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A finite volume scheme for the local sensing chemotaxis model
Maxime Herda1; Ariane Trescases2; Antoine Zurek3
1 Univ. Lille, CNRS, Inria, UMR 8524–Laboratoire Paul Painlevé, 59000 Lille
2 Institut de Mathématiques de Toulouse; UMR5219 - Université de Toulouse ; CNRS - UPS, F-31062 Toulouse Cedex 9, France
3 CNRS - Université de Montréal CRM - CNRS, Montréal, Canada and Université de Technologie de Compiègne, LMAC, 60200 Compiègne, France
The SMAI Journal of computational mathematics, Volume 11 (2025), pp. 637-676

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.

Published online:
DOI: 10.5802/smai-jcm.137
Classification: 35Q92, 35K51, 65M12, 65M08, 92C17
Keywords: finite volume scheme, entropy preserving, asymptotic preserving, Keller–Segel, local sensing, chemotaxis

Maxime Herda 1; Ariane Trescases 2; Antoine Zurek 3

1 Univ. Lille, CNRS, Inria, UMR 8524–Laboratoire Paul Painlevé, 59000 Lille
2 Institut de Mathématiques de Toulouse; UMR5219 - Université de Toulouse ; CNRS - UPS, F-31062 Toulouse Cedex 9, France
3 CNRS - Université de Montréal CRM - CNRS, Montréal, Canada and Université de Technologie de Compiègne, LMAC, 60200 Compiègne, France 
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Maxime Herda; Ariane Trescases; Antoine Zurek. A finite volume scheme for the local sensing chemotaxis model. The SMAI Journal of computational mathematics, Volume 11 (2025), pp. 637-676. doi: 10.5802/smai-jcm.137

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