Subquadrature expansions for TSRK methods

The representation of order conditions for general linear methods formulated using an algebraic theory by Butcher, and the alternative using B‐series by Hairer and Wanner for treating vector initial value problems in ordinary differential equations are well‐known. Each relies on a recursion over rooted trees; yet tractable forms–for example, those which may be solved to yield particular methods–often are obtained only after extensive computation. In contrast, for Runge–Kutta methods, tractable forms have been used by various authors for obtaining methods. Here, the corresponding recursion formula for two‐step Runge–Kutta methods is revised to yield tractable forms which may be exploited to derive such methods and to motivate the selection of efficient algorithms in an obvious way. The new recursion formula is utilized in a MAPLE code.