In this talk, I will present some recent work on the tumor growth equation along with various models for the nutrient component, including the in vitro model and the in vivo model. At the cell density level, the spatial availability of the tumor density n is governed by the Darcy law via the pressure p(n) = nm. As m goes to infi
nity, the cell density models formally converge to Hele-Shaw ow models, which determine the free boundary dynamics of the tumor tissue in the incompressible limit. We derive several analytical solutions to the Hele-Shaw ow models, which serve as benchmark solutions to the geometric motion of tumor front propagation. Also, we develop 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 goes to in
finity, and with proper spacial discretization, the fully discrete scheme has improved stability, preserves positivity, and implements without nonlinear solvers. This is a joint work with Jian-Guo Liu, Min Tang and Li Wang.