**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30174

##### MEGSOR Iterative Scheme for the Solution of 2D Elliptic PDE's

**Authors:**
J. Sulaiman,
M. Othman,
M. K. Hasan

**Abstract:**

Recently, the findings on the MEG iterative scheme has demonstrated to accelerate the convergence rate in solving any system of linear equations generated by using approximation equations of boundary value problems. Based on the same scheme, the aim of this paper is to investigate the capability of a family of four-point block iterative methods with a weighted parameter, ω such as the 4 Point-EGSOR, 4 Point-EDGSOR, and 4 Point-MEGSOR in solving two-dimensional elliptic partial differential equations by using the second-order finite difference approximation. In fact, the formulation and implementation of three four-point block iterative methods are also presented. Finally, the experimental results show that the Four Point MEGSOR iterative scheme is superior as compared with the existing four point block schemes.

**Keywords:**
MEG iteration,
second-order finite difference,
weighted parameter.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1334564

**References:**

[1] A.R. Abdullah, "The Four Point Explicit Decoupled Group (EDG) Method: A Fast Poisson Solver". International Journal of Computer Mathematics. 38, pp. 61-70, 1991.

[2] A.R. Abdullah, and N.H.M. Ali, "A comparative study of parallel strategies for the solution of elliptic pde-s". Parallel Algorithms and Applications. 10, pp. 93-103, 1996.

[3] A. Ibrahim, and A.R. Abdullah, "Solving the two-dimensional diffusion equation by the four point explicit decoupled group (EDG) iterative method". International Journal of Computer Mathematics. 58, pp. 253- 256, 1995.

[4] A.Hadjidimos, "Accelerated OverRelaxation Method", Maths Comp., 32, pp. 149-157, 1978.

[5] D.J. Evans. "Group Explicit Iterative methods for solving large linear systems", Int. J. Computer Maths., 17, pp. 81-108, 1985.

[6] D.J. Evans and W. S. Yousif, "Explicit Group Iterative Methods for solving elliptic partial differential equations in 3-space dimensions". Int. J. Computer Maths., 18, pp. 323-340, 1986

[7] D.J. Evans and W. S. Yousif, "The Explicit Block Relaxation method as a grid smoother in the Multigrid V-cycle scheme". Int. J. Computer Maths., 34, pp. 71-78, 1990

[8] D.J. Evans, and M.S. Sahimi, "The Alternating Group Explicit iterative method (AGE) to solve parabolic and hyperbolic partial differential equations". Ann. Rev. Num. Fluid Mechanic and Heat Trans. 2, pp. 283- 389, 1988.

[9] D.M.Young, "Iterative Methods for solving Partial Difference Equations of Elliptic Type", Trans. Amer. Math. Soc.,76, pp. 92-111, 1954.

[10] D.M.Young, "Iterative solution of large linear systems". London: Academic Press, 1971.

[11] D.M.Young, "Second-degree iterative methods for the solution of large linear systems". Journal of Approximation Theory. 5, pp. 37-148, 1972.

[12] D.M.Young, Iterative solution of linear systems arising from finite element techniques. In The Mathematics of Finite Elements and Applications II, J.R. Whiteman. (Eds.), London: Academic Press. 1976, pp. 439-464..

[13] J.B. Rosser, "Nine points difference solution for Poisson-s equation". Comp. Math. Appl., 1, pp. 351-360, 1995.

[14] M. Othman, and A.R. Abdullah, "The Halfsweeps Multigrid Method As A Fast Multigrid Poisson Solver". International Journal of Computer Mathematics. 69: 219-229, 1998.

[15] M. Othman, and A.R. Abdullah, "An Effcient Multigrid Poisson Solver". International Journal of Computer Mathematics. 71, pp. 541- 553, 1999.

[16] M. Othman, and A.R. Abdullah, "An Efficient Four Points Modified Explicit Group Poisson Solver". International Journal Computer Mathematics. 76, pp. 203-217, 2000.

[17] M. Othman, A.R. Abdullah, and D.J. Evans, "A Parallel Four Point Modified Explicit Group Iterative Algorithm on Shared Memory Multiprocessors", Parallel Algorithms and Applications, 19(1), pp. 1-9, 2004. (On January 01, 2005 this publication was renamed International Journal of Parallel, Emergent and Distributed Systems).

[18] J. Sulaiman, M.K. Hasan, and M. Othman, The Half-Sweep Iterative Alternating Decomposition Explicit (HSIADE) method for diffusion equations, In Computational and Information Science, J. Zhang, J.H. He and Y.Fu (Eds). Berlin: Springer-Verlag. 2004, pp.57-63.

[19] J. Sulaiman, M. Othman, and M.K. Hasan, "Quarter-Sweep Iterative Alternating Decomposition Explicit algorithm applied to diffusion equations", International Journal of Computer Mathematics. 81(12), pp. 1559-1565, 2004.

[20] M. M.Gupta, J. Kouatchou and J. Zhang. "Comparison of second and fourth order discretizations for multigrid Poisson solvers". Journal of Computational Physics. 132(2) , pp. 226-232, 1997

[21] M.M. Martins, W.S. Yousif and D.J. Evans. "Explicit Group AOR method for solving elliptic partial differential equations". Neura Parallel, and Science Computation. 10 (4) , pp. 411-422, 2002

[22] N.H.M. Ali. "The design and analysis of some parallel algorithms for the iterative solution of partial differential equations". Ph.D Thesis. Universiti Kebangsaan Malaysia.1998.

[23] N.H.M Ali and S.T. Ling. "A comparative study of parallel strategies for the solution of elliptic pde-s", Parallel Algorithms and Applications 206, pp. 425-437, 2008.

[24] N.H.M. Ali., Y. Yunus and M. Othman. "A New Nine-Point Multigrid V-Cycle algorithm". Sains Malaysiana. 31, pp. 135-147, 2002.

[25] R.H. Shariffudin and A.R. Abdullah, "Hamiltonian circuited simulations of elliptic partial differential equations using a Spark". Applied Mathematics Letter. 14, pp. 413-418, 2001.

[26] Y. Saad, Iterative method for sparse linear systems. Boston: International Thomas Publishing, 1996.

[27] W.S. Yousif, and D. J. Evans, "Explicit de-coupled group iterative methods and their implementations". Parallel Algorithms and Applications. 7, pp. 53-71, 1995.

[28] W. Hackbusch. Iterative solution of large sparse systems of equations. New York: Springer-Verlag, 1995.