On the order conditions of Runge-Kutta methods with higher derivatives

SummaryRunge-Kutta methods have been generalized to procedures with higher derivatives of the right side ofy′=f(t,y) e.g. by Fehlberg 1964 and Kastlunger and Wanner 1972. In the present work some sufficient conditions for the order of consistence are derived for these methods using partially the degree of the corresponding numerical integration formulas. In particular, methods of Gauß, Radau, and Lobatto type are generalized to methods with higher derivatives and their maximum order property is proved. The applied technique was developed by Crouzeix 1975 for classical Runge-Kutta methods. Examples of simple explicit and semi-implicit methods are given up to order 7 and 6 respectively.