2024/12/17 by Maxime Herda, Herda, Maxime, Ariane Trescases +3
Mathematics · Medicine · #35K51 #35Q92 #65M08 #65M12 #92C17 #FOS: Mathematics #MRI in cancer diagnosis #Mathematical Biology Tumor Growth #Medical Imaging Techniques and Applications #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2412.13143
openalex publication_date 2024/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we design, analyze and simulate a finite volume scheme for a cross-diffusion system which models chemotaxis with local sensing. This system has the same Lyapunov function (or entropy) as the celebrated minimal Keller-Segel system, but unlike the latter, its solutions are known to exist globally in 2D. The long-time behavior of solutions is only partially understood which motivates numerical exploration with a reliable numerical method. We propose a linearly implicit, two-point flux finite volume approximation of the system. We show that the scheme preserves, at the discrete level, the main features of the continuous system, namely mass conservation, non-negativity of solution, entropy dissipation, and duality estimates. These properties allow us to prove the well-posedness, unconditional stability and convergence of the scheme. We also show rigorously that the scheme possesses an asymptotic preserving (AP) property in the quasi-stationary limit. We complement our analysis with thorough numerical experiments investigating convergence and AP properties of the scheme as well as its reliability with respect to stability properties of steady solutions.