Maxwellian decay for well-balanced approximations of a super-characteristic chemotaxis model

Laurent Gosse (1)

1 IAC - Istituto per le Applicazioni del Calcolo "Mauro Picone"

We focus on the numerical simulation of a one-dimensional so--called Cattaneo model of chemotaxis dynamics in a bounded domain by means of a previously introduced well-balanced (WB) and asymptotic-preserving (AP) scheme \cite{siam}. We are especially interested in studying the decay onto numerical steady-states for two reasons: 1/ conventional upwind schemes have been shown to stabilize onto spurious non-Maxwellian states (with a very big mass flow rate, see {\it e.g.} \cite{gmnr}) and 2/ the initial data lead to a dynamic which is mostly super-characteristic in the sense of \cite{jk} thus the stability results of \cite{siam} don't apply. A reflecting boundary condition which is compatible with the well-balanced character is presented; a mass-preservation property is proved and some results on super-characteristic relaxation are recalled. Numerical experiments with coarse computational grids are presented in detail: they illustrate the bifurcation diagrams in \cite{gmnr} which relate the total initial mass of cells with the time-asymptotic values of the chemoattractant substance on each side of the domain. It is shown that the WB scheme stabilizes correctly onto zero-mass flow rate (hence Maxwellian) steady-states which agree with the aforementioned bifurcation diagrams. The evolution in time of residues is commented for every considered test-case.

Mots clés: Chemotaxis modeling; Discrete velocity kinetic model; Super-characteristic relaxation; Stabilization in time; WB scheme; Reflecting boundary conditions

Domaines: Analyse numérique [math.NA]; Biochimie, Biologie Moléculaire; Modélisation et simulation

Laurent Gosse. Maxwellian decay for well-balanced approximations of a super-characteristic chemotaxis model. SIAM Journal on Scientific Computing, 2012, 34 (1), pp.A520-A545. ⟨10.1137/10081753X⟩. ⟨hal-00551675v2⟩