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Parallel-in-time integration has been the focus of intensive research effortsover the past two decades due to the advent of massively parallel computerarchitectures and the scaling limits of purely spatial parallelization. Variousiterative parallel-in-time (PinT) algorithms have been proposed, like Parareal,PFASST, MGRIT, and Space-Time Multi-Grid (STMG). These methods have beendescribed using different notations, and the convergence estimates that areavailable are difficult to compare. We describe Parareal, PFASST, MGRIT andSTMG for the Dahlquist model problem using a common notation and give preciseconvergence estimates using generating functions. This allows us, for the firsttime, to directly compare their convergence. We prove that all four methodseventually converge super-linearly, and also compare them numerically. Thegenerating function framework provides further opportunities to explore andanalyze existing and new methods.

A unified analysis framework for iterative parallel-in-time algorithms