A finite volume scheme for the local sensing chemotaxis model

TL;DR

The paper develops and analyzes a TPFA-based finite-volume scheme with IMEX time stepping for a local-sensing chemotaxis model, preserving mass, positivity, and entropy dissipation at the discrete level. It proves well-posedness, convergence for $\varepsilon>0$, and an asymptotic-preserving property as $\varepsilon\to0$, with a detailed AP analysis relying on a discrete duality framework. The approach is validated by extensive 1D/2D numerical experiments demonstrating convergence rates, AP behavior under quasi-stationary limits, and 2D stability/instability phenomena as well as large-domain aggregation with non-exponential motility. The results provide a reliable numerical tool to explore long-time dynamics and quasi-stationary regimes in local-sensing chemotaxis, with potential impact on simulations of pattern formation and cell aggregation.

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.

Figures (8)

• Figure 1: Testcase 1. Time evolution of the entropy and related quantities. The mesh is taken as $N=3200$ and $\Delta t = 10^{-2}$.
• Figure 2: Testcase 2. Snapshots of the chemoattractant $v$ (red), and the density $u$ (dashed blue), at time $t\in\{0, 10, 20, 100\}$ for $\varepsilon = 10^{-1}$ (top line) and $\varepsilon = 0$ (bottom line). The initial condition is $u^0(x) = 15x^2(1-x)^2$, $v^0(x) = 0$.
• Figure 3: Testcase 2. Time derivative of the chemoattractant concentration in the quasi-stationary limit for ill-prepared \ref{['eq:ill_prepared_init']}, well-prepared \ref{['eq:well_prepared_init']}, and strongly well-prepared \ref{['eq:strong_well_prepared_init']} initial conditions.
• Figure 4: Testcase 2. Time evolution of the error (in $\log$-$\log$ scale) between the chemoattractant density $v$ for a given $\varepsilon$ and the quasi-stationary case $\varepsilon=0$. (Top left) $L^2(\Omega)$ distance vs time for the strongly well-prepared (SWP) initial data \ref{['eq:strong_well_prepared_init']}, (top right) $L^2(\Omega)$ distance vs time for the well-prepared (WP) initial data \ref{['eq:well_prepared_init']} and (bottom left) $L^2(\Omega)$ distance vs time for the ill-prepared (IP) initial data \ref{['eq:ill_prepared_init']}. On the bottom right plot we show the corresponding $L^2(Q_T)$ and $L^\infty(0,T; L^2(\Omega))$ errors with respect to $\varepsilon$ for the three initial data.
• Figure 5: Testcase 3. Snapshots of the solution density $u$ (left) and chemoattractant $v$ (right) at different times $t=2000, 5000, 50000$ (top to bottom). Initial data is \ref{['eq:initJ1']} with (c) $\mu = 4\mu_c^s$.

Theorems & Definitions (41)

• Remark 1
• Definition 1: Weak solutions to \ref{['1.equ']}--\ref{['1.IC']}
• Remark 2
• Definition 2: Weak-strong solution for the quasi-stationary system
• Proposition 3: Uniqueness of weak-strong solutions
• Remark 4
• Theorem 5: Well-posedness of the scheme
• Remark 6
• Theorem 7: Convergence of the scheme
• Remark 8