  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.
