Symmetry-preserving numerical schemes

TL;DR

This paper reviews two methods for creating symmetry-preserving finite difference schemes for differential equations, highlighting their advantages, applications to specific equations, and introducing innovative techniques and future research directions.

Contribution

It compares Lie symmetry-based and equivariant moving frame methods for constructing invariant numerical schemes, providing new insights and techniques.

Findings

Both methods effectively preserve symmetries in numerical schemes.

Numerical simulations demonstrate improved accuracy with invariant schemes.

The paper introduces innovative techniques for enhancing invariant numerical methods.

Abstract

In these lectures we review two procedures for constructing finite difference numerical schemes that preserve symmetries of differential equations. The first approach is based on Lie's infinitesimal symmetry generators, while the second method uses the novel theory of equivariant moving frames. The advantages of both techniques are discussed and illustrated with the Schwarzian differential equation, the Korteweg-de Vries equation and Burgers' equation. Numerical simulations are presented and innovative techniques for obtaining better invariant numerical schemes are introduced. New research directions and open problems are indicated at the end of these notes.