Boundary Value Problems on Triangular Domains and MKSOR Methods

I. K. Youssef; Sh. A. Meligy

doi:doi:10.11648/j.acm.20140303.14

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Boundary Value Problems on Triangular Domains and MKSOR Methods

I. K. Youssef, Sh. A. Meligy

Published in Applied and Computational Mathematics (Volume 3, Issue 3)

Received: 29 May 2014 Accepted: 16 June 2014 Published: 30 June 2014

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Abstract

The performance of six variants of the successive overrelaxation methods (SOR) are considered for an algebraic system arising from a finite difference treatment of an elliptic equation of Partial Differential Equations (PDEs) on a triangular region. The consistency of the finite difference representation of the system is achieved. In the finite difference method one obtains an algebraic system corresponding to the boundary value problem (BVP). The block structure of the algebraic system corresponding to four different labeling (the natural, the red- black and green (RBG), the electronic and the spiral) of the grid points is considered. Also, algebraic systems obtained from BVP with mixed derivatives are well established. Determination of the optimal relaxation parameters on the bases of the graphical representation of the spectral radius of the iteration matrices for the SOR, the Modified Successive over relaxation (MSOR) and their new variants KSOR, MKSOR, MKSOR1 and MKSOR2 are considered. Application of the treatment to two numerical examples is considered.

Published in	Applied and Computational Mathematics (Volume 3, Issue 3)
DOI	10.11648/j.acm.20140303.14
Page(s)	90-99
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2014. Published by Science Publishing Group

Keywords

SOR, KSOR, MKSOR, MKSOR1, Triangular Grid and Grid Labeling

References

[1]	D. M. Young, “Iterative Methods for Solving Partial Difference Equations of Elliptic Type,” Transctions of the American Mathematical Society, Vol. 76, No. 1, pp. 92-111, Jan. 1954.
[2]	D. M. Young, “Iterative Solution of Large Linear Systems,”Academic Press, London, 1971.
[3]	J. W. Sheldon, “On the Numerical Solution of Elliptic Difference Equations,” Mathematical Tables and Other Aids to Computation, Vol. 9, No. 51, pp. 101 - 112, Jul. 1955.
[4]	B. L. Buzbee, G. H. Golub and C. W. Nielson, “On Direct Methods for Solving Poisson’s Equations,” SIAM J. Numer. Anal., Vol. 7, No. 4, Dec. 1970.
[5]	G. Birkhoff, “The Numerical Solution of Elliptic Equations,” Society for Industrial and Applied Mathematics, 1972.
[6]	R. A. Nicolaides, “On the Observed Rate of Convergence of an Iterative Method Applied to a Model Elliptic Difference Equation,” Mathematics of Computation, Vol. 32, No. 141, pp. 127-133, Jan 1978.
[7]	G. D. Smith, “Numerical Solution of Partial Differential Equations,” Oxford university press, 1985.
[8]	S. M. Khamis, I. K. Youssef, M. H. El- Dewik and B. I. Bayoumi, “A Computational Finite Difference Treatment for PDEs Including the Mixed Derivative Term with High Accuracy on Curved Domains,” Journal of the Egyptian Mathematical Society, Vol. 17(1), pp. 15-33, 2009.
[9]	I. K. Youssef, “On Strongly Coupled Linear Elliptic Systems with Application to Otolith Membrane Distortion” Journal of Mathematics and Statistics 4 (4): 236-244, 2008.
[10]	D. J. Evans, “The Solution of Poisson’s Equation in a triangular region,”International Journal of Computer Math., Vol. 39 pp. 81 – 98,1991.
[11]	I. K. Youssef, “On the Successive Overrelaxation Method,” Journal of Mathematics and Statistics 8 (2): 176-184, 2012.
[12]	I. K. Youssef and A. A. Taha, “On the Modified Successive Overrelaxation Method,” Applied Mathematics and Computation 219, pp. 4601-4613, 2013.

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APA Style

I. K. Youssef, Sh. A. Meligy. (2014). Boundary Value Problems on Triangular Domains and MKSOR Methods. Applied and Computational Mathematics, 3(3), 90-99. https://doi.org/10.11648/j.acm.20140303.14

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ACS Style

I. K. Youssef; Sh. A. Meligy. Boundary Value Problems on Triangular Domains and MKSOR Methods. Appl. Comput. Math. 2014, 3(3), 90-99. doi: 10.11648/j.acm.20140303.14

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AMA Style

I. K. Youssef, Sh. A. Meligy. Boundary Value Problems on Triangular Domains and MKSOR Methods. Appl Comput Math. 2014;3(3):90-99. doi: 10.11648/j.acm.20140303.14

