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M. Bessemoulin-Chatard, A. Jüngel:
"A finite volume scheme for a Keller-Segel model with additional cross-diffusion";
IMA Journal of Applied Mathematics, 34 (2014), 96 - 122.

A finite volume scheme for the (Patlak-) Keller-Segel model in two space dimensions with an additional
cross-diffusion term in the elliptic equation for the chemical signal is analysed. The main feature of the
model is that there exists a new entropy functional yielding gradient estimates for the cell density and
chemical concentration. The main features of the numerical scheme are positivity preservation, mass
conservation, entropy stability and-under additional assumptions-entropy dissipation. The existence
of a discrete solution and its numerical convergence to the continuous solution is proved. Furthermore,
temporal decay rates for convergence of the discrete solution to the homogeneous steady state is shown
using a new discrete logarithmic Sobolev inequality. Numerical examples point out that the solutions
exhibit intermediate states and that there exist nonhomogeneous stationary solutions with a finite cell
density peak at the domain boundary.

Siehe englisches Abstract.

finite volume method; chemotaxis; cross-diffusion model

http://dx.doi.org/10.1093/imanum/drs061