Abstract: 
We discuss two models in which the "consistent
discretizations" approach is used to provide
discrete models that approximate continuum situations
of interest, as one does in numerical relativity.
The first model is a mechanical system that nevertheless
is challenging given the global structure of phase space.
The second model are the single polarized Gowdy cosmologies.
We will show that one can use the consistent discretizations
as good numerical approximations with convergence and
stability.
