Parallel implementation of non-linear evolution problems using parabolic domain decomposition

Abstract

We present implementation of parallel algorithms for the numerical solution of nonlinear time-dependent partial differential equations of parabolic type arising from complex large scale problems. The parallelization is achieved by using domain decomposition (DD) techniques. The essential feature of this algorithm is that the spatial discretization in each subdomain is performed by using spectral method with the Local Fourier Basis (LFB) [1]. Our solutions are based on a special projection technique that employed to localize functions in a smooth way on the extended subdomain. The current paper continue the flow of our previous results [1,4,12,13] on spectral multidomain algorithm. The application of the Parabolic Domain Decomposition (PDD) approach along with the LFB is shown to be very efficient when applied to a 2-dimensional domain splitted into strips and rectangular cells. In this case, all matching relations become completely uncoupled (at the price of some overlapping of subdomains required by the LFB implementation). Thus, all communication is reduced to interactions between neighbouring elements and thus fits to scalable message-passing multiprocessor. The continuity of a global solution is attained by using a direct point-wise matching of the local subsolutions on the interfaces. The implementation of the LFB technique enables us to trade a 2-D problem with the overall coupling of the interface unknown into a set of uncoupled 1-D differential equations with simple matching relations. 2-D Navier-Stokes type modeling equation is implemented on the Meiko message-passing type scalable MIMD multiprocessor. Detailed performance analysis is presented. The algorithm is scalable in the sense that problems of equal size have the same speedup when the number of processors increase because the communication is reduced from global to local and, thus, the solution depends on direct neighboring processors. When the size if each sub-domain in each processor is large enough a linear speedup is achieved. The implementation strategy of the algorithms can easily be changed to reflect the potential of having different resolution in each subdomain, which makes it valuable as an adaptive algorithm. The same results are derived when the the alternate direction implicit (ADI) method is used as an efficient time-discretization scheme.