高性能间断Galerkin谱元方法解欠分辨不可压湍流的效率
Efficiency of high-performance discontinuous Galerkin spectral element methods for under-resolved turbulent incompressible flows
摘要：
本文采用高阶间断Galerkin方法离散不可压缩纳维-斯托克斯方程，研究湍流的数值解。高阶方法的效率时，适用于欠解决的问题是一个开放的问题，在文学。本文以三维Taylor-Green涡问题为例，对这一问题进行了深入的研究。我们的实现是基于一个通用的高性能框架，无矩阵评价有限元算子与目前已知的最好的实现之一。我们提出了一种方法来系统地分析不可压缩Navier-Stokes求解器的效率高多项式次数。由于缺乏最佳的收敛速度在下解决制度，我们的研究结果表明，证明提高效率的高阶方法是一项具有挑战性的任务，最佳的计算复杂性的求解器，预处理器，和矩阵的实现是必要的成分，以实现更好的解决方案的质量在相同的计算成本已经为一个几何简单的问题，如泰勒-格林涡的目标。虽然分析是针对笛卡尔几何形状进行的，但我们的方法是通用的，可以应用于任意几何形状。我们在基于高速缓存的现代计算机架构上提供了出色的性能数据，在一个28核的英特尔Haswell节点上实现了3e 8至1 e9 DoF/秒的吞吐量，可供操作员评估。与过去5年中发表的可压缩纳维-斯托克斯方程高阶DG离散化的性能结果相比，我们的方法降低了计算成本，对于相同的设置超过一个数量级。
英文摘要：
The present paper addresses the numerical solution of turbulent flows with high-order discontinuous Galerkin methods for discretizing the incompressible Navier-Stokes equations. The efficiency of high-order methods when applied to under-resolved problems is an open issue in literature. This topic is carefully investigated in the present work by the example of the 3D Taylor-Green vortex problem. Our implementation is based on a generic high-performance framework for matrix-free evaluation of finite element operators with one of the best realizations currently known. We present a methodology to systematically analyze the efficiency of the incompressible Navier-Stokes solver for high polynomial degrees. Due to the absence of optimal rates of convergence in the under-resolved regime, our results reveal that demonstrating improved efficiency of high-order methods is a challenging task and that optimal computational complexity of solvers, preconditioners, and matrix-free implementations are necessary ingredients to achieve the goal of better solution quality at the same computational costs already for a geometrically simple problem such as the Taylor-Green vortex. Although the analysis is performed for a Cartesian geometry, our approach is generic and can be applied to arbitrary geometries. We present excellent performance numbers on modern, cache-based computer architectures achieving a throughput for operator evaluation of 3e8 up to 1e9 DoFs/sec on one Intel Haswell node with 28 cores. Compared to performance results published within the last 5 years for high-order DG discretizations of the compressible Navier-Stokes equations, our approach reduces computational costs by more than one order of magnitude for the same setup.
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