Sains Malaysiana 42(6)(2013): 837–844

Investigation of Steady State Problems via Quarter Sweep Schemes

(Kajian Terhadap Masalah Berkeadaan Mantap Menggunakan Skema Sapuan Suku)

Ng Yit Hoe* & Mohammad Khatim Hasan

School of Information Technology, Faculty of Information Science and Technology

Universiti Kebangsaan Malaysia, 43650 Bangi, Selangor D.E. Malaysia

Received: 19 July 2012/Accepted: 14 November 2012

ABSTRACT

BSTRACT

Numerical application helps researchers in simulating various problems and used for solving partial differential equation. Half sweep and quarter sweep approach have been applied onto iterative method to gain approximation solution. In this paper, the implementation of full sweep successive over relaxation (FSSOR), half sweep successive over relaxation (HSSOR) and quarter sweep successive over relaxation (QSSOR) methods and full sweep accelerated over relaxation (FSAOR), half sweep accelerated over relaxation (HSAOR) and quarter sweep accelerated over relaxation (QSAOR) for its numerical engines are shown. QSSOR and QSAOR method was the fastest among FSSOR, HSSOR, FSAOR and HSAOR methods. Additionally, QSAOR performance is more accurate than QSSOR.

Keywords: Elliptic problem; iterative scheme; numerical simulation; Poisson; quarter sweep approach

ABSTRAK

Aplikasi berangka membantu penyelidik dalam mensimulasikan pelbagai masalah dan digunakan untuk menyelesaikan persamaan terbitan separa. Pendekatan sapuan separuh dan sapuan suku telah digunakan ke atas kaedah lelaran untuk mendapatkan penyelesaian penghampiran. Dalam kertas ini, implementasi kaedah-kaedah pengenduran berlebihan berturut-turut sapuan penuh (FSSOR), pengenduran berlebihan berturut-turut sapuan separuh (HSSOR) dan pengenduran berlebihan berturut-turut sapuan suku (QSSOR) dan pemecutan berlebihan berturut-turut sapuan penuh (FSAOR), pemecutan berlebihan berturut-turut sapuan separuh (HSAOR) dan pemecutan berlebihan berturut-turut sapuan suku (QSAOR) untuk enjin berangka ditunjukkan. Kaedah QSSOR dan QSAOR adalah terpantas dalam kalangan kaedah-kaedah FSSOR, HSSOR, FSAOR dan HSAOR. Di samping itu, prestasi QSAOR adalah lebih tepat daripada QSSOR.

Kata kunci: Masalah eliptik; pendekatan sapuan suku; poisson; simulasi berangka; skim lelaran

REFERENCES

Ali, N.H. & Evans, D.J. 2004. Preconditioned rotated iterative methods in the solution of elliptic partial differential equation. International Journal of Computer Mathematics 81(9): 1163-1174.

Ali, N.H. & Sam, T.L. 2006. On implementing several krylov preconditioned iterative schemes on rotated grid. 2nd Annual Proceedings of the IMT-GT Regional Conference on Mathematics, Statistics and Application.

Hadjidimos, A. 1978. Accelerated overrelaxation method. Mathematics of Computation 32: 149-157.

Lee, S.C. 2007. Preconditioned 5 rotated scheme in solving the Poisson’s equation. 17th Annual Proceedings of the Simposium Kebangsaan Sains Matematik pp.154-159.

Moussavi, S. 2009. One step closer to an optimal two-parameter SOR method - Ageometric approach. Journal of Mathematical Sciences 21(1): 56-64.

Othman, M., Rakhimov, S.I. & Sulaiman, J. 2009. An AOR four points modified explicit group iterative algorithms for solving 2D Poisson equation. 5th Annual Proceedings of the Asian Mathematical Conference pp. 276-279.

Sulaiman, J., Othman, M. & Hasan, M.K. 2006. An optimal ordering strategy of the point iterative algorithm for solving 2D convection-diffusion problem. Borneo Science 18: 1-8.

Sulaiman, J., Othman, M. & Hasan, M.K. 2008a. Half-sweep algebraic multigrid (HSAMG) method applied to diffusion equations in modeling. Simulation and Optimization of Complex Processes, Springer-Verlag, Berlin Heidelberg. pp. 547-556.

Sulaiman, J., Othman, M. & Hasan, M.K. 2007. Red-black EDGSOR iterative method using triangle finite element approximation for 2D poisson equations. LNCS 4707 Part III: 298-308.

Sulaiman, J., Saudi, A., Abdullah, M.H., Hasan, M.K. & Othman, M. 2008b. Quarter-sweep arithmetic mean algorithm for water quality model, Proceeding International Symposium on Information Technology 3: 1859-1863.

Wang, G., Wen, H., Li, L. & Li, X. 2011. Convergence of GAOR method for doubly diagonally dominant matrices. Journal of Applied Mathematics and Computation 217: 7509-7514.

Young, D.M. 1950. Iterative methods for solving partial difference equations of elliptic type, APh.D. Thesis of Historical Importance, Harvard University Cambrige. Mass (unpublished).

*Corresponding author; email: fido_ng87@yahoo.com