A finite volume scheme for the Patlak-Keller-Segel chemotaxis model

A finite volume method is presented to discretize the Patlak-Keller-Segel (PKS) model for chemosensitive movements. On the one hand, we prove existence and uniqueness of a numerical solution to the proposed scheme. On the other hand, we give a priori estimates and establish a threshold on the initial mass, for which we show that the numerical approximation convergences to the solution to the PKS system when the initial mass is lower than this threshold. Finally, numerical simulations are performed to verify accuracy and the properties of the proposed scheme. We also investigate blow-up of the solution for large mass.