Towards new schemes: A Lie-group approach of the CBKDV and its derived equations

TL;DR

This paper introduces a Lie-group based approach to develop numerical schemes that preserve symmetries of differential equations, demonstrated on the CBKDV and Burgers equations.

Contribution

It proposes a novel method for constructing symmetry-preserving numerical schemes using Lie-group theory, applied to the CBKDV equation.

Findings

Semi-invariant scheme for Burgers equation presented

Method preserves Lie symmetries of the original equations

Potential for improved numerical accuracy and structure preservation

Abstract

The aim of this paper is to propose methods that enable us to build new numerical schemes, which preserve the Lie symmetries of the original differential equations. To this purpose, the compound Burgers-Korteweg-de Vries (\textit{CBKDV}) equation is considered. The particular case of the Burgers equation is taken as a numerical example, and the resulting semi-invariant scheme is exposed.