Abstract
A parallel scheme for distributed memory hierarchy system is presented to solve the large-scale three-dimensional heat equation. Since managing interprocess communications and coordination is the main difficulty with the system, the local physics/global algebraic object paradigm is introduced. Domain decomposition method is used to partition the modeling area, as well as the intensive computational effort and large memory requirement. Efficient storage and assembly of sparse matrix and parallel iterative solution of linear system are considered and developed. The efficiency and scalability of the parallel program are demonstrated by completing two experiments on Linux cluster, in which different preconditioning methods are tested and analyzed. And the results demonstrate this method could achieve desirable parallel performance.
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References
Dawson, C.N., Du, Q., Dupont, T.F.: A Finite Difference Domain Decomposition Algorithm for Numerical solution of the Heat Equation. Mathematics of Computation 195, 63–71 (1991)
Lions, J.L., Maday, Y., Turinici, G.: A “parareal” in time discretization of PDE’s. Comptes Rendus de l’Académie des Sciences - Series I – Mathematics 332, 661–668 (2001)
Contassot-Vivier, S., Couturier, R., Denis, C., Jézéquel, F.: Efficiently solving large sparse linear systems on a distributed and heterogeneous grid by using the multisplitting-direct method. In: Fourth International Workshop on Parallel Matrix Algorithms and Applications, PMAA’06, pp. 21–22 (2006)
Amestoy, P.R., Duff, I.S., Pralet, S., Vömel, C.: Adapting a parallel sparse direct solver to architectures with clusters of SMPs. Parallel Computing 29, 1645–1668 (2003)
Antoine, G., Kahou, A., Grigori, L., Sosonkina, M.: A partitioning algorithm for block-diagonal matrices with overlap. Parallel Computing 34, 332–344 (2008)
Dağ, H.: An approximate inverse preconditioner and its implementation for conjugate gradient method. Parallel Computing 33, 83–91 (2007)
Couturier, R., Denis, C., Jézéquel, F.: GREMLINS: a large sparse linear solver for grid environment. Parallel Computing 34, 380–391 (2008)
Hoefler, T., Gottschling, P., Lumsdaine, A., Rehm, W.: Optimizing a conjugate gradient solver with non-blocking collective operations. Parallel Computing 33, 624–633 (2007)
Hernandez, V., Roman, J.E., Tomas, A.: Parallel Arnoldi eigensolvers with enhanced scalability via global communications rearrangement. Parallel Computing 33, 521–540 (2007)
Mo, Z.Y., Xu, X.W.: Relaxed RS0 or CLJP coarsening strategy for parallel AMG. Parallel Computing 33, 174–185 (2007)
Balay, S., Buschelman, K., Eijkhout, V., Gropp, W., Kaushik, D., Knepley, M., Curfman McInnes, L., Smith, B., Zhang, H.: PETSc Users Manual, http://www-unix.mcs.anl.gov/petsc/petscas/documentation/index.html#Manual
Hypre- the LLNL preconditioner library, http://www.llnl.gov/CASC/hypre
Karypis, G., Schloegel, K., Kumar, V.: ParMETIS 1.0: Parallel Graph Partitioning and Sparse Matrix Ordering Library. Technical Report TR-97-060. Department of Computer Science, University of Minnesota (1997)
Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. SIAM Press, Philadelphia (1995)
Cheng, T.P., Ji, X.H., Wang, Q.: An Efficient Parallel Method for Large-scale Groundwater Flow Equation based on PETSc. In: IEEE Youth Conference on Information, Computing and Telecommunications, pp. 190–193 (2009)
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Cheng, T., Wang, Q., Ji, X., Li, D. (2010). A Parallel Solution of Large-Scale Heat Equation Based on Distributed Memory Hierarchy System. In: Hsu, CH., Yang, L.T., Park, J.H., Yeo, SS. (eds) Algorithms and Architectures for Parallel Processing. ICA3PP 2010. Lecture Notes in Computer Science, vol 6082. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13136-3_42
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DOI: https://doi.org/10.1007/978-3-642-13136-3_42
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