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

Roberto Natalini (1) , Magali Ribot (2, 3) , Monika Twarogowska (4)

1 IAC - Istituto per le Applicazioni del Calcolo "Mauro Picone"
2 LJAD - Laboratoire Jean Alexandre Dieudonné
3 COFFEE - COmplex Flows For Energy and Environment
4 OPALE - Optimization and control, numerical algorithms and integration of complex multidiscipline systems governed by PDE

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.

Mots clés: hyperbolic system with source; chemotaxis; stationary solutions with vacuum; finite volume methods; well-balanced scheme

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Roberto Natalini, Magali Ribot, Monika Twarogowska. A well-balanced numerical scheme for a one dimensional quasilinear hyperbolic model of chemotaxis. Communications in Mathematical Sciences, 2014, 12 (1), pp.13-39. ⟨hal-00764086⟩