A finite volume scheme for the local sensing chemotaxis model

TL;DR

This paper introduces a finite volume numerical scheme for a chemotaxis model with local sensing, ensuring key physical properties and stability, and explores its long-term behavior through rigorous analysis and simulations.

Contribution

The paper presents a novel linearly implicit finite volume scheme for a chemotaxis system, with proven stability, convergence, and asymptotic preserving properties, tailored for local sensing models.

Findings

Scheme preserves mass, non-negativity, and entropy dissipation.

Proven unconditional stability and convergence of the numerical method.

Numerical experiments confirm the scheme's reliability and asymptotic properties.

Abstract

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.