大规模动力系统的Krylov方法：在流体动力学中的应用
Krylov methods for large-scale dynamical systems: Application in fluid dynamics
摘要：
在流体动力学中，预测和表征从非定常开始到过渡到湍流的分叉对于学术和工业应用都是至关重要的。动力系统理论的不同工具可以用于此目的。在这篇综述中，我们提出了一个简洁的理论和数值框架，侧重于实际方面的计算和稳定性分析的稳定和时间周期的解决方案，重点是高维系统，如空间离散化的纳维-斯托克斯方程。使用基于Krylov方法的无矩阵方法，我们扩展了开源高性能基于光谱元素的时间步进器Nek5000的功能。讨论的数值方法在nekStab中实现，这是一个开源和用户友好的附加工具箱，致力于研究复杂三维几何形状中流动的稳定性。的性能和准确性的方法进行说明和检查使用标准的基准从流体力学文献。由于其灵活性和域不可知的性质，在这项工作中提出的方法可以应用于开发类似的工具箱，为其他求解器，最重要的是流体力学领域之外。
英文摘要：
In fluid dynamics, predicting and characterizing bifurcations, from the onset of unsteadiness to the transition to turbulence, is of critical importance for both academic and industrial applications. Different tools from dynamical systems theory can be used for this purpose. In this review, we present a concise theoretical and numerical framework focusing on practical aspects of the computation and stability analyses of steady and time-periodic solutions, with emphasis on high-dimensional systems such as those arising from the spatial discretization of the Navier-Stokes equations. Using a matrix-free approach based on Krylov methods, we extend the capabilities of the open-source high-performance spectral element-based time-stepper Nek5000. The numerical methods discussed are implemented in nekStab, an open-source and user-friendly add-on toolbox dedicated to the study of stability properties of flows in complex three-dimensional geometries. The performance and accuracy of the methods are illustrated and examined using standard benchmarks from the fluid mechanics literature. Thanks to its flexibility and domain-agnostic nature, the methodology presented in this work can be applied to develop similar toolboxes for other solvers, most importantly outside the field of fluid mechanics.
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