Title: The parareal in time algorithm for fast simulations of time dependent PDE's : Basics and new developments and directions

Author: Yvon Maday- University of Paris

Yvon Maday is the Director of the Jacques-Louis Lions Laboratory, which is part of the Mathematics Department at the Pierre and Marie Curie University, where he is also a professor. He is also Director of the doctoral school of mathematical sciences and on the editorial board of 4 international journals, including the Journal of Scientific Computing. His research interests involve areas including numerical analysis of spectral and finite element methods, fluid structure interaction phenomena, domain decomposition, and parallel-in-time discretization.

Abstract:

Domain decomposition methods for solving partial differential equations consist of breaking the computation domain into subdomains (with overlap or without) and solving iteratively over each subdomain independently using different processors. Parallelisation in time has not been considered as much, but it is possible following about the same strategy.

The parareal in time algorithm is a predictor corrector method that allows parallelisation of the time dimension. It consists of segmenting intervals of time integration into slabs, and then proposing iterative seed values at the left hand side of each of the slabs. The derivation of the seed values is done sequentially and involves a propagator that should mimic the equation to be solved but should be very cheap. Next, parallel accurate propagations of each seed value over each slab are performed. We give the basics of the algorithm and illustrate through various examples the pertinency of the approach to get at least impressive speed up in the restitution time and most often full efficiency of the parallel method. We shall present new realizations and provide some open problems and ways to may be overcome them.