Journal
JOURNAL OF COMPUTATIONAL PHYSICS
Volume 364, Issue -, Pages 73-94Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jcp.2018.03.013
Keywords
Hele-Shaw equation; Free boundary problem; Front capturing scheme; Tumor growth model; Prediction correction method
Funding
- KI-Net NSF RNMS [11-07444]
- NSF [DMS-1514826, DMS-1620135]
- Science Challenge Project [TZZT2017-A3-HT003-F]
- KI-Net [RNMS11-07444]
- SUNY Buffalo
- Peking University
- NSFC [91330203]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1620135] Funding Source: National Science Foundation
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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) = m/m-1 rho(m-1), and when m >> 1, the cell density rho may evolve its support according to a pressure-driven geometric motion with sharp interface along its boundary. The nonlinearity and degeneracy in the diffusion bring great challenges in numerical simulations. Prior to the present paper, there is lack of standard mechanism to numerically capture the front propagation speed as m >> 1. In this paper, we develop 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 -> infinity. With proper spatial discretization, the fully discrete scheme has improved stability, preserves positivity, and can be implemented without nonlinear solvers. Finally, extensive numerical examples in both one and two dimensions are provided to verify the claimed properties in various applications. (C) 2018 Elsevier Inc. All rights reserved.
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