Multi-step Runge-Kutta-Nyström methods for special second-order initial value problems

In this paper, multi-step Runge-Kutta-Nyström methods for the numerical integration of special second-order initial value problems are proposed and studied. These methods include classical Runge-Kutta-Nyström methods as special cases. General order conditions are derived by using the theory of B-series based on the set of special Nyström-trees, and two explicit methods with order five and six, respectively, are constructed. Numerical results show that our new methods are more efficient in comparison with classical Runge-Kutta-Nyström methods and other well-known high quality methods proposed in the scientific literature.