Runge--Kutta methods determined from extended phase space methods for Hamiltonian systems

TL;DR

This paper analyzes two extended phase space integrators for Hamiltonian systems, revealing their symplectic and pseudosymplectic properties, and classifies them as specific types of Runge--Kutta methods.

Contribution

It provides a detailed classification of existing extended phase space integrators as Runge--Kutta methods with symplectic and pseudosymplectic properties.

Findings

Midpoint projection method is pseudosymplectic and pseudosymmetric Runge--Kutta.

Symmetric projection method is a monoimplicit symplectic Runge--Kutta.

Both methods preserve geometric properties of Hamiltonian systems.

Abstract

We study two existing extended phase space integrators for Hamiltonian systems, the {\em midpoint projection method} and the {\em symmetric projection method}, showing that the first is a pseudosymplectic and pseudosymmetric Runge--Kutta method and the second is a monoimplicit symplectic Runge--Kutta method.