A survey on well-balanced and asymptotic preserving schemes for hyperbolic models for chemotaxis

Magali Ribot

doi:10.1051/proc/201861068

Abstract

Chemotaxis is a biological phenomenon widely studied these last years, at the biological level but also at the mathematical level. Various models have been proposed, from microscopic to macroscopic scales. In this article, we consider in particular two hyperbolic models for the density of organisms, a semi-linear system based on the hyperbolic heat equation (or dissipative waves equation) and a quasi-linear system based on incompressible Euler equation. These models possess relatively stiff solutions and well-balanced and asymptotic-preserving schemes are necessary to approximate them accurately. The aim of this article is to present various techniques of well-balanced and asymptotic-preserving schemes for the two hyperbolic models for chemotaxis.

Highlights

We have presented two numerical schemes for an hyperbolic semi-linear system of chemotaxis (5) : an asymptotic high order scheme and a well-balanced scheme
The two schemes are based on the same idea, that is to say a balance between the flux term and the source term in order to preserve stationary states, which results in stabilizing the numerical flux
We may point out some differences between the two schemes : concretely, the Asymptotic High Order (AHO) schemes result in a change in the discretization of the source term, whereas the well-balanced technique relies on a change of discretization of the flux

Summary

Introduction

The movement of these bacteria may be modeled by a diffusion equation, linked to random walk. When these bacteria feel some chemical fluctuations in the medium, the random walk is biased and the run phases may be extended or shortened, see [25, 95] for examples of models for this phenomenon. Another bacteria, Bacillus subtilis, shows some complex patterns

Objectives

Methods

Conclusion