On “Optimal” h‐independent convergence of Parareal and multigrid‐reduction‐in‐time using Runge‐Kutta time integration

Summary Although convergence of the Parareal and multigrid‐reduction‐in‐time (MGRIT) parallel‐in‐time algorithms is well studied, results on their optimality is limited. Appealing to recently derived tight bounds of two‐level Parareal and MGRIT convergence, this article proves (or disproves) hx‐ and ht‐independent convergence of two‐level Parareal and MGRIT, for linear problems of the form u′(t)+Lu(t)=f(t), where L is symmetric positive definite and Runge‐Kutta time integration is used. The theory presented in this article also encompasses analysis of some modified Parareal algorithms, such as the θ‐Parareal method, and shows that not all Runge‐Kutta schemes are equal from the perspective of parallel‐in‐time. Some schemes, particularly L‐stable methods, offer significantly better convergence than others as they are guaranteed to converge rapidly at both limits of small and large htξ, where ξ denotes an eigenvalue of L and ht time‐step size. On the other hand, some schemes do not obtain h‐optimal convergence, and two‐level convergence is restricted to certain regimes. In certain cases, an O(1) factor change in time step ht or coarsening factor k can be the difference between convergence factors ρ≈0.02 and divergence! The analysis is extended to skew‐symmetric operators as well, which cannot obtain h‐independent convergence and, in fact, will generally not converge for a sufficiently large number of time steps. Numerical results confirm the analysis in practice and emphasize the importance of a priori analysis in choosing an effective coarse‐grid scheme and coarsening factor. A Mathematica notebook to perform a priori two‐grid analysis is available at https://github.com/XBraid/xbraid‐convergence‐est.