On computing smooth solutions of problems with large Lipschitz constants

We investigate how one can construct numerical methods for computing smooth solutions of ODE's which potentially possess fast growing or decaying solutions. We do not want to use a global method (which computes a solution on the entire relevant interval first), but rather a procedure that obtains numerical values in a marching algorithm. It is shown how this can be achieved by both implicit and explicit integrators, for which some detailed analysis is given. Some numerical examples are also included.