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

Ordinary differential equations (ODEs) and ordinary difference systems (O$Δ$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$Δ$Ss, or conversely, the O$Δ$Ss are invariant discretizations of the ODEs. The invariant O$Δ$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.