Multi-step variant of the parareal algorithm: convergence analysis and numerics

Katia Ait-Ameur, Y. Maday
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Abstract

In this paper, we consider the problem of accelerating the numerical simulation of time dependent problems involving a multi-step time scheme by the parareal algorithm. The parareal method is based on combining predictions made by a coarse and cheap propagator, with corrections computed with two propagators: the previous coarse and a precise and expensive one used in a parallel way over the time windows. A multi-step time scheme can potentially bring higher approximation orders than plain one-step methods but the initialisation of each time window needs to be appropriately chosen. Our main contribution is the design and analysis of an algorithm adapted to this type of discretisation without being too much intrusive in the coarse or fine propagators. At convergence, the parareal algorithm provides a solution that coincides with the solution of the fine solver. In the classical version of parareal, the local initial condition of each time window is corrected at every iteration. When the fine and/or coarse propagators is a multi-step time scheme, we need to choose a consistent approximation of the solutions involved in the initialisation of the fine solver at each time windows. Otherwise, the initialisation error will prevent the parareal algorithm to converge towards the solution with fine solver’s accuracy. In this paper, we develop a variant of the algorithm that overcome this obstacle. Thanks to this, the parareal algorithm is more coherent with the underlying time scheme and we recover the properties of the original version. We show both theoretically and numerically that the accuracy and convergence of the multi-step variant of parareal algorithm are preserved when we choose carefully the initialisation of each time window.
准噶尔算法的多步变体:收敛分析和数值计算
在本文中,我们考虑的问题是如何利用准噶尔算法加速涉及多步时间方案的时间相关问题的数值模拟。parareal方法的基础是将一个粗略而廉价的传播者所做的预测与两个传播者计算的修正结合起来:前一个粗略传播者和一个精确而昂贵的传播者在时间窗上并行使用。与简单的一步法相比,多步时间方案可能带来更高的近似阶数,但每个时间窗的初始化需要适当选择。我们的主要贡献在于设计和分析了一种适应这种离散化的算法,而不会对粗传播器或细传播器造成过多干扰。在收敛时,parareal 算法提供的解与精细求解器的解相吻合。在经典版本的 parareal 算法中,每次迭代都会修正每个时间窗口的局部初始条件。当精细和/或粗放传播器为多步时间方案时,我们需要在每个时间窗口选择与精细求解器初始化所涉及的解一致的近似值。否则,初始化误差将阻碍准线性算法收敛到具有精细求解器精度的解。在本文中,我们开发了一种算法变体来克服这一障碍。因此,抛物线算法与底层时间方案更加一致,并恢复了原始版本的特性。我们从理论和数值上证明,当我们仔细选择每个时间窗口的初始化时,parareal 算法的多步变体的精度和收敛性都能得到保持。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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