Adaptive Parareal for Systems of ODEs

Abstract

Summary. The parareal scheme (resp. PITA algorithm) proposed in [3] (resp. [2]) 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 interface between the time subdomains, the solution makes a jump between 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, we study 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 recalled and it is shown that it is a particular case of the multiple shooting method of Deuflhard [1]. Then in section 2, the definition of the size 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 and enables us 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 for this stiff problem.