Fractional Steps methods for transient problems on commodity computer architectures

Physics of the Earth and Planetary Interiors

Volumn 171, Issue 1-4, 2008, Pages 122-136

  Krotkiewski, M ^a   Dabrowski, M ^a   Podladchikov, Y Y ^a  

^a UNIVERSITY OF OSLO   (Norway)

Author keywords

ADI; Commodity; Fractional Steps; Hyperbolic; Locally One Dimensional; Parabolic

Indexed keywords

CARBON FIBER REINFORCED PLASTICS; COMPUTATIONAL COMPLEXITY; COMPUTER ARCHITECTURE; COMPUTER SCIENCE; COMPUTER SYSTEMS; COMPUTERS; INCLUSIONS; ONE DIMENSIONAL; THERMAL CONDUCTIVITY; THERMOELECTRICITY; THREE DIMENSIONAL;

ADI; COMMODITY; FRACTIONAL STEPS; HYPERBOLIC; LOCALLY ONE-DIMENSIONAL; PARABOLIC;

PROBLEM SOLVING;

COMPUTER SYSTEM; FLUID FLOW; RESERVOIR; SOFTWARE; THERMAL CONDUCTIVITY; THREE-DIMENSIONAL MODELING;

EID: 55649084808     PISSN: 00319201     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.pepi.2008.08.008     Document Type: Article

References (39)

