Robust Convergence of Parareal Algorithms with Arbitrarily High-order Fine Propagators

The aim of this paper is to analyze the robust convergence of a class of parareal algorithms for solving parabolic problems. The coarse propagator is fixed to the backward Euler method and the fine propagator is a high-order single step integrator. Under some conditions on the fine propagator, we show that there exists some critical $J_*$ such that the parareal solver converges linearly with a convergence rate near $0.3$, provided that the ratio between the coarse time step and fine time step named $J$ satisfies $J \ge J_*$. The convergence is robust even if the problem data is nonsmooth and incompatible with boundary conditions. The qualified methods include all absolutely stable single step methods, whose stability function satisfies $|r(-\infty)|<1$, and hence the fine propagator could be arbitrarily high-order. Moreover, we examine some popular high-order single step methods, e.g., two-, three- and four-stage Lobatto IIIC methods, and verify that the corresponding parareal algorithms converge linearly with a factor $0.31$ and the threshold for these cases is $J_* = 2$. Intensive numerical examples are presented to support and complete our theoretical predictions.