Abstract
In this paper we introduce an iterative scheme for solving the Convection Diffusion equation. In fact, the method can be applied to any Partial Differential Equation which uses iterative schemes for their numerical solution. Parallelism is introduced by decoupling the mesh points with the use of red—black ordering for the 5—point stencil. The optimum set of values for the parameters involved, when the Jacobi iteration operator possesses real or imaginary eigenvalues, is determined. The performance of the method is illustrated.by its application to the numerical solution of the convection diffusion equation. It is found that the proposed method is significantly more efficient than local SOR when the absolute value of the smallest eigenvalue of the Jacobi operator is larger than unity. Finally, the parallel implementation of the method is discussed and results are presented for distributed memory processors with a mesh topology.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adams, L. M., Leveque, R. J. and Young, D. (1988), Analysis of the SOR iteration for the 9-point Laplacian, SIAM J. Num. Anal., 9, pp. 1156–1180.
Botta, E. F. and Veldman, A. P. (1981), On local relaxation methods and their application to convection-diffusion equations, J. Comput. Phys., 48, pp. 127–149.
Boukas, L. A. (1998), Parallel Iterative Methods for Solving Partial Differential Equations on MIMD Multiprocessors, Ph.D Thesis, Department of Informatics, University of Athens, Athens.
Boukas, L. A. and Missirlis, N. M. (1998), Convergence theory of Extrapolated Iterative Methods for Solving the Convection Diffusion Equation, Technical Report, Department of Informatics, University of Athens, Athens.
Boukas, L. A. and Missirlis, N. M. (1998), A parallel local modified SOR for nonsymmetric linear systems, Intern. J. of Comput. Math., 68, pp.153–174.
Brandt, A. (1977), Multi—level adaptive solutions to boundary—value problems, Math. Comput., 31, pp. 333–390.
Chan, T F and Elman, H. C. (1989), Fourier analysis of iterative methods for solving elliptic problems, SIAM Review, 31, pp. 20–49.
Ehrlich, L. W. (1981), An Ad-Hoc SOR method, J. Comput. Phys., 42, pp. 31–45.
Ehrlich, L. W. (1984), The Ad-Hoc SOR method: A local relaxation scheme, in Elliptic Problem Solvers II, Academic Press, New York, pp. 257–269.
Hageman, L. A. and Young, D. M. (1981), Applied Iterative Methods, Academic Press, New York.
Kuo, C.-C. J., Levy B. C. and Musicus, B. R. (1987) A local relaxation method for solving elliptic PDE’s on mesh-connected arrays, SIAM J. Sci. Statist. Comput., 8, pp. 530–573.
Missirlis, N. M. (1984), Convergence theory of Extrapolated Iterative methods for a certain class of non—symmetric linear systems, Numer. Math., 45, pp. 447–458.
Niethammer, W. (1979), On different splittings and the associated iteration method, SIAM J. Numer. Anal. 16, pp. 186–200.
Ortega, J. M. and Voight, R. G. (1985), Solution of Partial Differential Equations on Vector and Parallel Computers, SIAM, Philadelphia.
Varga, R. S. (1962), Matrix Iterative Analysis,Prentice-Hall, Englewood Cliffs, NJ.
Young, D. M. (1971), Iterative Solution of Large Linear Systems, Academic Press,New York.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Boukas, L.A., Missirlis, N.M. (1999). A Parallel Iterative Scheme for Solving the Convection Diffusion Equation on Distributed Memory Processors. In: Zlatev, Z., et al. Large Scale Computations in Air Pollution Modelling. NATO Science Series, vol 57. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4570-1_8
Download citation
DOI: https://doi.org/10.1007/978-94-011-4570-1_8
Publisher Name: Springer, Dordrecht
Print ISBN: 978-0-7923-5678-3
Online ISBN: 978-94-011-4570-1
eBook Packages: Springer Book Archive