Effective integrators for nonlinear second-order oscillatory systems with a time-dependent frequency matrix

Abstract

In this paper extended Runge–Kutta–Nyström (ERKN) methods are proposed for the system of oscillatory second-order initial value problems q″=-M(t)q+f(t,q),q(t0)=q0,q′(t0)=q0′, where M(t) is a time-dependent frequency matrix. If M(t) is symmetric and positive semi-definite and f(t,q) is the negative gradient of a real-valued function with respect to q whose second derivatives are continuous, then the system is a time-dependent Hamiltonian system. Since the classical variation-of-constants approach does not work for the system with a time-dependent frequency matrix M(t), we propose a recipe for adapting it to the system, namely, an equivalent system is introduced. Based on the approach, we formulate and analyze the ERKN methods for solving the system of oscillatory second-order initial value problems, or the time-dependent Hamiltonian system. In particular, the symplectic ERKN integrators developed very recently are applied to the equivalent system. Furthermore, the novel and important analysis within the broader framework of the main subject is made in an extended phase space for the nonautonomous Hamiltonian system. Accordingly, the ERKN schemes for the nonautonomous Hamiltonian system are presented in the extended phase space. Numerical experiments are accompanied in comparison with a symmetric and symplectic RKN method and a symmetric and symplectic composition method in the literature.