Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests

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.

References in corpus (4)

• Difference schemes with point symmetries and their numerical tests
• Invariant difference schemes and their application to $SL(2,\mathbb{R})$ invariant ordinary differential equations
• The Korteweg-de Vries equation and its symmetry-preserving discretization
• First integrals of ordinary difference equations beyond Lagrangian methods

Cited by in corpus (5)

• Multi-Component Extension of CAC Systems
• Difference systems in bond and face variables and non-potential versions of discrete integrable systems
• Symmetry preserving discretization of ordinary differential equations. Large symmetry groups and higher order equations
• Conformally invariant elliptic Liouville equation and its symmetry preserving discretization
• Explicit isomorphisms for the symmetry algebras of continuous and discrete isotropic oscillators