Finite difference schemes and partial differential equations
Finite difference schemes and partial differential equations
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Evaluating derivatives: principles and techniques of algorithmic differentiation
Evaluating derivatives: principles and techniques of algorithmic differentiation
Analysis of the Parareal Time-Parallel Time-Integration Method
SIAM Journal on Scientific Computing
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'Parareal' attempts to speed up the solution of ordinary differential equations (ODEs) by parallelizing the time dimension. Different solutions are obtained in parallel on different subintervals. An iterative procedure is employed to match the solution values at the end of each subinterval with those at the beginning of the next one. It has been shown that insufficient 'parareal' iterations lead to an inaccurate solution, while too many iterations waste CPU cycles without bringing any improvement to the solution. In this paper we discuss an error control mechanism for both the classical 'parareal' time discretization method and its adaptive variant. This mechanism can be effectively used as a mean of controlling the number of 'parareal' iterations. We show that bounding the difference between the solution of the fine integrator and the 'parareal' solution by the local truncation error of the fine grid leads to a sufficient convergence criterion that guaranties a solution that is accurate enough in a minimum number of iterations. Tests on a nonlinear problem illustrate the effectiveness of the proposed error control mechanism on the classical automatic adaptive time stepping formulation of an embedded method on the two-level 'parareal' algorithm.