Improving the accuracy of the AVF method

TL;DR

This paper introduces two higher-order, energy-preserving modifications of the AVF method, enhancing accuracy for Hamiltonian systems, with applications demonstrated on spherically symmetric potentials including the Coulomb-Kepler problem.

Contribution

The paper presents AVF-LEX and AVF-SLEX, new locally exact, energy-preserving modifications of the AVF method achieving third and fourth order accuracy.

Findings

AVF-LEX achieves third-order accuracy while preserving energy.

AVF-SLEX achieves fourth-order accuracy with energy preservation.

Explicit scheme for Coulomb-Kepler problem provided.

Abstract

The Average Vector Field (AVF) method is a B-series scheme of the second order. As a discrete gradient method it preserves exactly the energy integral for any canonical Hamiltonian system. We present and discuss two locally exact and energy-preserving modifications of the AVF method: AVF-LEX (of the third order) and AVF-SLEX (of the fourth order). Applications to spherically symmetric potentials are given, including a compact explicit expression for the AVF scheme for the Coulomb-Kepler problem.