### Abstract

In this paper, we propose two iterative methods, a Jacobi-type iteration (JI) and a Gauss-Seidel-type iteration (GSI), for the computation of energy states of the time-independent vector Gross-Pitaevskii equation (VGPE) which describes a multi-component Bose-Einstein condensate (BEC). A discretization of the VGPE leads to a nonlinear algebraic eigen-value problem (NAEP). We prove that the GSI method converges locally and linearly to a solution of the NAEP if and only if the associated minimized energy functional problem has a strictly local minimum. The GSI method can thus be used to compute ground states and positive bound states, as well as the corresponding energies of a multi-component BEC. Numerical experience shows that the GSI converges much faster than JI and converges globally within 10-20 steps.

Original language | English |
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Pages (from-to) | 367-390 |

Number of pages | 24 |

Journal | Journal of Computational Physics |

Volume | 202 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2005 |

### Keywords

- Gauss-Seidel-type iteration
- Gross-Pitaevskii equation
- Multi-component Bose-Einstein condensate
- Nonlinear eigenvalue problem

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## Cite this

*Journal of Computational Physics*,

*202*(1), 367-390. https://doi.org/10.1016/j.jcp.2004.07.012