  We consider a class of tumor growth models under the combined effects of
density-dependent pressure and cell multiplication, with a free boundary model
as its singular limit when the pressure-density relationship becomes highly
nonlinear. In particular, the constitutive law connecting pressure $p$ and
density $\rho$ is $p(\rho)=\frac{m}{m-1} \rho^{m-1}$, and when $m \gg 1$, the
cell density $\rho$ may evolve its support due to a pressure-driven geometric
motion with sharp interface along the boundary of its support. The nonlinearity
and degeneracy in the diffusion bring great challenges in numerical
simulations, let alone the capturing of the singular free boundary limit. Prior
to the present paper, there is lack of standard mechanism to numerically
capture the front propagation speed as $m\gg 1$. In this paper, we develope a
numerical scheme based on a novel prediction-correction reformulation that can
accurately approximate the front propagation even when the nonlinearity is
extremely strong. We show that the semi-discrete scheme naturally connects to
the free boundary limit equation as $m \rightarrow \infty$, and with proper
spacial discretization, the fully discrete scheme has improved stability,
preserves positivity, and implements without nonlinear solvers. Finally,
extensive numerical examples in both one and two dimensions are provided to
verify the claimed properties and showcase good performance in various
applications.
