Estimating the error in the classical Runge–Kutta methods

It is well known that it is impossible to construct embedded firth-order methods for estimating the error in four-stage Runge–Kutta methods of order four. In this paper, a technique for error estimating with no additional calculations of the right-hand sides of equations is proposed. The proposed estimate is of fifth order and is based on the data provided by three successive steps of the method. The main results of the paper are formulas for evaluating the local error based on two and three steps of the method, respectively. The main conclusion of the paper is that an automatic stepsize control should not necessarily be based on embedded methods. Such a control can be implemented for an arbitrary method.