论文标题
整体多族型多族，用于不可压缩流体流动的隐式runge-kutta
Monolithic multigrid for implicit Runge-Kutta discretizations of incompressible fluid flow
论文作者
论文摘要
大多数对时间依赖性PDE的预处理的研究都集中在隐式多步或对角线多阶段的多阶段时间离散上。在本文中，我们考虑了完全尺寸的多阶段runge-kutta（RK）时间集成方法的单层多族预处理。这些时间离散化具有非常有吸引力的准确性和稳定性，但是它们将空间自由度跨越了多个时间级，需要解决非常大的线性系统的解决方案。我们将经典的Vanka放松方案扩展到隐式RK鞍点问题的离散化。我们在两个和三个维度上呈现了不可压缩的Stokes，Navier-Stokes和电阻性磁动力动力学方程的数值结果，证实了这些弛豫方案可导致可靠且可扩展的单片多质物方法，以使其具有挑战性的不可压缩流体模型范围。
Most research on preconditioners for time-dependent PDEs has focused on implicit multi-step or diagonally-implicit multi-stage temporal discretizations. In this paper, we consider monolithic multigrid preconditioners for fully-implicit multi-stage Runge-Kutta (RK) time integration methods. These temporal discretizations have very attractive accuracy and stability properties, but they couple the spatial degrees of freedom across multiple time levels, requiring the solution of very large linear systems. We extend the classical Vanka relaxation scheme to implicit RK discretizations of saddle point problems. We present numerical results for the incompressible Stokes, Navier-Stokes, and resistive magnetohydrodynamics equations, in two and three dimensions, confirming that these relaxation schemes lead to robust and scalable monolithic multigrid methods for a challenging range of incompressible fluid-flow models.
