Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow

TL;DR

This paper proves the energy diminishing property of a normalized gradient flow for computing Bose-Einstein condensate ground states, introduces two effective numerical discretization methods, and demonstrates their superior performance through extensive simulations.

Contribution

It provides a rigorous mathematical justification for the imaginary time method and introduces two energy-preserving numerical schemes for BEC ground state computation.

Findings

BEFD and TSSP methods preserve energy diminishing property.

Numerical results confirm effectiveness in 1D, 2D, and 3D simulations.

Normalized gradient flow can compute first excited states with specific initial data.

Abstract

In this paper, we prove the energy diminishing of a normalized gradient flow which provides a mathematical justification of the imaginary time method used in physical literatures to compute the ground state solution of Bose-Einstein condensates (BEC). We also investigate the energy diminishing property for the discretization of the normalized gradient flow. Two numerical methods are proposed for such discretizations: one is the backward Euler centered finite difference (BEFD), the other one is an explicit time-splitting sine-spectral (TSSP) method. Energy diminishing for BEFD and TSSP for linear case, and monotonicity for BEFD for both linear and nonlinear cases are proven. Comparison between the two methods and existing methods, e.g. Crank-Nicolson finite difference (CNFD) or forward Euler finite difference (FEFD), shows that BEFD and TSSP are much better in terms of preserving energy diminishing property of the normalized gradient flow. Numerical results in 1d, 2d and 3d with magnetic trap confinement potential, as well as a potential of a stirrer corresponding to a far-blue detuned Gaussian laser beam are reported to demonstrate the effectiveness of BEFD and TSSP methods. Furthermore we observe that the normalized gradient flow can also be applied directly to compute the first excited state solution in BEC when the initial data is chosen as an odd function.