Scalable Algorithms for High Order Approximations on Compact Stencils
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The recent development of parallel technologies on modern desktop computers makes parallelization of the proposed numerical approaches a priority in algorithmic research. The main performance improvement in the upcoming years will be made based on the increasing number of cores on modern CPUs. This shifts the focus of the algorithmic research from the development of the sequential numerical methods to the parallel methodology. In this paper, we present an efficient parallel direct algorithm for the compact high-order approximation of the 3D Helmholtz equation. The developed method is based on a combination of the separation of variables technique and a Fast Fourier Transform (FFT) type method. The results of the implementation of this method in OpenMP, MPI and Hybrid programming environments on the multicores computers and multiple node clusters are presented. We considered a generalization of the presented algorithm to the solution of linear systems obtained from approximation on the compact 27-point 3D stencils on the rectangular grids with similar stencil coefficients. As an example of the diversity of applications, the direct parallel implementation of a compact fourth-order approximation scheme for a convection-diffusion equation is considered. The developed parallel algorithms present efficient direct solvers for many important applications, but they can be used as highly efficient preconditioners for a variety of iterative numerical methods in more general settings. In many situations, the efficiency of the iterative algorithms is determined by the robustness of the preconditioning technique, the presented methods have a wide range of applications. In this paper, we demonstrate the scalability of the developed numerical algorithms on a series of representative test problems.
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