Asymptotic-Preserving and Well-Balanced schemes for the 1D Cattaneo model of chemotaxis movement in both hyperbolic and diffusive regimes

The original Well-Balanced (WB) framework of Greenberg and LeRoux (1996) [24] and Gosse (2002) [18] relying on Non-Conservative (NC) products (see LeFoch and Tzavaras (1999) [41]) is set up in order to efficiently treat the so-called Cattaneo model of chemotaxis in 1D (see Hillen and Stevens (2000) [29]). It proceeds by concentrating the source terms onto Dirac masses: this allows to handle them by NC jump relations based on steady-state equations which can be integrated explicitly. A Riemann solver is deduced and the corresponding WB Godunov scheme completed with the standard Hoff–Smoller theory (see Hillen and Stevens (2000) [29]) for the diffusion–reaction equation ruling the evolution of the chemoattractant concentration is studied in detail. Later, following former results of Gosse and Toscani (2002, 2003) [21,22], a simple rewriting of the NC jump relations allows to generate another version of the same Godunov scheme which is well adapted to the parabolic scaling involving a small parameter ε. The standard BV framework is used to study the uniform stability of this Asymptotic-Preserving (AP) scheme with respect to ε allows to pass to the limit and derive a simple centered discretization of the Keller–Segel model. Finally, results by Filbet (2006) [14] permit to pass to the complementary limit when the space-step h is sent to zero. Numerical results are included to illustrate the feasibility and the efficiency of the method.