Sains Malaysiana 31: 135-147 (2002)                                                                           Pengajian Kuantitatif /

Quantitative Studies

A New Nine-Point Multigrid V-Cycle Algorithm

Norhashidah Hj. Mohd. Ali Yuzaimi Yunus

School of Mathematical Sciences

Universiti Sains Malaysia

11800 Minden, Pulau Pinang

Mohamed Othman

Department of Computer Science

Universiti Putra Malaysia

43400 UPM Serdang

Selangor D.E., Malaysia

ABSTRACT

A new multigrid scheme using half sweep nine-point finite difference approxi­mation in solving the two dimensional Poisson equation is presented. The concept of half sweep multigrid was initiated by Othman and Abdullah (1997) where promising results was established and confirmed. The five-point method was shown to be very much faster compared to the fullsweep multigrid method due to Gupta et al. (1995). In this paper, we apply the multigrid V-cycle algorithm on the nine-point finite difference approximation derived from the rotated nine-point stencil (Ali & Abdullah 1998). This nine­-point finite difference approximation has been proven to be a viable Poisson solver with second order accuracy. Using different grid sizes, the efficiency of this multigrid scheme is compared with the fullsweep multigrid derived from the standard nine-point stencil (Adams et al. 1988) in terms of execution times and maximum error.

ABSTRAK

Satu skema multigrid baru menggunakan penganggaran beza terhingga separuh sapuan sembilan titik dalam menyelesaikan persamaan Poisson berdimensi dua adalah dibentangkan. Konsep multigrid setengah sapuan ini telah dipelopori oleh Othman dan Abdullah (1997) di mana keputusan yang memberangsangkan telah dibentuk dan disahkan. Kaedah lima-titik ini telah ditunjukkan lebih pantas dibandingkan dengan kaedah multigrid sapuan penuh oleh Gupta et al. (1995). Dalam kertas ini, kita mengaplikasikan algoritma kitar- V multigrid pada penganggaran beza terhingga sembilan-­titik yang diterbitkan dari stensil sembilan-titik putaran (Ali & Abdullah 1998). Penganggaran beza terhingga sembilan-titik ini telah dibuktikan sebagai satu penyelesai Poisson yang berupaya dengan kejituan peringkat dua. Dengan menggunakan saiz grid berbeza, keefisienan skema multigrid ini dibandingkan dengan multigrid sapuan penuh yang diterbitkan dari stensil sembilan-titik piawai (Adams et al. 1988) dari segi masa perlaksanaan dan ralat maksimum.

RUJUKAN/REFERENCES

Adams, L.M., Leveque, R.J. & Young, D.M. 1988, Analysis of the SOR Iteration for the 9-Point Laplacian. SIAM Journal of Numerical Analysis 25 (5): 1156-1180.

Ali, N.H.M & Abdullah, A.R. 1998. New Parallel Nine-Point Poisson Solver. Journal of Information Technology 10 (2): 1-9

Bramble, J.H. 1995. Multigrid Methods. Pitman Research Notes in Mathematics Series. Great Britain: Longman Scientific & Technical.

Briggs, L. William. 1987. A Multigrid Tutorial. Society for Industrial and Applied Mathematics Philadelphia, Pennsylvania.

Dahlquist, G. & Bjorck, A. 1974. Numerical Methods. New York: Englewood Cliffs, Prentice Hall.

Gupta, M.M., Kouatchou, J. & Zhang, J. 1995. 'Comparison of 2nd and 4th Order Discretizations for Multigrid Poisson Solvers', http://www.cs.yale.edu directory mgnet/papers/Gupta.

Hackbush, W. 1984. Parabolic Multi-grid Methods. Computing Methods in Applied Sciences and Engineering, VI. North Holland, Amsterdam

Hackbush,W & Trottenberg, U. 1991. Multigrid Method III. International Series of Numerical Mathematics, vo1. 98. Basel: Birkhauser Verlag.

Othman, M. &Abdullah A.R. 1997. The Halfsweeps Multigrid Method As A Fast Multigrid Poisson Solver. Int. J. Comp. Math. 69: 319-329

Yousif, W.S., 1984, New Block Iterative Methods for the Numerical Solution of Boundary Value Problems. PhD Thesis, Loughborough University of Technol­ogy.