  We discuss adaptive mesh point selection for the solution of scalar IVPs. We
consider a method that is optimal in the sense of the speed of convergence, and
aim at minimizing the local errors. Although the speed of convergence cannot be
improved by using the adaptive mesh points compared to the equidistant points,
we show that the factor in the error expression can be significantly reduced.
We obtain formulas specifying the gain achieved in terms of the number of
discretization subintervals, as well as in terms of the prescribed level of the
local error. Both nonconstructive and constructive versions of the adaptive
mesh selection are shown, and a numerical example is given.
