A Fast, Parallel Performance of Fourth Order Iterative Algorithm on Shared Memory Multiprocessors (SMP) Architecture
The rotated fourth order iterative algorithm of O(h 4) accuracy which was applied to the linear system was introduced by Othman et al. [OTH01] and it was shown to be the fastest compared to the standard fourth order iterative algorithm. Meanwhile the para
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Department of Communication Technology and Network, University Putra Malaysia, 43400 UPM Serdang, Selangor D.E., Malaysia [email protected] School of Science and Technology, Universiti Malaysia Sabah, Locked Bag 2073, 88999 Kota Kinabalu, Sabah, Malaysia [email protected]
Abstract The rotated fourth order iterative algorithm of O(h4 ) accuracy which was applied to the linear system was introduced by Othman et al. [OTH01] and it was shown to be the fastest compared to the standard fourth order iterative algorithm. Meanwhile the parallel standard fourth order iterative algorithms with difference strategies were implemented successfully by many researchers for solving large scientific and engineering problems. In this paper, the implementation of the parallel rotated fourth order iterative algorithm on SMP architecture is discussed. The performance results of all the parallel algorithms were compared in order to show their outstanding performances.
1 Introduction The parallel fourth order iterative algorithm which incorporate the standard fourth order scheme known as compact high order scheme for solving a large and sparse linear system was implemented successfully by many researchers, see [YOU95, SPO98]. One of the most outstanding parallel algorithm was proposed by Spotz, et al., [SPO98]. Theoretically, the standard fourth order scheme of O(h4 ) was derived by Coltaz, [COL60]. Based on the scheme, several experiments were carried out and the results showed it has high accuracy, [GUP84]. In 2001, Othman et al. derived a new nine points scheme, known as a rotated fourth order scheme for solving the 2D Poisson’s equation. From the experimental results, they found that the new scheme ∗
The author is also an associate researcher at the Laboratory of Computational Sciences and Informatics, Institute of Mathematical Science Research (INSPEM), University Putra Malaysia
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M. Othman and J. Sulaiman
has a drastic improvement in execution time and relatively good accuracy compared to the standard fourth order scheme, see [OTH01, ALI02, ZHA02].
2 Derivation of A Rotated Fourth Order Scheme Let us consider the 2D Poisson’s equation, which can be represented mathematically as (1) uxx + uyy = f (x, y), for all (x, y) ∈ Ω h subject to the Dirichlet boundary condition and satisfying the exact solution, u(x, y) = g(x, y) for (x, y) ∈ Ω h . The discretization resulted in a large and sparse linear system. Hence, the iterative method is the best approach for solving such a linear system. Consider Eq. (1) on a unit square, Ω h with the grid spacing h in both directions, xi = x0 +ih and yj = y0 +jh for all i, j = 0, 1, . . . , n. Assume that due to the continuity of u(x, y) on Ω h . Based on the cross orientation approximation and central difference formula, the displacements i and j which correspond √ with ∆x and ∆y respectively change to 2h. Eq. (2) can be approximated at any points u(xi , yj ) using the finite difference formula and yields ui+1,j+1 + ui−1,j+1 + ui+1,j−1 + ui−1,j−1 + 4ui,j ∼ = 2h2 (uxx + uyy )i,j h4 + 6 (uxxx
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