Extended Runge-Kutta-like formulae

In this paper we present a new family of extended Runge-Kutta formulae in which, just like in Enright's methods, it is assumed that the user will evaluate both f and f' readily when solving the autonomous system y' = f(y) numerically. This means that we introduce some new parameters in the extended Runge-Kutta-like formulae in order to enhance the order of accuracy of the solutions using evaluations of both f and f', instead of the evaluations of f only. Moreover, if f' is approximated by a difference quotient of past and current evaluations of f, the order of convergence can be retained. The resulting two-step Runge-Kutta method can be regarded as replacing the function evaluations of f' with approximations of f'. Specifically, the proposed formulae with f' are more efficient for cases where f' is not more expensive to evaluate than f and the proposed 'derivative-free' formulae are more attractive for use when past values of f are available. Furthermore error estimates and step-choose strategies are considered for the 'derivative-free' extended Runge-Kutta methods.