Communications in Mathematical Sciences

Volume 12 (2014)

Number 1

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

Pages: 13 – 39

DOI: https://dx.doi.org/10.4310/CMS.2014.v12.n1.a2

Authors

Roberto Natalini (Istituto per le Applicazioni del Calcolo Mauro Picone, Consiglio Nazionale delle Ricerche, Roma, Italy)

Magali Ribot (Laboratoire J.A. Dieudonné, Université de Nice-Sophia Antipolis, Nice, France; Project Team COFFEE, INRIA Sophia Antipolis, France)

Monika Twarogowska (INRIA Sophia Antipolis - Méditerranée, OPALE Project-Team, Sophia Antipolis, France)

Abstract

We introduce a numerical scheme to approximate a quasilinear hyperbolic system which models the movement of cells under the in uence of chemotaxis. Since we expect to find solutions which contain vacuum parts, we propose an upwinding scheme which properly handles the presence of vacuum and 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.

Keywords

hyperbolic system with source, chemotaxis, stationary solutions with vacuum, finite volume methods, well-balanced scheme

2010 Mathematics Subject Classification

Primary 65M08. Secondary 35L60, 92B05, 92C17.

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Published 20 August 2013