Applied and Computational Mathematics
Volume 3, Issue 3, June 2014, Pages: 90-99
Received: May 29, 2014;
Accepted: Jun. 16, 2014;
Published: Jun. 30, 2014
Views 3539 Downloads 253
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
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.
I. K. Youssef,
Sh. A. Meligy,
Boundary Value Problems on Triangular Domains and MKSOR Methods, Applied and Computational Mathematics.
Vol. 3, No. 3,
2014, pp. 90-99.
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.
D. M. Young, “Iterative Solution of Large Linear Systems,”Academic Press, London, 1971.
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.
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.
G. Birkhoff, “The Numerical Solution of Elliptic Equations,” Society for Industrial and Applied Mathematics, 1972.
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.
G. D. Smith, “Numerical Solution of Partial Differential Equations,” Oxford university press, 1985.
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.
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.
D. J. Evans, “The Solution of Poisson’s Equation in a triangular region,”International Journal of Computer Math., Vol. 39 pp. 81 – 98,1991.
I. K. Youssef, “On the Successive Overrelaxation Method,” Journal of Mathematics and Statistics 8 (2): 176-184, 2012.
I. K. Youssef and A. A. Taha, “On the Modified Successive Overrelaxation Method,” Applied Mathematics and Computation 219, pp. 4601-4613, 2013.