Bessemoulin-Chatard, M., & Jüngel, A. (2014). A finite volume scheme for a Keller–Segel model with additional cross-diffusion. IMA Journal of Numerical Analysis, 34(1), 96–122. https://doi.org/10.1093/imanum/drs061
Applied Mathematics; General Mathematics; Computational Mathematics; finite volume method; chemotaxis; cross-diffusion model
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Abstract:
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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.