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.

翻译：针对含时偏微分方程预处理子的研究大多集中在隐式多步法或对角隐式多阶段时间离散方法上。本文考虑完全隐式多阶段龙格-库塔时间积分方法的整体多重网格预处理子。这类时间离散方法具有非常吸引人的精度和稳定性特性，但会将空间自由度耦合到多个时间层上，从而需要求解大型线性系统。我们将经典的Vanka松弛方案推广至鞍点问题的隐式RK离散化。针对二维和三维不可压缩斯托克斯方程、纳维-斯托克斯方程和电阻磁流体动力学方程给出了数值结果，证实这些松弛方案能为一系列具有挑战性的不可压缩流体流动模型提供鲁棒且可扩展的整体多重网格方法。