Symplectic and symmetric trigonometrically-fitted ARKN methods

Abstract

Adapted Runge–Kutta–Nyström (ARKN) methods for solving oscillatory problems q″(t)+w2q(t)=f(q(t),q′(t)) have been investigated by several authors. Recently, Yang et al. (2008) [28] proposed trigonometrically-fitted ARKN (TFARKN) methods by introducing frequency depending coefficients into the terms in the internal stages. In applications the function f(q) often does not contain q′ explicitly and satisfies f(q)−w2q=−∇V(q) for some smooth function V(q). Then the problem can be considered as a separable Hamiltonian system. In this paper we investigate the symplecticity and symmetry of TFARKN methods for separable Hamiltonian systems and derive necessary and sufficient conditions for an TFARKN method to be symplectic and symmetric. Based on these conditions, two explicit symplectic and symmetric TFARKN method with order two and four, respectively, are constructed. Some numerical experiments are provided to confirm the theoretical expectations.