SIAM Journal on Numerical Analysis, Volume 62, Issue 5, Page 2308-2330, October 2024.
Abstract. In this paper, we study the convergence properties of the parareal algorithm with uniform coarse time grid and arbitrarily distributed (nonuniform) fine time grid, which may be changed at each iteration. We employ the backward-Euler method as the coarse propagator and a general single-step time-integrator as the fine propagator. Specifically, we consider two implementations of the coarse grid correction: the standard time-stepping mode and the parallel mode via the so-called diagonalization technique. For both cases, we prove that under certain conditions of the stability function of the fine propagator, the convergence factor of the parareal algorithm is not larger than that of the associated algorithm with a uniform fine time grid. Furthermore, we show that when such conditions are not satisfied, one can indeed observe degenerations of the convergence rate. The model that is used for performing the analysis is the Dahlquist test equation with nonnegative parameter, and the numerical results indicate that the theoretical results hold for nonlinear ODEs and linear ODEs where the coefficient matrix has complex eigenvalues.

中文翻译：

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具有非均匀精细时间网格的 Parareal 算法的收敛分析


SIAM 数值分析杂志，第 62 卷，第 5 期，第 2308-2330 页，2024 年 10 月。

抽象。在本文中，我们研究了具有均匀粗略时间网格和任意分布（非均匀）精细时间网格的 parareal 算法的收敛特性，该算法在每次迭代中都可能发生变化。我们采用 backward-Euler 方法作为粗略传播器，采用一般的单步时间积分器作为精细传播器。具体来说，我们考虑了粗略网格校正的两种实现方式：标准时间步进模式和通过所谓的对角化技术的并行模式。对于这两种情况，我们证明了在精细传播器的稳定性函数的某些条件下，parareal 算法的收敛因子并不大于具有均匀精细时间网格的关联算法的收敛因子。此外，我们表明，当这些条件不满足时，人们确实可以观察到收敛率的退化。用于执行分析的模型是具有非负参数的 Dahlquist 测试方程，数值结果表明，理论结果适用于系数矩阵具有复杂特征值的非线性 ODE 和线性 ODE。