The Korteweg-de Vries equation and its symmetry-preserving discretization

TL;DR

This paper develops invariant discretization schemes for the Korteweg-de Vries equation in both Lagrangian and Eulerian forms, preserving symmetries like Galilean invariance, with numerical tests showing comparable accuracy to standard methods.

Contribution

It introduces new invariant discretization schemes for the Korteweg-de Vries equation that preserve key symmetries such as Galilean invariance, enhancing numerical fidelity.

Findings

Invariant schemes preserve Galilean transformations.

Accuracy comparable to standard schemes.

Numerical validation confirms symmetry preservation benefits.

Abstract

The Korteweg-de Vries equation is one of the most important nonlinear evolution equations in the mathematical sciences. In this article invariant discretization schemes are constructed for this equation both in the Lagrangian and in the Eulerian form. We also propose invariant schemes that preserve the momentum. Numerical tests are carried out for all invariant discretization schemes and related to standard numerical schemes. We find that the invariant discretization schemes give generally the same level of accuracy as the standard schemes with the added benefit of preserving Galilean transformations which is demonstrated numerically as well.