ABSTRACT
Runge–Kutta methods have been popular with practitioners for many years. Originally developed by Runge towards the end of the nineteenth century and generalised by Kutta in the early twentieth century, these methods are very convenient to implement, when compared with the Taylor polynomial scheme, which requires the formation and evaluation of higher derivatives. The program is designed to implement any explicit Runge-Kutta process, a feature which, although rarely seen in published programs, can be achieved rather easily without significant loss of computational efficiency. Comparing Runge-Kutta formulae of different orders, the highest order method must be the most efficient when the step–size is small enough. Of course, problems with a moderate error requirement may be best approached with a formula of intermediate order. The chapter deals with the derivation of Runge–Kutta formulae of increasing orders. It is necessary to check, by direct computation, the properties of the methods constructed.
