A variable step-size implementation of a variational method for stiff differential equations

In some recent works, we have proposed a new approximation of stiff systems of differential equations based on an analysis of a certain error functional associated, in a natural way, with the original problem. By using standard descent schemes, the procedure can never get stuck in local minima (since the error functional only has a minimum), but will always and steadily decrease the error until getting to the original solution (the only minimum). In this paper, we propose a variable step-size implementation of our variational approach. We are able to perform this implementation without any extra functional evaluation. We show the efficacy of this new implementation on some standard problems found in the literature.