Lie groups and numerical solutions of differential equations: Invariant discretization versus differential approximation

TL;DR

This paper compares symmetry-preserving discretization and differential approximation methods for solving ODEs using Lie group theory, demonstrating how invariant schemes maintain original symmetries.

Contribution

It highlights the effectiveness of invariant discretization schemes in preserving Lie group symmetries in numerical solutions of ODEs.

Findings

Invariant discretization schemes preserve original symmetries.

Differential approximation methods may not maintain symmetries.

Symmetry preservation improves numerical solution properties.

Abstract

We briefly review two different methods of applying Lie group theory in the numerical solution of ordinary differential equations. On specific examples we show how the symmetry preserving discretization provides difference schemes for which the "first differential approximation" is invariant under the same Lie group as the original ordinary differential equation.