On "Optimal" h-Independent Convergence of Parareal and MGRIT using Runge-Kutta Time Integration
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Parareal and multigrid-reduction-in-time (MGRIT) are two popular parallel-in-time algorithms. The idea of both algorithms is to combine the (fine-grid) time-stepping scheme of interest with a "coarse-grid" time-integration scheme that approximates several steps of the fine-grid time-stepping method. Convergence of Parareal and MGRIT has been studied in a number of papers. Research on the optimality of both methods, however, is limited, with results existing only for specific time-integration schemes. This paper focuses on analytically showing h_x- and h_t-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 analysis is based on recently derived tight bounds of two-level Parareal and MGRIT convergence that allow for analyzing arbitrary coarse- and fine-grid time integration schemes, coarsening factors, and time-step sizes. The theory presented in this paper 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. On the other hand, some schemes do not obtain h-optimal convergence, and two-level convergence is restricted to certain parameter regimes. In certain cases, an O(1) factor change in time step h_t can be the difference between convergence factors ρ≈0.02 and divergence! Numerical results confirm the analysis in the practical setting and, in particular, 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.
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