Stability of Runge–Kutta–Nyström methods

In this paper, a general and detailed study of linear stability of Runge–Kutta–Nyström (RKN) methods is given. In the case that arbitrarily stiff problems are integrated, we establish a condition that RKN methods must satisfy so that a uniform bound for stability can be achieved. This condition is not satisfied by any method in the literature. Therefore, a stable method is constructed and some numerical comparisons are made.