Asymptotic-Preserving and Well-Balanced scheme for the 1D Cattaneo model of chemotaxis movement in both hyperbolic and diffusive regimes
Résumé
The original well-balanced (WB) framework \cite{grl,mcom} relying on nonconservative (NC) products \cite{lft} is set up in order to efficiently treat the so--called Cattaneo model of chemotaxis in 1D \cite{hs}. 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 \cite{hs} for the diffusion-reaction equation ruling the evolution of the chemotractant concentration is studied in detail. Later, following former results \cite{CRAS,GT}, 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 $\ve$. The standard BV framework is used to study the uniform stability of this Asymptotic-Preserving (AP) scheme with respect to $\ve$ allows to pass to the limit and derive a simple centered discretization of the Keller-Segel model. Finally, results by Filbet \cite{f} 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.
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