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.

中文翻译：

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具有任意高阶精细传播子的 Parareal 算法的鲁棒收敛

本文的目的是分析一类用于解决抛物线问题的拟实算法的鲁棒收敛性。粗传播器固定为后向欧拉方法，精传播器是高阶单步积分器。在精细传播器的某些条件下，我们证明存在一些临界 $J_*$ 使得平行解算器以接近 $0.3$ 的收敛率线性收敛，前提是粗时间步长和细时间步长之间的比率名为 $J $ 满足 $J \ge J_*$。即使问题数据不平滑且与边界条件不兼容，收敛也是稳健的。合格的方法包括所有绝对稳定的单步方法，其稳定性函数满足$|r(-\infty)|<1$，因此精细传播子可以是任意高阶的。而且，我们检查了一些流行的高阶单步方法，例如，两阶段、三阶段和四阶段 Lobatto IIIC 方法，并验证相应的拟实算法线性收敛于因子 $0.31$，并且这些情况的阈值是 $J_* = 2 美元。提供了大量的数值例子来支持和完成我们的理论预测。