Explicit symplectic RKN methods for perturbed non-autonomous oscillators: Splitting, extended and exponentially fitting methods

Abstract

We consider the numerical integration of perturbed non-autonomous oscillatory systems using high order methods. The autonomous case has been efficiently integrated using explicit and symplectic Runge–Kutta–Nyström (RKN) methods like extended RKN methods, exponentially fitting RKN methods and splitting methods for perturbed systems. Recently, it has been shown that explicit and symplectic extended RKN methods and exponentially fitting RKN methods are equivalent (Wu et al., 2012) and in this work we show that these methods are also equivalent to splitting methods for perturbed oscillators. We provide a constructive proof which at the same time allows us to build for the first time new explicit and symplectic extended RKN methods for the non-autonomous problem (for multidimensional time dependent frequencies). The new methods obtained, while built from splitting methods, are different in the treatment of the time-dependent terms and can be superior in some cases. We build some new methods and show their performance on numerical examples.