A class of implicit symmetric symplectic and exponentially fitted Runge–Kutta–Nyström methods for solving oscillatory problems

Abstract The construction of implicit Runge–Kutta–Nyström (RKN) method is considered in this paper. Based on the symmetric, symplectic, and exponentially fitted conditions, a class of implicit RKN integrators is obtained. The new integrators called ISSEFMRKN integrate exactly differential systems whose solutions are linear combinations of functions from the set {exp(λt),exp(−λt),λ∈C} $\{\exp(\lambda t), \exp(-\lambda t), \lambda\in\mathbb{C}\}$. In addition, their final stages also preserve the quadratic invariants {exp(2λt),exp(−2λt)} $\{\exp(2\lambda t), \exp(-2\lambda t)\}$. Especially, we derived two methods: ISSEFMRKNs1o2 and ISSEFMRKNs2o4 which are of order 2 and 4, respectively. And the method ISSEFMRKNs2o4 has variable nodes. The derived method ISSEFMRKNs2o4 reduces to the classical RKN method (Qin and Zhu in Comput. Math. Appl. 22(9):85–95, 1991) as λh→0 $\lambda h\rightarrow0$. The numerical results show that our methods possess the efficiency and competence compared with some implicit RKN methods in the literature. Especially, ISSEFMRKNs2o4 improves the accuracy compared with unmodified method ISSEFRKNs2o4 proposed in (Zhai and Chen in Numer. Algebra Control Optim. 9(1):71–84, 2019)