Long-time analysis of extended RKN integrators for Hamiltonian systems with a solution-dependent high frequency
Provides theoretical long-time stability guarantees for numerical integration of highly oscillatory Hamiltonian systems, which is important for computational physics and engineering.
The paper proves that symmetric extended RKN integrators approximately conserve modified action and total energy over long times for Hamiltonian systems with solution-dependent high frequency, supported by numerical experiments.
In this paper, we analyse the long-time behaviour of the extended RKN (ERKN) integrators for solving highly oscillatory Hamiltonian systems with a slowly varying, solution-dependent high frequency. We prove that a symmetric ERKN integrator approximately conserves a modified action and a modified total energy over long time intervals based on the technique of varying-frequency modulated Fourier expansion. An illustrative numerical experiment is carried out and the numerical results strongly support the theoretical analysis presented in this paper. As a byproduct of this work, similar long-time behaviour is also investigated for an RKN method.