A well-balanced numerical scheme for a one dimensional quasilinear hyperbolic model of chemotaxis

TL;DR

This paper presents a novel upwinding numerical scheme for a one-dimensional quasilinear hyperbolic model of chemotaxis, effectively handling vacuum regions and accurately capturing long-term behavior.

Contribution

The authors develop and analyze a well-balanced upwinding scheme tailored for hyperbolic chemotaxis models with vacuum, including stability and asymptotic state approximation.

Findings

The scheme accurately approximates steady states.

It remains stable near key steady states.

Numerical simulations demonstrate parameter-dependent asymptotic behavior.

Abstract

We introduce a numerical scheme to approximate a quasi-linear hyperbolic system which models the movement of cells under the influence of chemotaxis. Since we expect to find solutions which contain vacuum parts, we propose an upwinding scheme which handles properly the presence of vacuum and, besides, which gives a good approximation of the time asymptotic states of the system. For this scheme we prove some basic analytical properties and study its stability near some of the steady states of the system. Finally, we present some numerical simulations which show the dependence of the asymptotic behavior of the solutions upon the parameters of the system.