Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests

TL;DR

This paper develops and compares three structure-preserving discretizations of the Liouville equation, each maintaining different symmetry properties, and evaluates their numerical performance on boundary value problems.

Contribution

It introduces three novel discretization schemes that preserve specific symmetry structures of the Liouville equation and analyzes their numerical effectiveness.

Findings

One discretization preserves linearizability.

Another preserves the infinite-dimensional symmetry group.

A third maintains the maximal finite-dimensional symmetry subgroup.

Abstract

The main purpose of this article is to show how symmetry structures in partial differential equations can be preserved in a discrete world and reflected in difference schemes. Three different structure preserving discretizations of the Liouville equation are presented and then used to solve specific boundary value problems. The results are compared with exact solutions satisfying the same boundary conditions. All three discretizations are on four point lattices. One preserves linearizability of the equation, another the infinite-dimensional symmetry group as higher symmetries, the third one preserves the maximal finite-dimensional subgroup of the symmetry group as point symmetries. A 9-point invariant scheme that gives a better approximation of the equation, but significantly worse numerical results for solutions is presented and discussed.