these theoretical results, demonstrating excellent agreement with established reference solutions.

Paper Structure

This paper contains 17 sections, 5 theorems, 95 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Mathematical formulation and the numerical scheme
  3. Gradient flow with a Lagrange multiplier
  4. A proposed numerical scheme
  5. Energy stability analysis
  6. Convergence analysis and error estimate
  7. A preliminary $\ell^\infty (0, T; \ell^2) \cap \ell^2 (0, T; H_h^1)$ error estimate at the intermediate stage
  8. A preliminary $\ell^\infty (0, T; H_h^2) \cap \ell^2 (0, T; H_h^3)$ error estimate at the intermediate stage
  9. A refined error estimate
  10. Numerical results
  11. One-dimensional numerical results
  12. Energy stability test
  13. Convergence test
  14. Two-dimensional numerical results
  15. Conclusions
  16. ...and 2 more sections

Key Result

Lemma 1

\newlabellem 1: preliminary0 The following estimates are valid:

Figures (4)

  • Figure 1: Numerical energy evolution in terms of time step, with $h=\frac{1}{128}$ and $\tau =\frac{1}{4}$ for Example \ref{['Ex_omg01D']}.
  • Figure 2: Convergence rates for Example \ref{['Ex_omg01D']}.
  • Figure 3: Numerical energy evolution in terms of time step, with $h=\frac{1}{16}$ for Example \ref{['Ex:2Domg0']}.
  • Figure 4: Convergence rates for Example \ref{['Ex:2Domg0']}.

Theorems & Definitions (11)

  • Lemma 1
  • Proof 1
  • Proposition 1
  • Proof 2
  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Remark 1
  • Remark 2
  • Example 1
  • ...and 1 more