An asymptotic preserving scheme for a tumor growth model of porous medium type

TL;DR

This paper develops and analyzes an asymptotic preserving finite difference scheme for a porous medium tumor growth model, demonstrating stability and exploring pressure regularity and singularity formation through numerical simulations.

Contribution

It introduces a stable, asymptotic preserving scheme for tumor growth models with pressure-dependent density evolution, advancing numerical analysis in this area.

Findings

The scheme is proven to be asymptotic preserving in the incompressible limit.

Numerical simulations reveal pressure regularity and the formation of singularities.

The scheme effectively captures the focusing solution with finite-time bubble closure.

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 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.