Upwind-Difference Potentials Method for Patlak-Keller-Segel Chemotaxis Model

We develop a novel upwind-difference potentials method for the Patlak-Keller-Segel chemotaxis model that can be used to approximate problems in complex geometries. The chemotaxis model under consideration is described by a system of two nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction-diffusion equation for the chemoattractant concentration. Chemotaxis is an important process in many medical and biological applications, including bacteria/cell aggregation and pattern formation mechanisms, as well as tumor growth. Furthermore modeling of real biomedical problems often has to deal with the complex structure of computational domains. There is consequently a need for accurate, fast, and computationally efficient numerical methods for different chemotaxis models that can handle arbitrary geometries. The upwind-difference potentials method proposed here handles complex domains with the use of only Cartesian meshes, and can be easily combined with fast Poisson solvers. In the numerical tests presented below we demonstrate the robustness of the proposed scheme.