A parallel DSDADI method for solution of the steady state diffusion equation

Parallel Computing

Volumn 23, Issue 13, 1997, Pages 2041-2065

  Lambert, Michael A ^a,b   Rodrigue, Garry H ^a,b   Hewett, Dennis W ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b LAWRENCE LIVERMORE NATIONAL LABORATORY   (United States)

Author keywords

ADI method; Conjugate gradient method; CRAY T3D; Distributed memory; Domain decomposition; Elliptic PDE; MIMD; Parallel processors; Tridiagonal system

Indexed keywords

DATA STORAGE EQUIPMENT; DATABASE SYSTEMS; LEAST SQUARES APPROXIMATIONS; PARALLEL ALGORITHMS; PARTIAL DIFFERENTIAL EQUATIONS; PROBLEM SOLVING; STORAGE ALLOCATION (COMPUTER);

CARTESIAN GRIDS; DIAGONALLY SCALED CONJUGATE GRADIENTS (DSCG); DIAGONALLY SCALED DYNAMIC ALTERNATING DIRECTION IMPLICIT (DSDADI); DISTRIBUTED MEMORY MULTIPROCESSORS; SPATIAL DOMAIN DECOMPOSITION (SDD);

PARALLEL PROCESSING SYSTEMS;

EID: 0031336277     PISSN: 01678191     EISSN: None     Source Type: Journal    
DOI: 10.1016/s0167-8191(97)00065-3     Document Type: Article

References (28)

1. G. Arfken, Mathematical Methods for Physicists, 3rd ed., Academic Press, New York, 1985.
2. R.F. Boisvert, Algorithms for special tridiagonal systems, SIAM J. Sci. Stat. Comput. 12 (2) (1991) 423-442.
3. S. Bondeli, Divide and conquer: A parallel algorithm for the solution of a tridiagonal linear system of equations, Parallel Comput. 17 (1991) 419-434.
4. P.L.T. Brian, A finite-difference method of high-order accuracy for the solution of three dimensional transient heat conduction problems, AIChE J. 7 (3) (1961) 367-370.
5. L. Brugnano, A parallel solver for tridiagonal linear systems for distributed memory parallel computers, Parallel Comput. 17 (1991) 1017-1023.
6. CRAY T3D System Architecture Overview, Rev. 1B, Cray Research Inc., Chippewa Falls, WI.
7. S. Doss, K. Miller, Dynamic ADI methods for elliptic methods, SIAM J. Numer. Anal. 16 (5) (1979) 837-856.
8. J. Douglas Jr., Alternating direction methods for three space variables, Num. Math. 4 (1963) 41-63.
9. C.F. Fischer, R.A. Usmani, Properties of some tridiagonal matrices and their application to boundary value problems, SIAM J. Numer. Anal. 6 (1) (1969) 127-143.
10. E. Gallopoulos, Y. Saad, Efficient Solution of Parabolic Equations by Krylov Approximation Methods, Technical Report 90.14, Research Institute for Advanced Computer Science, NASA Ames, March 1990.
11. A. Greenbaum, C. Li, H.Z. Chao, Parallelizing preconditioned conjugate gradient algorithms, Comput. Phys. Commun. 53 (1989) 295-309.
12. A. Hadjidimos, On a generalised alternating direction implicit method for solving Laplace's equation, Comp. J. 11 (1968) 324-328.
13. J. Hofhaus, E. Van De Velde, Alternating-direction line-relaxation methods on multicomputer, SIAM J. Sci. Comp. 17 (2) (1996) 454-478.
14. S.L. Johnsson, Y. Saad, M.H. Schultz, Alternating direction methods on multiprocessors, SIAM J. Sci. Stat. Comput. 8 (5) (1987) 686-700.
15. A. Krechel, H. Plum, K. Stuben, Parallelization and vectorization aspects of the solution of tridiagonal linear systems, Parallel Comput. 15 (1990) 31-49.
16. S.P. Kumar, Solving tridiagonal systems on the butterfly parallel computer, Int. J. Supercomputing App. 3 (1) (1989) 75-81.
17. N. Mattor, T.J. Williams, D.H. Hewett, Algorithm for solving tridiagonal matrix problems in parallel, Parallel Comput. 21 (1995) 1769-1782.
18. G. Meurant, A review on the inverse of symmetric tridiagonal and block tridiagonal matrices, SIAM J. Matrix Anal. Appl. 13 (3) (1992) 707-728.
19. D.W. Peaceman, H.H. Rachford Jr., The numerical solution of parabolic and elliptic differential equations, SIAM J. 3 (1955) 28-41.
20. G.H. Rodrigue, N. Madsen, J. Karush, Odd-even reduction for banded linear equations, J. ACM 26 (1) (1979) 72-81.
21. G. Rodrigue, P. Dubois, An Analysis of the Recursive Doubling Algorithm, Proceedings of the Conference on High Speed Computing and Applications, Academic Press, New York, 1977.
22. R.A. Sweet, A cyclic reduction algorithm for solving block tridiagonal systems of arbitrary dimensions, SIAM J. Numer. Anal. 14 (4) (1977) 706-720.
23. H.A. Van der Vorst, Analysis of a parallel solution method for tridiagonal linear systems, Parallel Comput. 5 (1987) 303-311.
24. R.F. Van der Wijngaart, Efficient Implementation of a 3-Dimensional ADI Method on the iPSC/860, Supercomputing '93 Proceedings, Portland, OR, Nov. 15-19, IEEE Computer Society Press, pp. 102-111.
25. R.S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1962.
26. E. Wachspress, Optimum alternating direction implicit iteration parameters for a model problem, J. Soc. Ind. Appl. Math. 10 (1962) 339-350.
27. E. Wachspress, G. Habetler, An alternating-direction-implicit iteration technique, SIAM J. 8 (2) (1960) 403-424.
28. H.H. Wang, A parallel method for tridiagonal equations, ACM Trans. Math. Software 7 (2) (1981) 170-183.