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.
Keywords: Porous medium equation, finite difference method, incompressible limit, asymptotic preserving scheme, focusing solution, Hele-Shaw problem
@article{M2AN_2022__56_1_121_0,
author = {David, Noemi and Ruan, Xinran},
title = {An asymptotic preserving scheme for a tumor growth model of porous medium type},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {121--150},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {1},
doi = {10.1051/m2an/2021080},
mrnumber = {4376275},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2021080/}
}
TY - JOUR AU - David, Noemi AU - Ruan, Xinran TI - An asymptotic preserving scheme for a tumor growth model of porous medium type JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2022 SP - 121 EP - 150 VL - 56 IS - 1 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2021080/ DO - 10.1051/m2an/2021080 LA - en ID - M2AN_2022__56_1_121_0 ER -
%0 Journal Article %A David, Noemi %A Ruan, Xinran %T An asymptotic preserving scheme for a tumor growth model of porous medium type %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2022 %P 121-150 %V 56 %N 1 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2021080/ %R 10.1051/m2an/2021080 %G en %F M2AN_2022__56_1_121_0
David, Noemi; Ruan, Xinran. An asymptotic preserving scheme for a tumor growth model of porous medium type. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 1, pp. 121-150. doi: 10.1051/m2an/2021080
[1] , , and , Energy and implicit discretization of the Fokker-Planck and Keller-Segel type equations. Netw. Heterogen. Media 14 (2019) 23–41. | MR | DOI
[2] , The focusing problem for the porous medium equation: experiment, simulation and analysis. Nonlinear Partial Differential Equations, in honor of Juan Luis Vázquez for his 70th birthday. Nonlinear Anal. 137 (2016) 135–147. | MR | DOI
[3] and , A selfsimilar solution to the focusing problem for the porous medium equation. Eur. J. Appl. Math. 4 (1993) 65–81. | MR | Zbl | DOI
[4] , and , Limit behaviour of focusing solutions to nonlinear diffusions. Comm. Part. Differ. Equ. 23 (1998) 307–332. | MR | Zbl
[5] , and , A moving mesh finite element algorithm for the adaptive solution of time-dependent partial differential equations with moving boundaries. Appl. Numer. Math. 54 (2005) 450–469. | MR | Zbl | DOI
[6] , , and , Scale-invariant moving finite elements for nonlinear partial differential equations in two dimensions. Appl. Numer. Math. 56 (2006) 230–252. | MR | Zbl | DOI
[7] and , An interface tracking algorithm for the porous medium equation. Trans. Am. Math. Soc. 284 (1984) 463–500. | MR | Zbl | DOI
[8] and , A finite volume scheme for nonlinear degenerate parabolic equations. SIAM J. Sci. Comput. 34 (2012) B559–B583. | MR | Zbl | DOI
[9] , , and , Hele-Shaw limit for a system of two reaction-(cross-)diffusion equations for living tissues. Arch. Ration. Mech. Anal. 236 (2020) 735–766. | MR | DOI
[10] , , and , Self-similar numerical solutions of the porous-medium equation using moving mesh methods. Philos. T. Roy. Soc. A 357 (1999) 1047–1077. | MR | Zbl | DOI
[11] , and , Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms. J. Comp. Phys. 327 (2016) 186–202. | MR | DOI
[12] , , and , Numerical study of a particle method for gradient flows. Kinet. Relat. Mod. 10 (2017) 613–641. | MR | DOI
[13] , , and , A lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes. J. Sci. Comp. 75 (2018) 1463–1499. | MR | DOI
[14] , , and , High-order relaxation schemes for nonlinear degenerate diffusion problems. SIAM J. Numer. Analy. 45 (2007) 2098–2119. | MR | Zbl | DOI
[15] and , Free boundary limit of a tumor growth model with nutrient. J. Math. App. 155 (2021) 62–82. | MR
[16] and , On the incompressible limit for a tumour growth model incorporating convective effects. Preprint (2021). | arXiv | MR
[17] , and , Convergence rate for the incompressible limit of nonlinear diffusion-advection equations. Preprint (2021). | arXiv | MR
[18] , , and , Incompressible limit for a two-species model with coupling through Brinkman’s law in any dimension. J. Math. App. 145 (2021) 204–239. | MR
[19] , and , Incompressible limit of a continuum model of tissue growth for two cell populations. Netw. Heterogen. Media 15 (2020) 57–85. | MR | DOI
[20] , , and , Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Numer. Math. 92 (2002) 41–82. | MR | Zbl | DOI
[21] and , A finite difference approach to some degenerate nonlinear parabolic equation. SIAM J. Appl. Math. 20 (1971) 199–223. | MR | Zbl | DOI
[22] and , The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commu. Pure Appl. Math. 48 (1995) 235–277. | MR | Zbl | DOI
[23] and , Porous medium equation to Hele-Shaw flow with general initial density. Trans. Amer. Math. Soc. 370 (2018) 873–909. | MR | DOI
[24] and , On lagrangian schemes for porous medium type generalized diffusion equations: a discrete energetic variational approach. J. Comp. Phys. 417 (2020) 109566. | MR | DOI
[25] , and , High order finite difference weno schemes for nonlinear degenerate parabolic equations. SIAM J. Sci. Comp. 33 (2011) 939–965. | MR | Zbl | DOI
[26] , , and , An accurate front capturing scheme for tumor growth models with a free boundary limit. J. Comp. Phys. 364 (2018) 73–94. | MR | DOI
[27] , , and , Analysis and computation of some tumor growth models with nutrient: from cell density models to free boundary dynamics. Discrete Continuous Dyn. Syst. B 24 (2019) 3011. | MR | DOI
[28] , , , , and , Multiscale modelling and nonlinear simulation of vascular tumour growth. J. Math. Biol. 58 (2009) 765–798. | MR | DOI
[29] , An explicit finite-difference scheme for one-dimensional generalized porous medium equations: interface tracking and the hole filling problem. ESAIM: M2AN 50 (2016) 1011–1033. | MR | Zbl | Numdam | DOI
[30] , and , Relaxation schemes for partial differential equations and applications to degenerate diffusion problems. Surv. Math. Ind. 10 (2002) 315–343. | MR | Zbl
[31] and , A study on moving mesh finite element solution of the porous medium equation. J. Comp. Phys. 331 (2017) 357–380. | MR | DOI
[32] and , Incompressible limit of a mechanical model of tumour growth with viscosity. Philos. Trans. Roy. Soc. A 373 (2015) 20140283. | MR | DOI
[33] , and , The Hele-Shaw asymptotics for mechanical models of tumor growth. Arch. Ration. Mech. Anal. 212 (2014) 93–127. | MR | Zbl | DOI
[34] , , and , Derivation of a Hele-Shaw type system from a cell model with active motion. Interfaces Free Bound. 16 (2014) 489–508. | MR | Zbl | DOI
[35] , and , Traveling wave solution of the Hele-Shaw model of tumor growth with nutrient. Math. Models Methods Appl. Sci. 24 (2014) 2601–2626. | MR | Zbl | DOI
[36] , Numerical methods for flows through porous media. I. Math. Comp. 40 (1983) 435–467. | MR | Zbl | DOI
[37] and , Numerical simulation for porous medium equation by local discontinuous galerkin finite element method. J. Sci. Comp. 38 (2009) 127–148. | MR | Zbl | DOI
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