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 gradient flow structure 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, non-negativity of solution, entropy, 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.

翻译：本文针对局部感知趋化交叉扩散系统，设计、分析并模拟了一种有限体积格式。该系统与著名的极小Keller-Segel系统具有相同的梯度流结构，但不同于后者，其解在二维情况下已知是全局存在的。解的长时间行为目前仅得到部分理解，这促使我们采用可靠的数值方法进行数值探索。我们提出了一种线性隐式的两点通量有限体积近似格式。我们证明该格式在离散层面保持了连续系统的主要特征，即质量守恒性、解的非负性、熵估计以及对偶估计。这些性质使我们能够证明格式的适定性、无条件稳定性和收敛性。我们同时严格证明了该格式在准静态极限下具有渐近保持性质。我们通过详尽的数值实验补充理论分析，研究了格式的收敛性与渐近保持特性，并验证了其关于稳态解稳定性特征的可靠性。