A comparative study of explicit group iterative solvers on a cluster of workstations

Abstract

In this paper, a group iterative scheme based on rotated (cross) five-point finite difference discretisation, i.e. the four-point explicit decoupled group (EDG) is considered in solving a second order elliptic partial differential equation (PDE). This method was firstly introduced by Abdullah [“The four point EDG method: a fast poisson solver”, Int. J. Comput. Math., 38 (1991) 61–70], where the method was found to be more superior than the common existing methods based on the standard five-point finite difference discretisation. The method was further extended to different type of PDE's, where similar improved results were established [Ali, N.H.M., Abdullah, A.R. Four Point EDG: A Fast Solver For The Navier–Stokes Equation, M.H.Hamza (ed.) Proceedings of the IASTED International Conference on Modelling Simulation And Optimization, Gold Coast, Australia, May 6–9 (1996) (CD Rom-File 242-165.pdf), ISBN: 0-88986-197-8; Ali, N.H.M., Abdullah, A.R. New Parallel Point Iterative Solutions For the Diffusion-Convection Equation Proceedings of the International Conference on Parallel and Distributed Computing and Networks Singapore, Aug. 11–13 (1997) 136–139; Ali, N.H.M., Abdullah, A.R. “New rotated iterative algorithms for the solution of a coupled system of elliptic equations” Int. J. Comput. Math. 74 (1999) 223–251]. These new iterative algorithms had been developed to run on the Sequent Balance, a shared memory parallel computer [A.R. Abdullah, N.M. Ali, The Comparative Study of Parallel Strategies For The Solution of Elliptic PDE's Parallel Algorithms and Applications Vol. 10 (1996) 93–103; Ali, N.H.M., Abdullah, A.R. “Parallel four point explicit decoupled group (EDG) method for elliptic PDE's” Proceedings of the Seventh IASTED/ISMM International Conference on Parallel and Distributed Computing and Systems (1995) 302–304 (Washington DC); Ali, N.H.M., Abdullah, A.R. New Parallel Point Iterative Solutions For the Diffusion-Convection Equation Proceedings of the International Conference on Parallel and Distributed Computing and Networks, Singapore, Aug. 11–13 (1997) 136–139; Yousif, W.S., Evans, D.J.“Explicit decoupled group iterative methods and their parallel implementations” Parallel Algorithms and Applications 7 (1995) 53–71] where they were shown to be suitable to be implemented in parallel. In this work, the four-point group algorithm was ported to run on a cluster of Sun workstations using a parallel virtual machine (PVM) programming environment together with the four-point explicit group (EG) method [Evans, D.J., Yousif, W.S. “The implementation of the explicit block iterative methods on the balance 8000 parallel computer” Parallel Computing 16 (1990) 81–97]. We describe the parallel implementations of these methods in solving the Poisson equation and the results of some computational experiments are compared and reported. rosni@cs.usm.my kokjl@hotmail.com