Energy-preserving trigonometrically fitted continuous stage Runge-Kutta-Nyström methods for oscillatory Hamiltonian systems

Abstract

Recently, continuous-stage Runge-Kutta-Nystrom (CSRKN) methods for solving numerically second-order initial value problem $q^{\prime \prime }= f(q)$ have been proposed and developed by Tang and Zhang (Appl. Math. Comput. 323, 204–219, 2018). This problem is equivalent to a separable Hamiltonian system when f(q) = −∇U(q) with smooth function U(q). Symplecticity-preserving discretizations of this system were studied in that paper. However, as an important representation of the Hamiltonian system, energy preservation has not been studied. In addition, many Hamiltonian systems in practical applications often have oscillatory characteristics so we should design special algorithms adapted to this feature. In this paper, we propose and study energy-preserving trigonometrically fitted CSRKN methods for oscillatory Hamiltonian systems. We extend the theory of trigonometrical fitting to CSRKN methods and derive sufficient conditions for energy preservation. We also study the symmetry and stability of the methods. Two symmetric and energy-preserving trigonometrically fitted schemes of order two and four, respectively, are constructed. Some numerical experiments are provided to confirm the theoretical expectations.