Fast Iterative Solver for the All-at-Once Runge–Kutta Discretization
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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_RK = U Σ V^⊤. Supposing A_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^⊤ V (which is well defined due to the polar decomposition of A_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.
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