The 'Parareal in Time' Algorithm

In this paper we present the current status of a method, first introduced in 2001 authored by J.-L. Lions, Y. Maday and G. Turinici that allows for parallization in time for the simulation of systems of Ordinary Differential Equations or time dependent partial differential equations. Following the same strategy as the one that is used in domain decomposition methods for solving elliptic problems that consists in breaking the domain of computation into subdomains (with overlap or without) and solving iteratively over each subdomain independently using different processors, the “parareal in time” method proposes to break the global problem of time evolution into a series of independent evolution problems on smaller time intervals. The iterative algorithm is based on a predictor corrector approach that generally converges quite fast, and leads, when very many processors are available, to real time solution procedures. This reasoning led us to name ”parareal in time” (parallel in real time) this new algorithm.