Upwind-difference Potentials Method for Chemotaxis Models

We consider here a recently developed upwind-difference potentials method for chemotaxis models [8]. The introduced scheme 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 reactiondiffusion equation for the chemoattractant concentration. Chemotaxis is an important process in many medical and biological applications, such as bacteria/cell aggregation and pattern formation mechanisms. Modeling of real biomedical problems often has to deal with the complex structure of computational domains. The upwind-difference potentials method proposed in [8] handles complex domains with the use of only Cartesian meshes, and can be easily employed with fast Poisson solvers. Here, in several numerical tests applied to the simplified Patlak-Keller-Segel model with ’parabolicelliptic’ coupling, we further demonstrate the robustness of the scheme.