Invariant difference schemes and their application to sl(2, R) invariant ordinary differential equations

We present an exposition of a method of discretizing ordinary differential equations while preserving their Lie point symmetries. This method is very general and can be applied to any ordinary differential equations (ODE) with a nontrivial symmetry group. The method is applied to obtain numerical solutions of second- and third-order ODEs invariant under two different realizations of sl(2, R). The symmetry preserving method is shown to provide a better qualitative description of solutions than standard methods. In particular it provides solutions that are valid close to singularities and beyond them.