High-order Gauss-Legendre methods admit a composition representation and a conjugate-symplectic counterpart

TL;DR

This paper demonstrates that higher-order Gauss-Legendre methods can be expressed as compositions and have conjugate-symplectic counterparts, extending classical composition structures known for lower-order methods.

Contribution

The authors show that high-order Gauss-Legendre methods admit a composition representation and introduce a conjugate-symplectic counterpart, generalizing known structures from lower-order schemes.

Findings

Higher-order Gauss-Legendre methods can be represented as compositions.

A conjugate-symplectic counterpart to these methods exists.

The results extend classical composition structures to higher orders.

Abstract

One of the most classical pairs of symplectic and conjugate-symplectic schemes is given by the Midpoint method (the Gauss-Runge-Kutta method of order 2) and the Trapezoidal rule. These can be interpreted as compositions of the Implicit and Explicit Euler methods, taken in direct and reverse order, respectively. This naturally raises the question of whether a similar composition structure exists for higher-order Gauss-Legendre methods. In this paper, we provide a positive answer by first examining the fourth-order case and then outlining a generalization to higher orders.