h-Refined Schwarz-Krylov Solution for Hydrodinamics and Mass Transport in a PC Cluster
Resumo
This work presents a parallel solution, implemented in a PC cluster, using MPI library, to the simulation of hydrodynamics and mass transport in the Rio Guaiba. The governing equations of hydrodynamics and scalar transport of substances are defined in space-staggered grids, where the finer grid is nested in the coarser grid and built through interpolation. The PDEs are discretized using finite differences using upwind and centered difference techniques to generate a semi-implicit numerical scheme. To build the local subproblems a version of bissection algorithm was used to partitionate the numerical grid, and Schwartz additive method was used to build the overlapping Local solutions are obtained using Krylov subspace iterative methods.
Palavras-chave:
Semi-implicit scheme, local refinement, grid partitioning, Schwarz-Krylov method, PC Cluster
Referências
BOTI, A. Monotone Flux Limitation in the Area-preserving Flux-Form Advection Algorithm. Monthly Weather Review, vol. 120, November 1992, pp. 2592-2602.
CASULLI, Vincenzo. Semi-Implicit Finite Difference Methods for the Two-Dimensional Shallow Water Equations. Journal of Computational Physics, v.86. p.56-74, 1990.
CASULLI, V., CATIANl, E., Stability, Accuracy and Efficiency of a Semi-Implicit Method for Three-Dimensional Shallow Water Flow. Computers Math. Applic., vol. 27, nr. 4, pp. 99-112, 1994.
CHAN, T. .F.; MATHEW, T. P. Domain Decomposition Algorithms. Acta Numerica, p.61-143. 1994.
CHENG, R. Eulerian-Lagrangian Solution of the Convection-Dispersion Equation in Natural Coordinates. Water Resources Research, v.20. p.944-952. July 1984.
DEBREU, L., BLA YO, E. On the Schwarz Alternating Method for Oceanic Models on Parallel computers. Journal of computacional Physics, V. 141, p. 93-111. 1998.
DORNELES, R. V., RIZZI, R. L., ZEFERINO, C. A., DIVERIO, T. A., NAVAUX, P. O. A., BAMPl, S, SUZIN, A. A. A PC cluster Implementation of a Mass Transport Two Dimensional Model in: XII SBAC-PAD. 2000, São Pedro.
DORNELES, R. V. Particionamento de Domínio e Balanceamento Dinâmico de Carga em Arquiteturas Heterogêneas: Aplicação a Modelos Hidrodinâmicos e de Transporte de Massa 2-D e 3-D. Proposta de Tese. PPGC-UFRGS. 2001.
GROSS, E. S., CASULLI, V., BONAVENTURA, L., KOSEFF, JEFFREY. A Semi-Implicit Method for Vertical transport in Multidimensional Models. Int. Journal for Numerical Methods in Fluids. Vol. 28. pp. 157-186. 1998.
GAREY, M. R., JOHNSON, D. S. Computer and Intractability: a guide to the theory of NP-completeness. Freeman, San Francisco, 1979.
HARTEN, A. High resolution schemes for hyperbolic conservation laws. Journal of Computational Physics, vol. 49.pp. 357-393. 1983.
HIRSCH, C., Numerical Computation of Internal and External Flows. Vol. I: Fundamentals of Numerical discretization. John Wiley & Sons. Chichester. p. 514. 1992.
LEENDERTSE, J. J; GRITION, E. C. A Water-Quality Simulation Model for Well-Mixed Estuaries and Coastal Seas: vol. II, Computation Procedures. Technical Report R-708-NYC, Santa Monica: The Rand Corp, 1971 .
LEENDERTSE, J. J. A new approach to three-dimensional free-surface flow modeling. Technical Report R-3712-NETH/RC, Santa Monica, The Rand Corp., 1989.
MESSINGER, F. Numerical Methods: The Arakawa Approach, Horizontal Grid. Global and Limited-Area Modeling. Camp Springs: Academic Press. I 998.
