Maxwellian decay for well-balanced approximations of a supercharacteristic chemotaxis model

Abstract. We focus on the numerical simulation of a one-dimensional model of chemotaxis dynamics (proposed by Greenberg and Alt [22]) in a bounded domain by means of a previously introduced well-balanced (WB) and asymptotic-preserving (AP) scheme [16]. 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 flux, see e.g. [24]) and 2 / the initial data lead to a dynamic which is mostly super-characteristic in the sense of [32] thus the stability results of [16] do not 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 [24] 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. Key words. Chemotaxis modeling; discrete velocity kinetic model; stiffness; super-characteristi