Parallel-in-time integration has been the focus of intensive research efforts over the past two decades due to the advent of massively parallel computer architectures and the scaling limits of purely spatial parallelization. Various iterative parallel-in-time (PinT) algorithms have been proposed, like Parareal, PFASST, MGRIT, and Space-Time Multi-Grid (STMG). These methods have been described using different notations, and the convergence estimates that are available are difficult to compare. We describe Parareal, PFASST, MGRIT and STMG for the Dahlquist model problem using a common notation and give precise convergence estimates using generating functions. This allows us, for the first time, to directly compare their convergence. We prove that all four methods eventually converge super-linearly, and also compare them numerically. The generating function framework provides further opportunities to explore and analyze existing and new methods.

翻译：并行时间积分方法因大规模并行计算机架构的出现及纯空间并行化的扩展限制，在过去二十年成为研究热点。目前已提出多种迭代并行时间（PinT）算法，如Parareal、PFASST、MGRIT和时空多重网格（STMG）。这些方法采用不同符号体系描述，且现有的收敛估计难以相互比较。本文采用统一符号体系针对Dahlquist模型问题描述Parareal、PFASST、MGRIT及STMG方法，并通过生成函数给出精确的收敛估计。这使我们首次能够直接比较它们的收敛性。我们证明所有四种方法最终均呈超线性收敛，并进行了数值对比。该生成函数框架为探索与分析现有及新型方法提供了更多可能性。