In this article, we derive fast and robust preconditioned iterative methods for the all-at-once linear systems arising upon discretization of time-dependent PDEs. The discretization we employ is based on a Runge--Kutta method in time, for which the development of robust solvers is an emerging research area in the literature of numerical methods for time-dependent PDEs. By making use of classical theory of block matrices, one is able to derive a preconditioner for the systems considered. An approximate inverse of the preconditioner so derived consists in a fixed number of linear solves for the system of the stages of the method. We thus propose a preconditioner for the latter system based on a singular value decomposition (SVD) of the (real) Runge--Kutta matrix $A_{\mathrm{RK}} = U \Sigma V^\top$. Supposing $A_{\mathrm{RK}}$ is invertible, we prove that the spectrum of the system for the stages preconditioned by our SVD-based preconditioner is contained within the right-half of the unit circle, under suitable assumptions on the matrix $U^\top V$ (which is well defined due to the polar decomposition of $A_{\mathrm{RK}}$). We show the numerical efficiency of our SVD-based preconditioner by solving the system of the stages arising from the discretization of the heat equation and the Stokes equations, with sequential time-stepping. Finally, we provide numerical results of the all-at-once approach for both problems.

翻译：本文针对时间依赖偏微分方程离散化中出现的全时段线性系统，推导了快速且鲁棒的预处理迭代方法。所采用的离散化基于时间方向的龙格-库塔方法，针对此类格式开发鲁棒求解器是时间依赖偏微分方程数值方法领域的一个新兴研究方向。利用经典分块矩阵理论，我们能对所考虑的系统推导出预条件子。该预条件子的近似逆包含对方法各阶段系统进行固定次数的线性求解。为此，我们提出基于（实）龙格-库塔矩阵 $A_{\mathrm{RK}} = U \Sigma V^\top$ 奇异值分解的预条件子。假设 $A_{\mathrm{RK}}$ 可逆，我们证明在关于矩阵 $U^\top V$（由 $A_{\mathrm{RK}}$ 的极分解自然定义）的适当假设下，经基于奇异值分解的预条件子处理后的阶段系统谱包含在单位圆右半平面内。通过求解热方程和斯托克斯方程离散化过程中产生的阶段系统（采用顺序时间推进），我们展示了基于奇异值分解的预条件子的数值效率。最后，我们给出了针对这两个问题的全时段方法的数值结果。