Functionally-fitted energy-preserving methods for solving oscillatory nonlinear Hamiltonian systems

Abstract: In the last few decades, numerical simulation for nonlinear oscillators has received a great deal of attention, and many researchers have been concerned with the design and analysis of numerical methods for solving oscillatory problems. In this paper, from the perspective of the continuous finite element method, we propose and analyze new energy-preserving functionally fitted methods, in particular trigonometrically fitted methods of an arbitrarily high order for solving oscillatory nonlinear Hamiltonian systems with a fixed frequency. To implement these new methods in a widespread way, they are transformed into a class of continuous-stage Runge--Kutta methods. This paper is accompanied by numerical experiments on oscillatory Hamiltonian systems such as the FPU problem and nonlinear Schr\"odinger equation. The numerical results demonstrate the remarkable accuracy and efficiency of our new methods compared with the existing high-order energy-preserving methods in the literature.