Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T07:39:21.743Z Has data issue: false hasContentIssue false
  • English
  • Français

ANALYSIS OF BLOCK-SOR ITERATION FOR THE THREE-DIMENSIONAL LAPLACIAN

Part of: Numerical linear algebra

Published online by Cambridge University Press:  04 December 2009

WENJUN ZHENG  and
ZHIQIN ZHAO
WENJUN ZHENG
Affiliation:
School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan, PR China (email: [email protected])
ZHIQIN ZHAO*
Affiliation:
School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan, PR China (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The successive over-relaxation (SOR) iteration method for solving linear systems of equations depends upon a relaxation parameter. A well-known theory for determining this parameter was given by Young for consistently ordered matrices. In this paper, for the three-dimensional Laplacian, we introduce several compact difference schemes and analyse the block-SOR method for the resulting linear systems. Their optimum relaxation parameters are given for the first time. Analysis shows that the value of the optimum relaxation parameter of block-SOR iteration is very sensitive for compact stencils when solving the three-dimensional Laplacian. This paper provides a theoretical solution for determining the optimum relaxation parameter in real applications.

Keywords

compact stencilblock-SOR iterationoptimum relaxation parameterthree-dimensional Laplacian

MSC classification

Secondary: 65F10: Iterative methods for linear systems
Type
Research Article
Information
The ANZIAM Journal , Volume 50 , Issue 4 , April 2009 , pp. 501 - 512
Copyright
Copyright © Australian Mathematical Society 2009

References

[1]Adams, L. M., LeVeque, R. J. and Young, D. M., “Analysis of the SOR iteration for the 9-point Laplacian”, SIAM J. Numer. Anal. 25 (1988) 1156–1180.CrossRefGoogle Scholar
[2]Xie, D. and Adams, L., “New parallel SOR method by domain partitioning”, SIAM J. Sci. Comput. 20 (1999) 2261–2281.CrossRefGoogle Scholar
[3]Hadjidimos, A., “Successive overrelaxation (SOR) and related methods”, J. Comput. Appl. Math. 123 (2000) 177–199.CrossRefGoogle Scholar
[4]Leveque, R. J. and Trefethen, L. N., “Fourier analysis of the SOR iteration”, IMA J. Numer. Anal. 8 (1988) 273–279.CrossRefGoogle Scholar
[5]Van de Vooren, A. I. and Vliegenthart, A. C., “On the 9-point difference formula for Laplace’s equation”, J. Engrg. Math. 1 (1967) 187–202.CrossRefGoogle Scholar
[6]Young, D. M., “Iterative methods for solving partial difference equations of elliptic type”, Doctoral Thesis, Harvard University, Mathematical Department, Cambridge, Massachusetts, 1950.Google Scholar
[7]Young, D. M., “Iterative methods for solving partial difference equations of elliptic types”, Trans. Amer. Math. Soc. 76 (1954) 92–111.CrossRefGoogle Scholar
[8]Young, D. M., Iterative solution of large linear systems (Academic Press, New York–London, 1971).Google Scholar