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@article{10.11648/j.acm.20140303.14,
  author = {I. K. Youssef and Sh. A. Meligy},
  title = {Boundary Value Problems on Triangular Domains and MKSOR Methods},
  journal = {Applied and Computational Mathematics},
  volume = {3},
  number = {3},
  pages = {90-99},
  doi = {10.11648/j.acm.20140303.14},
  url = {https://doi.org/10.11648/j.acm.20140303.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140303.14},
  abstract = {The performance of six variants of the successive overrelaxation methods (SOR) are considered for an algebraic system arising from a finite difference treatment of an elliptic equation of Partial Differential Equations (PDEs) on a triangular region. The consistency of the finite difference representation of the system is achieved. In the finite difference method one obtains an algebraic system corresponding to the boundary value problem (BVP). The block structure of the algebraic system corresponding to four different labeling (the natural, the red- black and green (RBG), the electronic and the spiral) of the grid points is considered. Also, algebraic systems obtained from BVP with mixed derivatives are well established. Determination of the optimal relaxation parameters on the bases of the graphical representation of the spectral radius of the iteration matrices for the SOR, the Modified Successive over relaxation (MSOR) and their new variants KSOR, MKSOR, MKSOR1 and MKSOR2 are considered. Application of the treatment to two numerical examples is considered.},
 year = {2014}
}

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TY - JOUR
T1 - Boundary Value Problems on Triangular Domains and MKSOR Methods
AU - I. K. Youssef
AU - Sh. A. Meligy
Y1 - 2014/06/30
PY - 2014
N1 - https://doi.org/10.11648/j.acm.20140303.14
DO - 10.11648/j.acm.20140303.14
T2 - Applied and Computational Mathematics
JF - Applied and Computational Mathematics
JO - Applied and Computational Mathematics
SP - 90
EP - 99
PB - Science Publishing Group
SN - 2328-5613
UR - https://doi.org/10.11648/j.acm.20140303.14
AB - The performance of six variants of the successive overrelaxation methods (SOR) are considered for an algebraic system arising from a finite difference treatment of an elliptic equation of Partial Differential Equations (PDEs) on a triangular region. The consistency of the finite difference representation of the system is achieved. In the finite difference method one obtains an algebraic system corresponding to the boundary value problem (BVP). The block structure of the algebraic system corresponding to four different labeling (the natural, the red- black and green (RBG), the electronic and the spiral) of the grid points is considered. Also, algebraic systems obtained from BVP with mixed derivatives are well established. Determination of the optimal relaxation parameters on the bases of the graphical representation of the spectral radius of the iteration matrices for the SOR, the Modified Successive over relaxation (MSOR) and their new variants KSOR, MKSOR, MKSOR1 and MKSOR2 are considered. Application of the treatment to two numerical examples is considered.
VL - 3
IS - 3
ER -

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Author Information

I. K. Youssef

Dept. of Mathematics, Faculty of Science, Ain Shams Univeristy, Abbassia, 11566, Cairo, Egypt
Sh. A. Meligy

Dept. of Engineering Mathematics and Physics, Faculty of Engineering Shoubra, Benha Univeristy, Cairo, Egypt

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APA Style

I. K. Youssef, Sh. A. Meligy. (2014). Boundary Value Problems on Triangular Domains and MKSOR Methods. Applied and Computational Mathematics, 3(3), 90-99. https://doi.org/10.11648/j.acm.20140303.14

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ACS Style

I. K. Youssef; Sh. A. Meligy. Boundary Value Problems on Triangular Domains and MKSOR Methods. Appl. Comput. Math. 2014, 3(3), 90-99. doi: 10.11648/j.acm.20140303.14

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AMA Style

I. K. Youssef, Sh. A. Meligy. Boundary Value Problems on Triangular Domains and MKSOR Methods. Appl Comput Math. 2014;3(3):90-99. doi: 10.11648/j.acm.20140303.14

Copy | Download

@article{10.11648/j.acm.20140303.14,
  author = {I. K. Youssef and Sh. A. Meligy},
  title = {Boundary Value Problems on Triangular Domains and MKSOR Methods},
  journal = {Applied and Computational Mathematics},
  volume = {3},
  number = {3},
  pages = {90-99},
  doi = {10.11648/j.acm.20140303.14},
  url = {https://doi.org/10.11648/j.acm.20140303.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140303.14},
  abstract = {The performance of six variants of the successive overrelaxation methods (SOR) are considered for an algebraic system arising from a finite difference treatment of an elliptic equation of Partial Differential Equations (PDEs) on a triangular region. The consistency of the finite difference representation of the system is achieved. In the finite difference method one obtains an algebraic system corresponding to the boundary value problem (BVP). The block structure of the algebraic system corresponding to four different labeling (the natural, the red- black and green (RBG), the electronic and the spiral) of the grid points is considered. Also, algebraic systems obtained from BVP with mixed derivatives are well established. Determination of the optimal relaxation parameters on the bases of the graphical representation of the spectral radius of the iteration matrices for the SOR, the Modified Successive over relaxation (MSOR) and their new variants KSOR, MKSOR, MKSOR1 and MKSOR2 are considered. Application of the treatment to two numerical examples is considered.},
 year = {2014}
}

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