Symplecticity-preserving continuous-stage Runge-Kutta-Nystr\"{o}m methods

TL;DR

This paper introduces continuous-stage Runge-Kutta-Nyström methods designed to preserve symplectic structure when solving second order ODEs, especially those reducible to Hamiltonian systems, enhancing numerical stability and accuracy.

Contribution

The paper provides a sufficient condition for symplecticity-preserving csRKN methods and offers a straightforward construction approach using Legendre polynomial expansion.

Findings

Developed a condition for symplecticity preservation in csRKN methods.

Presented a simple construction method using Legendre polynomials.

Applicable to Hamiltonian systems and second order ODEs.

Abstract

We develop continuous-stage Runge-Kutta-Nystr\"{o}m (csRKN) methods for solving second order ordinary differential equations (ODEs) in this paper. The second order ODEs are commonly encountered in various fields and some of them can be reduced to the first order ODEs with the form of separable Hamiltonian systems. The symplecticity-preserving numerical algorithm is of interest for solving such special systems. We present a sufficient condition for a csRKN method to be symplecticity-preserving, and by using Legendre polynomial expansion we show a simple way to construct such symplectic RKN type method.