A Well-Balanced Finite Volume Scheme for a Mixed Hyperbolic/Parabolic System to Model Chemotaxis

This work is concerned by the numerical approximation of the weak solutions of a system of partial differential equations arising when modeling the movements of cells according to a chemoattractant concentration. The adopted PDE system turns out to couple a hyperbolic system with a diffusive equation. The solutions of such a model satisfy several properties to be preserved at the numerical level. Indeed, the solutions may contain vacuum, satisfy steady regimes and asymptotic regimes. By deriving a judicious approximate Riemann solver, a finite volume method is designed in order to exactly preserve the steady regimes of particular physical interest. Moreover, the scheme is able to deal with vacuum regions and it preserves the asymptotic regimes. Numerous numerical experiments illustrate the relevance of the proposed numerical procedure.