@article{9049,
  author = {Yvon Maday and Einar R{\o}nquist and Gunnar Staff},
  title = {The Parareal-in-Time Algorithm: Basics, Stability Analysis and More},
  abstract = {The parareal-in-time algorithm allows to take benefit of a parallel architecture of large scale computing resources in order to decrease the restitution time for the numerical simulation of time dependent problems. The method can be presented as a predictor corrector scheme where the predictor is based on a coarse grain and inexpensive simulation, solved in a serial manner and the expensive corrector can be spread on different processors. Like for domain decomposition techniques, the algorithm is based on the decomposition of the global simulation time interval into slabs, each slab being dedicated to a processor. After reminding the basics of the approximation, we discuss the stability of the algorithm for a system of autonomous differential equations. The stability function for the algorithm is derived and analyzed, based on various choices of schemes in time for the coarse and the fine propagator. We then present some complementary analysis that provides the frame for the definition of the cheap coarse simulation. Finally, numerical results for the viscous Burger{\textquoteright}s equation are presented.},
  year = {2006},
  journal = {journal},
}