On fifth order Runge-Kutta methods (Q1899930)

Runge-Kutta methods applied to a single equation \(z'= g(x, z)\) and to an autonomous system \(y'= f(y)\) are considered. To obtain a method of order 5 there are 16 order conditions for the single equation and 17 order conditions for the system. A particular method is derived which is order 5 for the single equation but only order 4 for the system, thus establishing that it is important to consider systems when deriving such methods. Numerical experiments are devised to show that the orders appear indeed to be 5 and 4 in practice.