Error estimates for Runge-Kutta type solutions to systems of ordinary differential equations

Means of estimating the truncation error of Runge-Kutta type methods are considered. A number of existing processes have various disadvantages, giving a valid estimate only for certain types of equation. The conditions are stated for a process to give a valid estimate when applied to a general system of differential equations, and a number of fifth order estimates are proposed for a fourth order Runge–Kutta type method. Some numerical tests are applied to these processes, which suggest their superiority over Merson's process.