Some procedures for the construction of high-order exponentially fitted Runge–Kutta–Nyström methods of explicit type Original Research Article

The construction of high-order exponentially fitted Runge–Kutta–Nyström (EFRKN) methods of explicit type for the numerical solution of oscillatory differential systems is analyzed. Based on two basic symmetric and symplectic EFRKN methods of reference we present two procedures for constructing high-order explicit methods. The first procedure is based on composition methods and it allows the construction of high-order explicit EFRKN methods which are symmetric and symplectic. The second procedure is based on combining different EFRKN methods in order to construct embedded pairs of explicit parallel EFRKN methods which can be implemented in variable-step codes without additional cost. The numerical experiments carried out show the qualitative behavior and the efficiency of the new EFRKN methods when they are compared with some standard methods proposed in the scientific literature for solving second-order nonstiff differential systems.