### 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 |
---|---|

Pages (from-to) | 367-390 |

Number of pages | 24 |

Journal | Journal of Computational Physics |

Volume | 202 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2005 Jan 1 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Numerical Analysis
- Modelling and Simulation
- Physics and Astronomy (miscellaneous)
- Physics and Astronomy(all)
- Computer Science Applications
- Computational Mathematics
- Applied Mathematics

### Cite this

*Journal of Computational Physics*,

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

**Gauss-Seidel-type methods for energy states of a multi-component Bose-Einstein condensate.** / Chang, Shu Ming; Lin, Wen Wei; Shieh, Shih-Feng.

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 202, no. 1, pp. 367-390. https://doi.org/10.1016/j.jcp.2004.07.012

}

TY - JOUR

T1 - Gauss-Seidel-type methods for energy states of a multi-component Bose-Einstein condensate

AU - Chang, Shu Ming

AU - Lin, Wen Wei

AU - Shieh, Shih-Feng

PY - 2005/1/1

Y1 - 2005/1/1

N2 - 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.

AB - 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.

KW - Gauss-Seidel-type iteration

KW - Gross-Pitaevskii equation

KW - Multi-component Bose-Einstein condensate

KW - Nonlinear eigenvalue problem

UR - http://www.scopus.com/inward/record.url?scp=8744291229&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=8744291229&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2004.07.012

DO - 10.1016/j.jcp.2004.07.012

M3 - Article

AN - SCOPUS:8744291229

VL - 202

SP - 367

EP - 390

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 1

ER -