Discrete gradient algorithms of high order for one-dimensional systems

TL;DR

This paper introduces high-order discrete gradient algorithms for one-dimensional systems that significantly improve accuracy and stability while preserving energy and trajectories, even with large time steps.

Contribution

The authors develop a method to increase the order of discrete gradient integrators without sacrificing their stability and conservation properties.

Findings

Higher accuracy by several orders compared to standard schemes

Exceptional stability and exact conservation of energy

High accuracy maintained with large time steps

Abstract

We show how to increase the order of one-dimensional discrete gradient numerical integrator without losing its advantages, such as exceptional stability, exact conservation of the energy integral and exact preservation of the trajectories in the phase space. The accuracy of our integrators is higher by several orders of magnitude as compared with the standard discrete gradient scheme (modified midpoint rule) and, what is more, our schemes have very high accuracy even for large time steps.