Discretization of partial differential equations preserving their physical symmetries

TL;DR

This paper introduces a method to discretize partial differential equations while preserving their Lie point symmetries, ensuring the discrete models retain the original equations' physical properties.

Contribution

It presents a systematic procedure for creating minimal invariant discretizations of PDEs that maintain all their Lie symmetries, demonstrated on several classical equations.

Findings

Invariant discretizations of heat, Burgers, and KdV equations are constructed.

Exact solutions of the discrete schemes are obtained.

The method preserves the physical symmetries of the original PDEs.

Abstract

A procedure for obtaining a "minimal" discretization of a partial differential equation, preserving all of its Lie point symmetries is presented. "Minimal" in this case means that the differential equation is replaced by a partial difference scheme involving N difference equations, where N is the number of independent and dependent variable. We restrict to one scalar function of two independent variables. As examples, invariant discretizations of the heat, Burgers and Korteweg-de Vries equations are presented. Some exact solutions of the discrete schemes are obtained.