Abstract
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 theL4-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.
Highlights
We consider a model of tumor growth describing the evolution of the cell population density n(x, t) through a porous medium equation with a source,∂n − ∇(n∇p) = nG(p),∂t x ∈ Rd, t > 0, (1)where p is the internal pressure of the tumor, defined by the law of state p = nγ, γ > 1. (2)The non-linearity and degeneracy of the diffusion term bring several difficulties to the numerical analysis of the model, and many numerical schemes have been proposed in the literature, cf. [25, This provisional PDF is the accepted version
We show that, as γ → ∞, the aforementioned scheme is asymptotic preserving and the solution converges to a solution of the following finite difference equation pi(δx2pi + G(pi)) = 0, where we denote δx2pi :=/|∆x|2
The numerical simulation of the tumor growth model (1) is challenging in two aspects, the lack of regularity of solutions near the free boundary, which is a common difficulty of porous medium equations, and the stiffness appearing in the Hele-Shaw limit γ → ∞
Summary
We consider a model of tumor growth describing the evolution of the cell population density n(x, t) through a porous medium equation with a source,. It is our aim to recover a discrete version of this lower bound for our scheme This purpose has been already addressed in the literature, in particular we refer the reader to [29] for a tracking front scheme for which the author proves the Aronson-Bénilan estimate for the classical porous medium equation (namely, with no reaction terms), and for any γ > 1. The numerical simulation of the tumor growth model (1) is challenging in two aspects, the lack of regularity of solutions near the free boundary, which is a common difficulty of porous medium equations, and the stiffness appearing in the Hele-Shaw limit γ → ∞. We report the results of the 2-dimensional simulations on the focusing solution which confirm the sharpness of the L4-uniform bound of ∇p
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