Derivation of Efficient, Continuous, Explicit Runge–Kutta Methods

Continuous, explicit Runge–Kutta methods with the minimal number of stages are considered. These methods are continuously differentiable if and only if one of the stages is the FSAL evaluation. A characterization of a subclass of these methods is developed for orders 3, 4, and 5. It is shown how the free parameters of these methods can be used either to minimize the continuous truncation error coefficients or to maximize the stability region. As a representative for these methods the fifth-order method with minimized error coefficients is chosen, supplied with an error estimation method, and analysed by using the DETEST software. The results are compared with a similar implementation of the Dormand–Prince 5(4) pair with interpolant, showing a significant advantage in the new method for the chosen problems.