TY - JOUR

T1 - Flux-corrected transport stabilization of an evolutionary cross-diffusion cancer invasion model

AU - Heydari, Shahin

AU - Knobloch, Petr

AU - Wick, Thomas

N1 - 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 .

PY - 2024/2/15

Y1 - 2024/2/15

N2 - 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.

AB - 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.

KW - Cancer invasion

KW - Cross-diffusion equation

KW - Existence of solutions

KW - FEM-FCT stabilization

KW - Positivity preservation

UR - http://www.scopus.com/inward/record.url?scp=85185827631&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2307.08096

DO - 10.48550/arXiv.2307.08096

M3 - Article

AN - SCOPUS:85185827631

VL - 499

JO - Journal of computational physics

JF - Journal of computational physics

SN - 0021-9991

M1 - 112711

ER -