Two-Level Convergence Theory for Multigrid Reduction in Time (MGRIT)

Abstract. In this paper we develop a two-grid convergence theory for the parallel-in-time scheme known as multigrid reduction in time (MGRIT), as it is implemented in the open-source XBraid package [25]. MGRIT is a scalable and multi-level approach to parallel-in-time simulations that non-intrusively uses existing time-stepping schemes, and that in a specific two-level setting is equivalent to the widely-known parareal algorithm. The goal of this paper is two-fold. First, we present a two-level MGRIT convergence analysis for linear problems where the spatial discretization matrix can be diagonalized, and use this to analyze our two basic model problems, the heat equation and the advection equation. One important assumption is that the coarse and fine time-grid propagators can be diagaonalized by the same set of eigenvectors, which is often the case when the same spatial discretization operator is used on the coarse and fine time grids. In many cases, the MGRIT algorithm is guaranteed to converge and we demonstrate numerically that the theoretically predicted convergence rates are sharp in practice for our model problems. The second goal of the paper is to explore how the convergence of MGRIT compares to the stability of the chosen time-stepping scheme. In particular, we demonstrate that a stable time-stepping scheme does not necessarily imply convergence of MGRIT, although MGRIT with FCF-relaxation always converges for the di↵usion dominated problems considered here.