Energy-preserving numerical schemes of high accuracy for one-dimensional Hamiltonian systems

TL;DR

This paper introduces high-accuracy, energy-preserving numerical schemes for 1D Hamiltonian systems, improving long-term energy conservation and accuracy over existing methods.

Contribution

It develops a new class of energy-preserving schemes based on modifications of the discrete gradient method, including high-order and locally exact variants.

Findings

Energy is preserved up to round-off error.

High-order schemes outperform standard methods in long-term energy conservation.

Numerical experiments demonstrate superior accuracy and stability.

Abstract

We present a class of non-standard numerical schemes which are modifications of the discrete gradient method. They preserve the energy integral exactly (up to the round-off error). The considered class contains locally exact discrete gradient schemes and integrators of arbitrary high order. In numerical experiments we compare our integrators with some other numerical schemes, including the standard discrete gradient method, the leap-frog scheme and a symplectic scheme of 4th order. We study the error accumulation for very long time and the conservation of the energy integral.