in partial differential equations can be preserved in a discrete world and reflected in difference schemes. Three different structure preserving discretizations of the hyperbolic Liouville equation are presented and then used to solve a specific boundary value problem. The results are compared with the 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.