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Abstract: Parareal is a recent algorithm able to parallelize the time dimension in spite of its sequential nature. It has been applied to several linear and nonlinear problems and, very recently, to a simulation of fully-developed, two-dimensional drift wave turbulence. The mere fact that parareal works in such a turbulent regime is in itself somewhat unexpected, due to the characteristic sensitivity of turbulence to any change in initial conditions. This fundamental property of any turbulent system should render the iterative correction procedure characteristic of the parareal method inoperative, but this seems not to be the case. In addition, the choices that must be made to implement parareal (division of the temporal domain, election of the coarse solver and so on) are currently made using trial-and-error approaches. Here, we identify the mechanisms responsible for the convergence of parareal of these simulations of drift wave turbulence. We also investigate which conditions these mechanisms impose on any successful parareal implementation. The results reported here should be useful to guide future implementations of parareal within the much wider context of fully-developed fluid and plasma turbulent simulations. [Copyright &y& Elsevier]

Abstract: Parareal is an iterative algorithm that, in effect, achieves temporal decomposition for a time-dependent system of differential or partial differential equations. A solution is obtained in a shorter wall-clock time, but at the expense of increased compute cycles. The algorithm combines a fine solver that solves the system to acceptable accuracy with an approximate coarse solver. The critical task for the successful implementation of parareal on any system is the development of a coarse solver that leads to convergence in a small number of iterations compared to the number of time slices in the full time interval, and is, at the same time, much faster than the fine solver. Very fast coarse solvers may not lead to sufficiently rapid convergence, and slow coarse solvers may not lead to significant gains even if the number of iterations to convergence is satisfactory. We find that the difficulty of meeting these conflicting demands can be substantially eased by using a data-driven, event-based implementation of parareal. As a result, tasks for one iteration do not wait for the previous iteration to complete, but are started when the needed data are available. For given convergence properties, the event-based approach relaxes the speed requirements on the coarse solver by a factor of ∼K, where K is the number of iterations required for a converged solution. This may, for many problems, lead to an efficient parareal implementation that would otherwise not be possible or would require substantial coarse solver development. In addition, the framework used for this implementation executes a task when the data dependencies are satisfied and computational resources are available. This leads to improved computational efficiency over previous approaches that pipeline or schedule groups of tasks to a particular processor or group of processors. [Copyright &y& Elsevier]