MEGSOR Iterative Scheme for the Solution of 2D Elliptic PDE's
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.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:56942", author = "J. Sulaiman and M. Othman and M. K. Hasan", title = "MEGSOR Iterative Scheme for the Solution of 2D Elliptic PDE's", 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.", volume = "4", number = "2", pages = "258-7", }