1. Allmann S., Rauber T., and Runger G. Cyclic reduction on distributed shared memory machines. Proceedings of the Ninth Euromicro Workshop on Parallel and Distributed Processing. Mantova, Italy (2001) 290-297
2. Birkhoff G., and Varga R.S. Implicit alternating direction methods. Transactions of the American Mathematical Society 92 1 (1959) 13
3. Brandt A. Multilevel adaptive solutions to boundary-value problems. Mathematics of Computation 31 138 (1977) 333-390
4. Browne S., Dongarra J., Garner N., Ho G., and Mucci P. A portable programming interface for performance evaluation on modern processors. International Journal of High Performance Computing Applications 14 3 (2000) 189-204
5. Bruggeman D.A.G. Calculation of various physical constants of heterogenous substances. II. Dielectricity constants and conductivity of non regular multi crystal systems. Annalen Der Physik 25 7 (1936) 0645-0672
6. Conte S.D., and Boor C.W.D. Elementary Numerical Analysis: An Algorithmic Approach (1980), McGraw-Hill Higher Education 408 pp.
7. Courant R., Friedric K., and Lewy H. On partial difference equations of mathematical physics. IBM Journal of Research and Development 11 2 (1967) 215
8. Dabrowski M., Krotkiewski M., and Schmid D.W. MILAMIN: MATLAB-based finite element method solver for large problems. Geochemistry Geophysics Geosystems (2008) 9
9. Douglas J. On the numerical integration of D2U - Dx2 + D2U - Dy2 = Du - D + implicit methods. Journal of the Society for Industrial and Applied Mathematics 3 1 (1955) 42-65
10. Douglas J. Alternating direction methods for three space variables. Numerische Mathematik 4 1 (1962) 41
11. Douglas Jr. J., Garder A.O., and Pearcy C. Multistage alternating direction methods. SIAM Journal on Numerical Analysis 3 4 (1966) 570
12. Douglas Jr. J., and Rachford Jr. H.H. On the numerical solution of heat conduction problems in two and three space variables. Transactions of the American Mathematical Society 82 2 (1956) 421
13. Douglas J., and Kim S. Improved accuracy for locally one-dimensional methods for parabolic equations. Mathematical Models & Methods in Applied Sciences 11 9 (2001) 1563-1579
14. Egecioglu O., Koc C.K., and Laub A.J. Recursive doubling algorithm for solution of tridiagonal systems on hypercube multiprocessors. Journal of Computational and Applied Mathematics 27 (1989) 95-108
15. Eshelby J.D. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Royal Society of London Series A: Mathematical and Physical Sciences 241 1226 (1957) 376-396
16. Gander W., and Golub G.H. Cyclic reduction-history and applications. Proceedings of the Workshop on Scientific Computing (1997), Springer-Verlag, Hong Kong
17. Hackbusch W. Fast numerical-solution of time-periodic parabolic problems by a multigrid method. SIAM Journal on Scientific and Statistical Computing 2 2 (1981) 198-206
18. Hockney R.W. A fast direct solution of Poissons equation using Fourier analysis. Journal of the ACM 12 1 (1965) 95-&
19. Hofhaus J., and VandeVelde E.F. Alternating-direction line-relaxation methods on multicomputers. Siam Journal on Scientific Computing 17 2 (1996) 454-478
20. Janenko N.N. The Method of Fractional Steps the Solution of Problems of Mathematical Physics in Several Variables vol. VIII (1971), Springer, Berlin 160 pp.
21. Kamil S., Husbands P., Oliker L., Shalf J., and Yelick K. Impact of modern memory subsystems on cache optimizations for stencil computations. Proceedings of the 2005 Workshop on Memory System Performance (2005), ACM, Chicago, IL
22. Kurzak J., and Dongarra J. Implementation of mixed precision in solving systems of linear equations on the cell processor. Concurrency and Computation-Practice & Experience 19 10 (2007) 1371-1385
23. Langou J., et al. Exploiting the performance of 32 bit floating point arithmetic in obtaining 64 bit accuracy (revisiting iterative refinement for linear systems). Proceedings of the 2006 ACM/IEEE conference on Supercomputing (2006), ACM, Tampa, Florida
24. Larsson S., Thomee V., and Zhou S.Z. On multigrid methods for parabolic problems. Journal of Computational Mathematics 13 3 (1995) 193-205
25. Lubich C., and Ostermann A. Multigrid dynamic iteration for parabolic equations. BIT 27 2 (1987) 216-234
26. Marchuk, G.I., 1990. Splitting and Alternating Direction Methods. Handbook of Numerical Analysis, vol. 1. Elsevier, Amsterdam, pp. 197-462.
27. McCalpin, J.D., 1991-2007. STREAM: Sustainable Memory Bandwidth in High Performance Computers. A Continually Updated Technical Report.
28. Overman A., and Rosendale J.V. Mapping robust parallel multigrid algorithms to scalable memory architectures. In: McCormick N.D.M.a.T.A.M.a.S.F. (Ed). Sixth Copper Mountain Conference on Multigrid Methods (1993), NASA, Hampton, VA 635-647
29. Peaceman D.W., and Rachford H.H. The numerical solution of parabolic and elliptic differential equations. Journal of the Society for Industrial and Applied Mathematics 3 1 (1955) 28-41
30. Pearcy C. On convergence of alternating direction procedures. Numerische Mathematik 4 1 (1962) 172
31. Povitsky A. Parallelization of pipelined algorithms for sets of linear banded systems. Journal of Parallel and Distributed Computing 59 1 (1999) 68-97
32. Rivera G., and Tseng C.-W. Tiling optimizations for 3D scientific computations. Proceedings of the 2000ACM/IEEE conference on Supercomputing (CDROM) (2000), IEEE Computer Society, Dallas, TX, United States
33. Stone H.S. An efficient parallel algorithm for the solution of a tridiagonal linear system of equations. Journal of ACM 20 1 (1973) 27-38
34. Strzodka R., and Göddeke D. Mixed precision methods for convergent iterative schemes. Proceedings of the 2006 Workshop on Edge Computing Using New Commodity Architectures (2006)
35. Wakatani A. A parallel and scalable algorithm for ADI method with pre-propagation and message vectorization. Parallel Computing 30 12 (2004) 1345-1359
36. Weber L., Dorn J., and Mortensen A. On the electrical conductivity of metal matrix composites containing high volume fractions of non-conducting inclusions. Acta Materialia 51 11 (2003) 3199-3211
37. Wesseling P. An Introduction to Multigrid Methods (2004)
38. Whaley R.C., Petitet A., and Dongarra J.J. Automated empirical optimization of software and the ATLAS project. Parallel Computing 27 1/2 (2001) 3-35
39. Widlund O.B. On the rate of convergence of an alternating direction implicit method in a noncommutative case. Mathematics of Computation 20 96 (1966) 500