Improving the accuracy of the AVF method

The average vector field (AVF) method is a BB-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.