Lie-point symmetries of the discrete Liouville equation

Abstract: The Liouville equation is well known to be linearizable by a point transformation. It has an infinite dimensional Lie point symmetry algebra isomorphic to a direct sum of two Virasoro algebras. We show that it is not possible to discretize the equation keeping the entire symmetry algebra as point symmetries. We do however construct a difference system approximating the Liouville equation that is invariant under the maximal finite subalgebra $ SL_x \lf 2 , \mathbb{R} \rg \otimes SL_y \lf 2 , \mathbb{R} \rg $. The invariant scheme is an explicit one and provides a much better approximation of exact solutions than comparable standard (non invariant) schemes.