| Issue |
ESAIM: M2AN
Volume 56, Number 1, January-February 2022
|
|
|---|---|---|
| Page(s) | 121 - 150 | |
| DOI | https://doi.org/10.1051/m2an/2021080 | |
| Published online | 07 February 2022 | |
An asymptotic preserving scheme for a tumor growth model of porous medium type
1
Sorbonne Université, Inria, CNRS, Université de Paris, Laboratoire Jacques-Louis, Lions UMR7598, F-75005 Paris, France
2
Dipartimento di Matematica, Universitá di Bologna, 40126 Bologna, Italy
3
School of Mathematical Sciences, Capital Normal University, 100048 Beijing, P.R. China
* Corresponding author: xinran.ruan@cnu.edu.cn
Received:
24
May
2021
Accepted:
6
December
2021
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 L4-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.
Mathematics Subject Classification: 35K57 / 35K65 / 35Q92 / 65M06 / 65M12
Key words: Porous medium equation / finite difference method / incompressible limit / asymptotic preserving scheme / focusing solution / Hele-Shaw problem
© The authors. Published by EDP Sciences, SMAI 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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