How to improve the accuracy of the discrete gradient method in the one-dimensional case

TL;DR

This paper introduces an improved numerical scheme for one-dimensional dynamical systems that enhances the accuracy of the discrete gradient method while maintaining its stability and energy conservation features.

Contribution

The authors propose a modified discrete gradient method that significantly increases accuracy without sacrificing stability or energy conservation.

Findings

Accuracy improved by several orders of magnitude

Maintains stability and energy conservation

Applicable to one-dimensional dynamical systems

Abstract

We present a new numerical scheme for one dimensional dynamical systems. This is a modification of the discrete gradient method and keeps its advantages, including the stability and the conservation of the energy integral. However, its accuracy is higher by several orders of magnitude.