Order conditions for a class of two‐step methods for y″ = f (x, y)

The theory of B‐series is used to investigate the order of convergence of a general class of two‐step hybrid methods for systems of differential equations of the special form y″ = f (x, y). The main result is a set of order conditions, analogous to those for Runge–Kutta methods, offering an alternative to the customary ad hoc Taylor expansions. A byproduct is a remarkably simple formula from which the order of dispersion of such methods is easily determined. Conditions under which the two‐step methods are symmetric are established, and particular examples are considered.