Summary: The theory of Runge-Kutta methods for problems of the form \(y^{\prime} = f(y)\) is extended to include the second derivative \(y^{\prime \prime} = g(y): = f^{\prime}(y)f(y)\). We present an approach to the order conditions based on Butcher’s algebraic theory of trees [cf. J. F. Butcher, Math. Comput. 26, 79–106 (1972; Zbl 0258.65070)], and derive methods that take advantage of cheap computations of the second derivatives. Only explicit methods are considered here where attention is given to the construction of methods that involve one evaluation of \(f\) and many evaluations of \(g\) per step. Methods with stages up to five and of order up to seven including some embedded pairs are presented.
The first part of the paper discusses a theoretical formulation used for the derivation of these methods which are also of wider applicability. The second part presents experimental results for non-stiff and mildly stiff problems. The methods include those with the computation of one second derivative (plus many first derivatives) per step, and embedded methods for changing stepsize as well as those involving one first derivative (plus many second derivatives) per step. The experiments have been performed on standard problems and comparisons made with some standard explicit Runge-Kutta methods.