Conjugate-symplecticity properties of Euler--Maclaurin methods and their implementation on the Infinity Computer

TL;DR

This paper investigates Euler-Maclaurin methods for Hamiltonian systems, showing they are conjugate-symplectic up to a certain order and exploring their implementation on the Infinity Computer for improved efficiency.

Contribution

It demonstrates the conjugate-symplecticity of Euler-Maclaurin methods and proposes a novel implementation approach using the Infinity Computer.

Findings

Euler-Maclaurin methods are conjugate-symplectic up to order p+2.

Implementation on the Infinity Computer can efficiently compute higher derivatives.

These methods are promising for geometric integration of Hamiltonian systems.

Abstract

Multi-derivative one-step methods based upon Euler-Maclaurin integration formulae are considered for the solution of canonical Hamiltonian dynamical systems. Despite the negative result that simplecticity may not be attained by any multi-derivative Runge--Kutta methods, we show that the Euler-MacLaurin method of order p is conjugate-symplectic up to order p+2. This feature entitles them to play a role in the context of geometric integration and, to make their implementation competitive with the existing integrators, we explore the possibility of computing the underlying higher order derivatives with the aid of the Infinity Computer.