Automatic step size and order control in explicit one-step extrapolation methods

A general theory is presented for explicit one-step extrapolation methods for ordinary differential equations. The emphasis is placed on the efficient use of extrapolation processes of this type in practice. The choice of the optimal step size and the order at each grid point is made in the automatic mode with the minimum computational work per step being the guiding principle. This principle makes it possible to find a numerical solution in the minimal time. The efficiency of the automatic step size and order control is demonstrated using test problems for which the well-known GBS method was used.