High order symplectic integrators based on continuous-stage Runge-Kutta Nystrom methods

TL;DR

This paper introduces a new technique using Legendre expansions to efficiently construct high-order symplectic integrators for second order Hamiltonian systems, simplifying the process compared to previous methods.

Contribution

It presents a novel approach employing Legendre expansions to design high-order symplectic integrators, improving efficiency over traditional step-by-step order condition analysis.

Findings

Efficient construction of high-order symplectic integrators

Use of Legendre expansions simplifies order condition analysis

Truncating orthogonal series enables practical integrator design

Abstract

On the basis of the previous work by Tang \& Zhang (Appl. Math. Comput. 323, 2018, p. 204--219), in this paper we present a more effective way to construct high-order symplectic integrators for solving second order Hamiltonian equations. Instead of analyzing order conditions step by step as shown in the previous work, the new technique of this paper is using Legendre expansions to deal with the simplifying assumptions for order conditions. With the new technique, high-order symplectic integrators can be conveniently devised by truncating an orthogonal series.