Optimal Mesh Size in the Numerical Integration of an Ordinary Differential Equation

In the integration of a system of ordin'~ry differential equations, the simplest approach is to use a fixed step size. However, over some parts of the range of i~tegra~ion it: is generally possible to ~ake a larger step size wilhoue, seriously affecd~g the "local ~nmcatiou error." This gives rise to the currently popular "halving and doubling" me~hod, in which one changes the step size ir~ such a way as to keep the local truncation error more or less constant. This, however, is not necessarily optimal since a smM1 local truncation error in some parts of the range of integration can give rise to a large total truncation error. The basic problem is then to choose the step size in an optimal way; i.e. for a fixed mmff)er of mesh points, how should one distribute the mesh points in o:der to achieve the smallest error ~ at the end of the range of integration. (One might ask instead that the integral of the square of the truncation error over the range of integration should be minimized instead of the error at the end of the interval. This is reasonable when the error over the entire range is of interest instead of simply the error at the end. This problem does not seem to have a simple closed fonn solution and we will not discuss it here.) The problem in this generality is extremely diffmult; hence we will first approximate it, by a simpler problem, and we will solve the simpler problem eom-pletely. Specifically, we use the results of P. Henriei [1] on the asymptotic behavior of the tnmcation error in order to get the simpler problem. In order to solve ~he simpler problem we make the following assumptions: (1 There is only one differential equation. (Otherwise, a simple closed form solution such as given here does not seem to exist; instead one has an unpleasant integral equation to solve.) (2) The (approximate) local truncation error has one sign throughout the range of integration. (Otherwise, the solution becomes very strange; one may find that it is necessary to make as large an error as possible over some parts of the range of integration.) (3) The functions involved are sufficiently smooth so that the results of Henrici are valid. We will also make some further smoothness assumptions as we go along. For practical investigation one can sometimes weaken these assumptions. Thus if there …