RIZZI, R. L., ZEFERINO, C. A., DORNELES, R. V., NAVAUX, P. O. A., BAMPI, S., SUZIN, A. A., DIVERIO, T. A. Fluvial Flowing of Guaiba River Estuary: A Parallel Solution for the Shallow Water Equations Model in: Proceedings of the Fourth Vecpar. 2000. Part III, June 23, p. 895-896, Portugal. 2000.
RIZZI, R.L. Modelo Computacional Paralelo para a Hidrodinâmica e para o Transporte de Massa 2-D e 3-D. Proposta de Tese. PPGC-UFRGS. 2001.
ROEST, M.R.T. Partitioning for Parallel Finite Difference Computations in Coastal Water Simulation. Delft: Technische Universiteit Delf, 1997 (Ph.D. Thesis).
SAAD, Y. Iterative Methods for Sparse Linear Systems. Boston, PWS Publishing Company, 447 p. 1996.
SILVA, R. S. et al. Iterative Local Solvers for Distributed Krylov-Schwarz Method Applied to Convection-Diffusion Problems. Computer Methods for Applied Mechanics and Engineering, Vol. 149,353-362, 1997.
SMlTH, Barry. et al. Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Carnbridge University. 1996.
SMOLARKIEWINCZ, P. K. A Fully Multidimensional Positive Definite Advection Transport Algorithm with Small lmplicit Diffusion. Journal of Computational Physics, vol. 54, pp. 325-362, 1984.
VOOLEBREGT, Edwin A. H. Parallel Software Development Techiques for Shallow Water Models. Delft: Technische Universiteit Delf, 1997 (Ph.D. Thesis).
WEIY AN, T. Shallow Water Hydrodynamics: Mathematical Theory and Numerical Solution for a Two-dimensional System of Shallow Water Equations. Water & Power Press, Beijing e Elsevier, Amsterdam, 434 p. 1992.
ZEEUW, D., POWELL, K. G. An Adaptively Refined Cartesian Mesh Solver for the Euler Equations. Journal of Computational Physics, vol. 104. pp. 56-68, 1993.
CASULLI, Vincenzo. Semi-Implicit Finite Difference Methods for the Two-Dimensional Shallow Water Equations. Journal of Computational Physics, v.86. p.56-74, 1990.
CASULLI, V., CATIANl, E., Stability, Accuracy and Efficiency of a Semi-Implicit Method for Three-Dimensional Shallow Water Flow. Computers Math. Applic., vol. 27, nr. 4, pp. 99-112, 1994.
CHAN, T. .F.; MATHEW, T. P. Domain Decomposition Algorithms. Acta Numerica, p.61-143. 1994.
CHENG, R. Eulerian-Lagrangian Solution of the Convection-Dispersion Equation in Natural Coordinates. Water Resources Research, v.20. p.944-952. July 1984.
DEBREU, L., BLA YO, E. On the Schwarz Alternating Method for Oceanic Models on Parallel computers. Journal of computacional Physics, V. 141, p. 93-111. 1998.
DORNELES, R. V., RIZZI, R. L., ZEFERINO, C. A., DIVERIO, T. A., NAVAUX, P. O. A., BAMPl, S, SUZIN, A. A. A PC cluster Implementation of a Mass Transport Two Dimensional Model in: XII SBAC-PAD. 2000, São Pedro.
DORNELES, R. V. Particionamento de Domínio e Balanceamento Dinâmico de Carga em Arquiteturas Heterogêneas: Aplicação a Modelos Hidrodinâmicos e de Transporte de Massa 2-D e 3-D. Proposta de Tese. PPGC-UFRGS. 2001.
GROSS, E. S., CASULLI, V., BONAVENTURA, L., KOSEFF, JEFFREY. A Semi-Implicit Method for Vertical transport in Multidimensional Models. Int. Journal for Numerical Methods in Fluids. Vol. 28. pp. 157-186. 1998.
