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

Authors:
Francis Filbet
Affiliations:
Mathématiques pour l’Industrie et la Physique, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse cedex 04, France
Venue:
Numerische Mathematik
Year:
2006

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Abstract

A finite volume method is presented to discretize the Patlak–Keller–Segel (PKS) modeling chemosensitive movements. First, we prove existence and uniqueness of a numerical solution to the proposed scheme. Then, we give a priori estimates and establish a threshold on the initial mass, for which we show that the numerical approximation converges to the solution to the PKS system when the initial mass is lower than this threshold. Numerical simulations are performed to verify accuracy and the properties of the scheme. Finally, in the last section we investigate blow-up of the solution for large mass.