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首页> 外文期刊>Journal of Computational Physics >Parallel direct Poisson solver for discretisations with one Fourier diagonalisable direction
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Parallel direct Poisson solver for discretisations with one Fourier diagonalisable direction

机译：平行直接泊松解算器，用于离散傅立叶对角线方向

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摘要

In the context of time-accurate numerical simulation of incompressible flows, a Poisson equation needs to be solved at least once per time-step to project the velocity field onto a divergence-free space. Due to the non-local nature of its solution, this elliptic system is one of the most time consuming and difficult to parallelise parts of the code. In this paper, a parallel direct Poisson solver restricted to problems with one uniform periodic direction is presented. It is a combination of a direct Schur-complement based decomposition (DSD) and a Fourier diagonalisation. The latter decomposes the original system into a set of mutually independent 2D systems which are solved by means of the DSD algorithm. Since no restrictions are imposed in the non-periodic directions, the overall algorithm is well-suited for solving problems discretised on extruded 2D unstructured meshes. The load balancing between parallel processes and the parallelisation strategy are also presented and discussed. The scalability of the solver is successfully tested using up to 8192 CPU cores for meshes with up to 10~9 grid points. Finally, the performance of the DSD algorithm as 2D solver is analysed by direct comparison with two preconditioned conjugate gradient methods. For this purpose, the turbulent flow around a circular cylinder at Reynolds numbers 3900 and 10,000 are used as problem models.
机译：在不可压缩流的时间精确数值模拟的情况下，泊松方程需要在每个时间步长至少求解一次，以将速度场投影到无散度空间上。由于其解决方案的非本地性质，因此该椭圆系统是最耗时且难于并行处理代码部分的系统之一。在本文中，提出了一种并行的直接泊松求解器，该求解器限于一个统一的周期方向的问题。它是直接基于Schur补码的分解（DSD）和傅立叶对角化的组合。后者将原始系统分解为一组相互独立的2D系统，这些系统通过DSD算法求解。由于在非周期性方向上没有施加限制，因此整个算法非常适合解决离散化的二维非结构化网格上离散的问题。还介绍并讨论了并行进程与并行化策略之间的负载平衡。求解器的可扩展性已成功使用多达8192个CPU内核成功测试了网格，网格数高达10〜9。最后，通过与两种预处理共轭梯度方法的直接比较，分析了DSD算法作为2D求解器的性能。为此，将雷诺数为3900和10,000的圆柱体周围的湍流用作问题模型。

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