  Ordinary differential equations (ODEs) and ordinary difference systems
(O$\Delta$Ss) invariant under the actions of the Lie groups $\mathrm{SL}_x(2)$,
$\mathrm{SL}_y(2)$ and $\mathrm{SL}_x(2)\times\mathrm{SL}_y(2)$ of projective
transformations of the independent variables $x$ and dependent variables $y$
are constructed. The ODEs are continuous limits of the O$\Delta$Ss, or
conversely, the O$\Delta$Ss are invariant discretizations of the ODEs. The
invariant O$\Delta$Ss are used to calculate numerical solutions of the
invariant ODEs of order up to five. The solutions of the invariant numerical
schemes are compared to numerical solutions obtained by standard Runge-Kutta
methods and to exact solutions, when available. The invariant method performs
at least as well as standard ones and much better in the vicinity of
singularities of solutions.
