Discretizations preserving all Lie point symmetries of the Korteweg-de Vries equation

TL;DR

This paper presents methods for discretizing the Korteweg-de Vries equation while preserving all its Lie point symmetries, ensuring the discrete model maintains the continuous symmetries.

Contribution

It introduces two approaches for symmetry-preserving discretizations of the Korteweg-de Vries equation using centered implicit schemes.

Findings

At least two symmetry-preserving discretization methods identified

Centered implicit schemes can maintain all Lie point symmetries

Enhances numerical methods by preserving fundamental symmetries

Abstract

We show how to descritize the Korteweg-de Vries equation in such a way as to preserve all the Lie point symmetries of the continuous differential equation. It is shown that, for a centered implicit scheme, there are at least two possible ways of doing so.