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A finite volume scheme for the (Patlak-) Keller-Segel model in two spacedimensions with an additional cross-diffusion term in the elliptic equation forthe chemical signal is analyzed. The main feature of the model is that thereexists a new entropy functional yielding gradient estimates for the celldensity and chemical concentration. The main features of the numerical schemeare positivity preservation, mass conservation, entropy stability, and - underadditional assumptions - entropy dissipation. The existence of a discretesolution and its numerical convergence to the continuous solution is proved.Furthermore, temporal decay rates for convergence of the discrete solution tothe homogeneous steady state is shown using a new discrete logarithmic Sobolevinequality. Numerical examples point out that the solutions exhibitintermediate states and that there exist nonhomogeneous stationary solutionswith a finite cell density peak at the domain boundary.

A finite volume scheme for a Keller-Segel model with additional cross-diffusion