  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 ODE with a nontrivial symmetry group. The
method is applied to obtain numerical slutions of second and third order ODEs
invariant under two different realizations of $SL(2,\mathbb{R})$. 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.
