The talk will consist of two parts. In the "group theory" part, we will
show how to discretize an ordinary differential equation and obtain
difference schemes invariant under the same Lie point symmetry group. In
the "numerical" part, we will consider several second and third order
nonlinear differential equations and show that the invariant schemes
provide a much higher precision than standard schemes, without
significantly increasing the the complexity of the computations.
Moreover invariant schemes make it possible to treat singular solutions
and to obtain numerical solutions valid up to and beyond singularities.