GAREY, M. R., JOHNSON, D. S. Computer and Intractability: a guide to the theory of NP-completeness. Freeman, San Francisco, 1979.
HARTEN, A. High resolution schemes for hyperbolic conservation laws. Journal of Computational Physics, vol. 49.pp. 357-393. 1983.
HIRSCH, C., Numerical Computation of Internal and External Flows. Vol. I: Fundamentals of Numerical discretization. John Wiley & Sons. Chichester. p. 514. 1992.
LEENDERTSE, J. J; GRITION, E. C. A Water-Quality Simulation Model for Well-Mixed Estuaries and Coastal Seas: vol. II, Computation Procedures. Technical Report R-708-NYC, Santa Monica: The Rand Corp, 1971 .
LEENDERTSE, J. J. A new approach to three-dimensional free-surface flow modeling. Technical Report R-3712-NETH/RC, Santa Monica, The Rand Corp., 1989.
MESSINGER, F. Numerical Methods: The Arakawa Approach, Horizontal Grid. Global and Limited-Area Modeling. Camp Springs: Academic Press. I 998.
RIZZI, R. L., ZEFERINO, C. A., DORNELES, R. V., NAVAUX, P. O. A., BAMPI, S., SUZIN, A. A., DIVERIO, T. A. Fluvial Flowing of Guaiba River Estuary: A Parallel Solution for the Shallow Water Equations Model in: Proceedings of the Fourth Vecpar. 2000. Part III, June 23, p. 895-896, Portugal. 2000.
RIZZI, R.L. Modelo Computacional Paralelo para a Hidrodinâmica e para o Transporte de Massa 2-D e 3-D. Proposta de Tese. PPGC-UFRGS. 2001.
ROEST, M.R.T. Partitioning for Parallel Finite Difference Computations in Coastal Water Simulation. Delft: Technische Universiteit Delf, 1997 (Ph.D. Thesis).
SAAD, Y. Iterative Methods for Sparse Linear Systems. Boston, PWS Publishing Company, 447 p. 1996.
SILVA, R. S. et al. Iterative Local Solvers for Distributed Krylov-Schwarz Method Applied to Convection-Diffusion Problems. Computer Methods for Applied Mechanics and Engineering, Vol. 149,353-362, 1997.
SMlTH, Barry. et al. Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Carnbridge University. 1996.
SMOLARKIEWINCZ, P. K. A Fully Multidimensional Positive Definite Advection Transport Algorithm with Small lmplicit Diffusion. Journal of Computational Physics, vol. 54, pp. 325-362, 1984.
VOOLEBREGT, Edwin A. H. Parallel Software Development Techiques for Shallow Water Models. Delft: Technische Universiteit Delf, 1997 (Ph.D. Thesis).
WEIY AN, T. Shallow Water Hydrodynamics: Mathematical Theory and Numerical Solution for a Two-dimensional System of Shallow Water Equations. Water & Power Press, Beijing e Elsevier, Amsterdam, 434 p. 1992.
ZEEUW, D., POWELL, K. G. An Adaptively Refined Cartesian Mesh Solver for the Euler Equations. Journal of Computational Physics, vol. 104. pp. 56-68, 1993.
Publicado
10/09/2001
Como Citar
DORNELES, Ricardo Vargas; RIZZI, Rogerio Luis; DIVERIO, Tiarajú A.; NAVAUX, Philippe O. A..
h-Refined Schwarz-Krylov Solution for Hydrodinamics and Mass Transport in a PC Cluster. In: INTERNATIONAL SYMPOSIUM ON COMPUTER ARCHITECTURE AND HIGH PERFORMANCE COMPUTING (SBAC-PAD), 13. , 2001, Pirenópolis.
Anais [...].
Porto Alegre: Sociedade Brasileira de Computação,
2001
.
p. 17-24.
DOI: https://doi.org/10.5753/sbac-pad.2001.22187.
