Flux-corrected transport stabilization of an evolutionary cross-diffusion cancer invasion model

authored by: Shahin Heydari, Petr Knobloch, Thomas Wick
Abstract: In the present work, we investigate a model of the invasion of healthy tissue by cancer cells which is described by a system of nonlinear PDEs consisting of a cross-diffusion-reaction equation and two additional nonlinear ordinary differential equations. We show that when the convective part of the system, the haptotaxis term, is dominant, then straightforward numerical methods for the studied system may be unstable. We present an implicit finite element method using conforming P₁ or Q₁ finite elements to discretize the model in space and the θ-method for discretization in time. The discrete problem is stabilized using a nonlinear flux-corrected transport approach. It is proved that both the nonlinear scheme and the linearized problems used in fixed-point iterations are solvable and positivity preserving. Several numerical experiments are presented in 2D to demonstrate the performance of the proposed method.
Organisation(s): Institute of Applied Mathematics
External Organisation(s): Charles University
Type: Article
Journal: Journal of computational physics
Volume: 499
ISSN: 0021-9991
Publication date: 15.02.2024
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Numerical Analysis, Modelling and Simulation, Physics and Astronomy (miscellaneous), General Physics and Astronomy, Computer Science Applications, Computational Mathematics, Applied Mathematics
Sustainable Development Goals: SDG 3 - Good Health and Well-being
Electronic version(s): https://doi.org/10.48550/arXiv.2307.08096 (Access: Open)
https://doi.org/10.1016/j.jcp.2023.112711 (Access: Closed)

BibTeX

@article{bbc7409679534acea65c1cc0ba8d46b4,
title = "Flux-corrected transport stabilization of an evolutionary cross-diffusion cancer invasion model",
abstract = "In the present work, we investigate a model of the invasion of healthy tissue by cancer cells which is described by a system of nonlinear PDEs consisting of a cross-diffusion-reaction equation and two additional nonlinear ordinary differential equations. We show that when the convective part of the system, the haptotaxis term, is dominant, then straightforward numerical methods for the studied system may be unstable. We present an implicit finite element method using conforming P1 or Q1 finite elements to discretize the model in space and the θ-method for discretization in time. The discrete problem is stabilized using a nonlinear flux-corrected transport approach. It is proved that both the nonlinear scheme and the linearized problems used in fixed-point iterations are solvable and positivity preserving. Several numerical experiments are presented in 2D to demonstrate the performance of the proposed method.",
keywords = "Cancer invasion, Cross-diffusion equation, Existence of solutions, FEM-FCT stabilization, Positivity preservation",
author = "Shahin Heydari and Petr Knobloch and Thomas Wick",
note = "Funding Information: This work was initiated during a research stay of the first author at the Institute of Applied Mathematics at the Leibniz University Hanover from November 2021 to April 2022 for which hospitality is still gratefully acknowledged. The work of Shahin Heydari was further supported through the grant No. 396921 of the Charles University Grant Agency and the grant SVV-2023-260711 of Charles University . The work of Petr Knobloch was supported through the grant No. 22-01591S of the Czech Science Foundation . ",
year = "2024",
month = feb,
day = "15",
doi = "10.48550/arXiv.2307.08096",
language = "English",
volume = "499",
journal = "Journal of computational physics",
issn = "0021-9991",
publisher = "Academic Press Inc.",
}