Symbolic derivation of Runge-Kutta-Nyström type order conditions and methods for solving y′′′=f(x,y)

In this work we study the Runge-Kutta-Nyström (RKN) type methods for the solution of a special third order initial value problems. Based on rooted trees the relative order conditions theory is presented introducing a new set of SN-trees named ⊤3 whose elements' enumeration is given. A Mathematica package, that furnishes instantly order conditions of high orders, is also listed. Finally, a new method of order 8 is constructed that outperforms by far the methods found in the literature.