A General Class of Two-Step Runge–Kutta Methods for Ordinary Differential Equations

A general class of two-step Runge–Kutta methods that depend on stage values at two consecutive steps is studied. These methods are special cases of general linear methods introduced by Butcher and are quite efficient with respect to the number of function evaluations required for a given order. General order conditions are derived using the approach proposed recently by Albrecht, and examples of methods are given up to the order 5. These methods can be divided into four classes that are appropriate for the numerical solution of nonstiff or stiff differential equations in sequential or parallel computing environments.