  The Parareal algorithm is used to solve time-dependent problems considering
multiple solvers that may work in parallel. The key feature is a initial rough
approximation of the solution that is iteratively refined by the parallel
solvers. We report a derivation of the Parareal method that uses a convergence
acceleration technique to improve the accuracy of the solution. Our approach
uses firstly an explicit ODE solver to perform the parallel computations with
different time-steps and then, a decomposition of the solution into specific
convergent series, based on an extrapolation method, allows to refine the
precision of the solution. Our proposed method exploits basic explicit
integration methods, such as for example the explicit Euler scheme, in order to
preserve the simplicity of the global parallel algorithm. The first part of the
paper outlines the proposed method applied to the simple explicit Euler scheme
and then the derivation of the classical Parareal algorithm is discussed and
illustrated with numerical examples.
