  The renormalization group equation describing the evolution of the metric of
the non linear sigma models poses some nice mathematical problems involving
functional analysis, differential geometry and numerical analysis. We describe
the techniques which allow a numerical study of the solutions in the case of a
two-dimensional target space (deformation of the $O(3)\; \sigma$--model. Our
analysis shows that the so-called sausages define an attracting manifold in the
U(1) symmetric case, at one-loop level. The paper describes i) the known
analytical solutions, ii) the spectral method which realizes the numerical
integrator and allows to estimate the spectrum of zero--modes, iii) the
solution of variational equations around the solutions, and finally iv) the
algorithms which reconstruct the surface as embedded in $R^3$.
