A finite volume scheme for a Keller-Segel model with additional cross-diffusion

TL;DR

This paper develops and analyzes a finite volume numerical scheme for a Keller-Segel model with cross-diffusion, ensuring stability, positivity, and convergence, and explores the model's complex solution behaviors.

Contribution

It introduces a novel finite volume scheme for the Keller-Segel model with cross-diffusion, proving its stability, convergence, and decay properties, and investigates solution phenomena.

Findings

The scheme preserves positivity and mass.

Discrete solutions converge to the continuous solution.

Solutions can form boundary peaks and exhibit intermediate states.

Abstract

A finite volume scheme for the (Patlak-) Keller-Segel model in two space dimensions with an additional cross-diffusion term in the elliptic equation for the chemical signal is analyzed. The main feature of the model is that there exists a new entropy functional yielding gradient estimates for the cell density and chemical concentration. The main features of the numerical scheme are positivity preservation, mass conservation, entropy stability, and - under additional assumptions - entropy dissipation. The existence of a discrete solution and its numerical convergence to the continuous solution is proved. Furthermore, temporal decay rates for convergence of the discrete solution to the homogeneous steady state is shown using a new discrete logarithmic Sobolev inequality. Numerical examples point out that the solutions exhibit intermediate states and that there exist nonhomogeneous stationary solutions with a finite cell density peak at the domain boundary.