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ADVANCES IN PARALLEL, DISTRIBUTED, GRID AND CLOUD COMPUTING FOR ENGINEERING
Edited by: P. Iványi, B.H.V. Topping and G. Várady
Parallel Semi-Aggregation Techniques for Solving Parabolic Partial Differential Equations
G.A. Gravvanis1, B.E. Moutafis1, C.K. Filelis-Papadopoulos1 and H.G. Theodosiou2
1Department of Electrical and Computer Engineering, School of Engineering, Democritus
University of Thrace, University Campus, Kimmeria, Xanthi, Greece
G.A. Gravvanis, B.E. Moutafis, C.K. Filelis-Papadopoulos and H.G. Theodosiou, "Parallel Semi-Aggregation Techniques for Solving Parabolic Partial Differential Equations", in P. Iványi, B.H.V. Topping and G. Várady, (Editors), "Advances in Parallel, Distributed, Grid and Cloud Computing for Engineering", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 8, pp 157-182, 2017. doi:10.4203/csets.40.8
Keywords: algebraic time domain decomposition method, space-time semi-aggregation, heat transfer, sparse linear systems, parallel hybrid solver.
Many engineering and scientific problems are described by sparse linear systems derived from the discretization of elliptic and parabolic partial differential equations (PDEs). Over the last decades, preconditioned Krylov subspace iterative methods have been extensively used for solving large sparse linear systems in order to improve convergence behavior and performance. The domain decomposition methods have been shown to be efficient and scalable for solving large sparse linear systems in modern parallel computer architectures. There are overlapping and non-overlapping domain decomposition methods, according to the partitioning scheme. The overlapping domain decomposition methods, which require more inter-node communications, usually have better convergence behavior compared to the non-overlapping methods. Recently, a new class of algebraic domain decomposition methods, based on a semi-coarse aggregation technique, namely multi-projection methods (MPM), has been proposed. These types of methods in conjunction with Krylov Subspace methods have been shown to have improved convergence behavior especially for large number of subdomains, whereas most of the extant domain decomposition methods present worse convergence behavior as the number of subdomains increases. The semi-aggregation techniques can be extended to time discretization for solving parabolic PDEs. The parabolic PDEs are discretized with the finite differences methods and through time domain decomposition techniques the coefficient matrix is formed. The solution of the resulting large sparse linear systems leads to the computation of multiple time steps per iteration, thus enhancing performance and increasing granularity. The effectiveness and applicability of the proposed semi-aggregation scheme for solving parabolic PDEs, modelling heat transfer phenomena, are examined and numerical results along with performance and scalability are given.
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