A general family of explicit Runge-Kutta pairs of orders 6(5)

Explicit Runge-Kutta formula pairs of different orders of accuracy form a class of efficient algorithms for treating nonstiff ordinary differential equations. So far, several Runge-Kutta pairs of order 6(5) have appeared in the literature. These pie use 8 function evaluations per step and belong to certain families of solutions of a set of 54 nonlinear algebraic equations in 44 or 45 coefficients, depending on the use of the FSAL (first stage as last) device. These equations form a set of necessary and sufficient conditions that a 6(5) Runge-Kutta pair must satisfy. The solution of the latter is achieved by employing various types of simplifying assumptions. In this paper we make use of the fact that all these families of pairs satisfy a common set of simplifying assumptions. Using only these simplifying assumptions we define a new family of 6(5) Runge-Kutta pairs. Its main characteristic, which is also a property that no other known family shares, is that all of its nodes (except the last one, which equals 1) are free parameters of the resulting solution. A search has been carried out among the pairs of the new family and two nearly optimum pairs, with respect to accuracy and stability characteristics, have been constructed. The new pairs, as is exhibited by several numerical examples, compare favorably with all other currently known similar pairs.