Adaptive Parareal for systems of ODEs

The parareal scheme (resp. PITA algorithm) proposed in [4] (resp. [1]) considers two levels of grids in time in order to split the domain in time-subdomains. A prediction of the solution is computed on the fine grid in parallel. Then at each end boundaries of time subdomains, the solution makes a jump with the previous initial boundary value (IBV) of the next time-subdomain . A correction of the IBV for the next fine grid iteration is then computed on the coarse grid in time. In this paper, it is proposed to investigate adaptivity in the time slice decomposition based on an a posteriori numerical estimation obtained from the time step behavior on coarse grids. The outline of this paper is as follows: in section 1, the original parareal method is reminded and it is shown that the latter is a particular case of the multiple shooting method of Deuflhard [2]. Then in section 2, the definition of the fineness of the grids is slightly modified in order to introduce adaptivity within the parareal algorithm for the time stepping, the number of subdomains, and the time decomposition. This adaptivity leads to an improvement of the method so as to solve moderately stiff nonlinear ODEs problems. Nevertheless for very stiff problems as the Oregonator model, it fails even with the introduced adaptivity. This leads to develop in section 3 an adaptive parallel extrapolation method, based on a posteriori numerical assessment, which obtains results on this stiff problem.