Remarks on Picard-Lindelöf iteration

The paper discusses Picard-Lindelöf iteration for systems of autonomous linear equations on finite intervals, as well as its numerical variants. Most of the discussion is under a model assumption which roughly says that the coupling terms are of moderate size compared with the slow time scales in the problem. It is shown that the speed of convergence is quite independent of the step sizes already for very large time steps. This makes it possible to design strategies in which the mesh gets gradually refined during the iteration in such a way that the iteration error stays essentially on the level of discretization error.This research was supported by the Academy of Finland and by the Institute for Mathematics and its Applications (IMA, Minneapolis with funds provided by NSF). Ulla Miekkala visited IMA with a grant from Tekniikan edisstämissäätiö and carried out the numerical tests.