Functionally Fitted Continuous Finite Element Methods for Oscillatory Hamiltonian Systems

Abstract

In recent decades, the numerical simulation for nonlinear oscillators has received much attention and a large number of integrators for oscillatory problems have been developed. In this chapter, based on the continuous finite element approach, we propose and analyse new energy-preserving functionally-fitted, in particular, trigonometrically-fitted methods of an arbitrarily high order for solving oscillatory nonlinear Hamiltonian systems with a fixed frequency. In order to implement these new methods in an accessable and efficient style, they are formulated as a class of continuous-stage Runge–Kutta methods. The numerical results demonstrate the remarkable accuracy and efficiency of the new methods compared with the existing high-order energy-preserving methods in the literature.