An accurate front capturing scheme for tumor growth models with a free boundary limit

TL;DR

This paper introduces a novel numerical scheme for tumor growth models with free boundary limits, accurately capturing the front propagation even under extreme nonlinearity and degeneracy, with proven stability and positivity preservation.

Contribution

A new prediction-correction numerical scheme that effectively approximates free boundary tumor growth models with high nonlinearity, connecting to the limit equation as nonlinearity intensifies.

Findings

Scheme accurately captures front propagation speed for large m

Method preserves positivity and improves stability

Numerical examples demonstrate effectiveness in 1D and 2D

Abstract

We consider a class of tumor growth models under the combined effects of density-dependent pressure and cell multiplication, with a free boundary model as its singular limit when the pressure-density relationship becomes highly nonlinear. In particular, the constitutive law connecting pressure $p$ and density $\rho$ is $p(\rho)=\frac{m}{m-1} \rho^{m-1}$, and when $m \gg 1$, the cell density $\rho$ may evolve its support due to a pressure-driven geometric motion with sharp interface along the boundary of its support. The nonlinearity and degeneracy in the diffusion bring great challenges in numerical simulations, let alone the capturing of the singular free boundary limit. Prior to the present paper, there is lack of standard mechanism to numerically capture the front propagation speed as $m\gg 1$. In this paper, we develope 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 $m \rightarrow \infty$, and with proper spacial discretization, the fully discrete scheme has improved stability, preserves positivity, and implements without nonlinear solvers. Finally, extensive numerical examples in both one and two dimensions are provided to verify the claimed properties and showcase good performance in various applications.