Starting algorithms for implicit Runge-Kutta-Nyström methods Original Research Article

In this work we study algorithms that produce initial values to solve the algebraic system of the internal stages of some implicit Runge-Kutta-Nyström (RKN) methods. Firstly, starting algorithms that do not require additional function evaluations are studied. By using Butcher-series with special Nyström rooted trees, the order equations for these algorithms are obtained. By imposing some simplifying assumptions on the coefficients of the s-stage RKN method, it is shown that this type of algorithms attains order s + 1. Next we study algorithms that need one or two additional function evaluations per step. After obtaining the corresponding order equations, we prove that starting algorithms of order s + 2 and s + 3 can be obtained, depending on the type of algorithm and of the simplifying conditions imposed on the coefficients of the RKN method. Special attention is devoted to starting algorithms for RKN methods induced by implicit Runge-Kutta formulas and for collocation RKN methods.