Multi-step zero approximations for stepsize control (Q1567635)

The one-step-two-half-step approach, based on Richardson extrapolation, has now been replaced by embedding methods as the preferred method of error estimation. This paper deals with several ways of deriving at least correct-order error estimations which can be used with low computational costs over a few steps. Compared with a lower-order error estimation under the same techniques, the corresponding multi-step zero approximation generally takes steps but is less accurate. The authors discuss the construction of ``zero approximations'' based on embedded pairs of Runge-Kutta methods. A stepsize control scheme that is suitable for the use with the new estimators is presented.