Starting algorithms for the iterations of the RKN–Gauss methods

Abstract

In this paper the construction of starting algorithms which produce highly accurate initial approximations to the internal stages of the 2- and 3-stage Runge–Kutta–Nyström Gauss methods is presented, in order to apply these formulas to the integration of non-stiff Hamiltonian systems. Firstly, we point out a natural way of obtaining starting algorithms for induced RKN methods and we show how their order can be studied. Together with starting algorithms that do not require any additional cost, other algorithms of higher order are constructed for the RKN–Gauss formulas by adding one or two function evaluations per step. The order of the iterated values of the stages and of the numerical solution is also given when simple iteration is used to compute the internal stages of an implicit RKN method. Finally, some numerical experiments are carried out to show the better performance of the starting algorithms of high order here constructed.