In this paper, we develop a new mesh selection strategy based on the computation of some conditioning parameters which allows to give information about the conditioning and the stiffness of the problem. The reliability of the proposed algorithm is demonstrated by some numerical experiments. We observe that “when an initial value problem is run on a computer, the results may appear plausible even if they are unreliable because of some unrecognized numerical instability” (Miller, 1967) [23]. The additional information about the behavior of the numerical solution provided by the new mesh selection algorithm are, therefore, of great interest for potential users of a numerical computer code.