  Mechanical models of tumor growth based on a porous medium approach have been
attracting a lot of interest both analytically and numerically. In this paper,
we study the stability properties of a finite difference scheme for a model
where the density evolves down pressure gradients and the growth rate depends
on the pressure and possibly nutrients. Based on the stability results, we
prove the scheme to be asymptotic preserving (AP) in the incompressible limit.
Numerical simulations are performed in order to investigate the regularity of
the pressure. We study the sharpness of the $L^4$-uniform bound of the
gradient, the limiting case being a solution whose support contains a bubble
which closes-up in finite time generating a singularity, the so-called focusing
solution.
