A sixth order averaged vector field method

TL;DR

This paper introduces a sixth order energy-preserving integrator for Hamiltonian systems based on an advanced rooted tree and B-series theory, extending the averaged vector field method to higher order accuracy.

Contribution

The paper develops a new sixth order AVF integrator using a novel substitution law for trees, enhancing energy preservation and accuracy in numerical simulations.

Findings

The sixth order AVF method precisely preserves energy in Hamiltonian systems.

Numerical experiments confirm the method's high accuracy and energy conservation.

The new integrator outperforms lower-order methods in efficiency and stability.

Abstract

In this paper, based on the theory of rooted trees and B-series, we propose the concrete formulas of the substitution law for the trees of order =5. With the help of the new substitution law, we derive a B-series integrator extending the averaged vector field (AVF) method to high order. The new integrator turns out to be of order six and exactly preserves energy for Hamiltonian systems. Numerical experiments are presented to demonstrate the accuracy and the energy-preserving property of the sixth order AVF method.