TY - JOUR
T1 - Robust Convergence of Parareal Algorithms with Arbitrarily High-Order Fine Propagators
AU - Yang , Jiang
AU - Yuan , Zhaoming
AU - Zhou , Zhi
JO - CSIAM Transactions on Applied Mathematics
VL - 3
SP - 566
EP - 591
PY - 2023
DA - 2023/04
SN - 4
DO - http://doi.org/10.4208/csiam-am.SO-2022-0025
UR - https://global-sci.org/intro/article_detail/csiam-am/21642.html
KW - Parareal algorithm, parabolic problems, arbitrarily high-order, single step integrator, convergence factor.
AB - <p style="text-align: justify;">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 $J ≥ J_∗.$ 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 $|r(−∞)|&lt;1,$ 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.</p>