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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 projectivetransformations of the independent variables $x$ and dependent variables $y$are constructed. The ODEs are continuous limits of the O$\Delta$Ss, orconversely, the O$\Delta$Ss are invariant discretizations of the ODEs. Theinvariant O$\Delta$Ss are used to calculate numerical solutions of theinvariant ODEs of order up to five. The solutions of the invariant numericalschemes are compared to numerical solutions obtained by standard Runge-Kuttamethods and to exact solutions, when available. The invariant method performsat least as well as standard ones and much better in the vicinity ofsingularities of solutions.

Symmetry preserving discretization of ordinary differential equations. Large symmetry groups and higher order equations