Sains Malaysiana 38(5)(2009): 729–733

An Accelerated Over-Relaxation Quarter-Sweep Point Iterative Method for

Two-Dimensional Poisson Equation

(Kaedah Lelaran Titik Suku Sapuan Pengenduran Berlebihan Terpecut

bagi Persamaan Poisson 2-Matra)

Shukhrat I. Rakhimov*

Institute for Mathematical Research, Universiti Putra Malaysia

43400 UPM Serdang, Selangor D.E., Malaysia

Mohamed Othman

Dept of Communication Tech and Network

Faculty of Computer Science and Information Technology, Universiti Putra Malaysia

43400 UPM Serdang, Selangor D.E., Malaysia

Received: 11 September 2008 / Accepted: 12 March 2009

ABSTRACT

Iterative methods, particularly over-relaxation methods, are efficiently and frequently used to solve large systems oflinear equations, because in the solutions of partial differential equations, these methods are applied to systems which are resulted from different iterative schemes to discrete equations. In this paper we formulate an accelerated over-relaxation (AOR) method with the quarter-sweep iterative scheme applied to the Poisson equation. To benchmark the new method we conducted experiments by comparing it with the previous AOR methods based on full- and half-sweep iterative schemes. The results of the experiments and the estimation of the computational complexity of the methods proved the superiority of the new method.

Keywords: Accelerated over-relaxation; point iterative methods; Poisson equation

ABSTRAK

Kaedah lelaran, khususnya kaedah pengenduran berlebihan, kerap digunakan secara cekap bagi menyelesaikan sistem persaman linear yang besar, kerana dalam penyelesaian persamaan pembezaan separa, kaedah lelaran ini digunakan ke atas sistem yang terhasil daripada skema lelaran berbeza untuk mengdiskretkan persamaan. Dalam kertas kerja ini, kami memformulasikan kaedah pengenduran berlebihan tercepat (PBT) dengan menggunakan pendekatan sapuan suku dan mengaplikasikannya kepada sistem persamaan Poisson. Sebagai penanda aras kaedah terbaru ini, kami melakukan beberapa kajian dan membandingkan dengan kaedah PBT terdahulu yang menggunakan pendekatan lelaran sapuan penuh dan separuh. Hasil ujikaji tersebut dan anggaran pengiraan kekompleksitian bagi semua kaedah tersebut telah membuktikan keberkesanan kaedah baru ini.

Kata kunci: Kaedah lelaran titik; pengenduran berlebihan terpecut; persamaan Poisson

REFERENCES

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Othman, M. & A. R. Abdullah. 1998. A new technique point iterative method for solving poisson equation on MIMD computer system. Sains Ma1aysiana 27: 107-117.

Othman, M. & A. R. Abdullah. 2000. An Efficient parallel quarter-sweep point iterative algorithm for solving poisson equation on SMP parallel computer. Pertanika J. Sci & Technol. 8(1): 161-174.

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*Corresponding author; email: sh.rakhimov@gmail.com