Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 080, 20 pages      arXiv:1504.01953      https://doi.org/10.3842/SIGMA.2015.080
Contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet

Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests

Decio Levi a, Luigi Martina b and Pavel Winternitz ac
a) Mathematics and Physics Department, Roma Tre University and Sezione INFN of Roma Tre, Via della Vasca Navale 84, I-00146 Roma, Italy
b) Dipartimento di Matematica e Fisica - Università del Salento and Sezione INFN of Lecce, Via per Arnesano, C.P. 193 I-73100 Lecce, Italy
c) Département de mathématiques et de statistique and Centre de recherches mathématiques, Université de Montréal, C.P. 6128, succ. Centre-ville, Montréal (QC) H3C 3J7, Canada (permanent address)

Received April 01, 2015, in final form September 22, 2015; Published online October 02, 2015

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.

Key words: Lie algebras of Lie groups; integrable systems; partial differential equations; discretization procedures for PDEs.

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