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首页> 外文期刊>Journal of Computational Physics >High-order accurate solution of the incompressible Navier-Stokes equations on massively parallel computers
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High-order accurate solution of the incompressible Navier-Stokes equations on massively parallel computers

机译：大规模并行计算机上不可压缩的Navier-Stokes方程的高阶精确解

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摘要

The emergence of "petascale" supercomputers requires us to develop today's simulation codes for (incompressible) flows by codes which are using numerical schemes and methods that are better able to exploit the offered computational power. In that spirit, we present a massively parallel high-order Navier-Stokes solver for large incompressible flow problems in three dimensions. The governing equations are discretized with finite differences in space and a semi-implicit time integration scheme. This discretization leads to a large linear system of equations which is solved with a cascade of iterative solvers. The iterative solver for the pressure uses a highly efficient commutation-based preconditioner which is robust with respect to grid stretching. The efficiency of the implementation is further enhanced by carefully setting the (adaptive) termination criteria for the different iterative solvers. The computational work is distributed to different processing units by a geometric data decomposition in all three dimensions. This decomposition scheme ensures a low communication overhead and excellent scaling capabilities. The discretization is thoroughly validated. First, we verify the convergence orders of the spatial and temporal discretizations for a forced channel flow. Second, we analyze the iterative solution technique by investigating the absolute accuracy of the implementation with respect to the different termination criteria. Third, Orr-Sommerfeld and Squire eigenmodes for plane Poiseuille flow are simulated and compared to analytical results. Fourth, the practical applicability of the implementation is tested for transitional and turbulent channel flow. The results are compared to solutions from a pseudospectral solver. Subsequently, the performance of the commutation-based preconditioner for the pressure iteration is demonstrated. Finally, the excellent parallel scalability of the proposed method is demonstrated with a weak and a strong scaling test on up to processing units and grid points.
机译：“千万亿次”超级计算机的出现要求我们用能够更好地利用所提供的计算能力的数值方案和方法，通过代码为（不可压缩的）流量开发当今的仿真代码。本着这种精神，我们提出了一种大规模并行的高阶Navier-Stokes求解器，用于解决三维三维中不可压缩的大型流动问题。控制方程是离散的，具有有限的空间差异和半隐式时间积分方案。这种离散化导致了一个大型的线性方程组，可以使用级联的迭代求解器进行求解。压力的迭代求解器使用高效的基于换向的预处理器，该预处理器对网格拉伸具有鲁棒性。通过为不同的迭代求解器仔细设置（自适应）终止标准，可以进一步提高实现效率。通过在所有三个维度上进行几何数据分解，将计算工作分配给不同的处理单元。这种分解方案可确保较低的通信开销和出色的缩放能力。离散化已得到充分验证。首先，我们验证了强制通道流的时空离散化的收敛阶。其次，我们通过研究关于不同终止标准的实现的绝对准确性来分析迭代解决方案技术。第三，模拟了平面Poiseuille流动的Orr-Sommerfeld和Squire特征模式，并将其与分析结果进行比较。第四，针对通道的过渡和湍流测试了实现的实际适用性。将结果与伪谱求解器的解进行比较。随后，演示了基于换向的预处理器在压力迭代中的性能。最后，通过对多达处理单元和网格点的弱缩放测试和强缩放测试，证明了所提出方法的出色并行可扩展性。

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