The minimal stage, energy preserving Runge-Kutta method for polynomial Hamiltonian systems is the Averaged Vector Field method

TL;DR

This paper proves that for polynomial Hamiltonian systems, the minimal-stage energy-preserving Runge-Kutta method is uniquely the Averaged Vector Field (AVF) method, establishing its optimality in this context.

Contribution

The paper demonstrates that the AVF method is the unique minimal-stage energy-preserving Runge-Kutta scheme for polynomial Hamiltonian systems.

Findings

The AVF method can be viewed as a Runge-Kutta method with a quadrature rule of degree at least that of the Hamiltonian.

When the number of stages is minimal, the Runge-Kutta scheme must be the AVF method.

The result characterizes the AVF method as the minimal-stage energy-preserving scheme for polynomial Hamiltonian systems.

Abstract

No Runge-Kutta method can be energy preserving for all Hamiltonian systems. But for problems in which the Hamiltonian is a polynomial, the Averaged Vector Field (AVF) method can be interpreted as a Runge-Kutta method whose weights $b_i$ and abscissae $c_i$ represent a quadrature rule of degree at least that of the Hamiltonian. We prove that when the number of stages is minimal, the Runge-Kutta scheme must in fact be identical to the AVF scheme.