Numerical simulation of a hyperbolic model for chemotaxis after blow up

Francois James (1, 2) , Nicolas Vauchelet (3, 4)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 FDP - Fédération de recherche Denis Poisson
3 BANG - Nonlinear Analysis for Biology and Geophysical flows
4 LJLL - Laboratoire Jacques-Louis Lions

A macroscopic system modelling the chemotactic motion of bacteria is considered. This model has been obtained in a previous work thanks to a hydrodynamical limit of a kinetic system. Existence and uniqueness of measure solutions for this system using the concept of duality solution have been proved by the authors in \cite{jamesnv}. In this paper, we investigate the numerical discretization of this system. A scheme based on a finite volume approach is proposed and the convergence of the numerical solution towards the unique duality solution is stated. Numerical simulations are provided that shows the behaviour of solutions after blow up.

Mots clés: chemotaxis; non-local conservation equations; numerical analysis; finite volume scheme; measure-valued solutions

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

Francois James, Nicolas Vauchelet. Numerical simulation of a hyperbolic model for chemotaxis after blow up. 14th International Conference on Hyperbolic Problems, F. Ancona, A. Bressan, P. Marcati, A. Marson, 2012, Padova, Italy. pp.693-700. ⟨hal-00772653⟩