On the numerical solution to a parabolic-elliptic system with chemotactic and periodic terms using Generalized Finite Differences

In the present paper we propose the Generalized Finite Difference Method (GFDM) for numerical solution of a cross-diffusion system with chemotactic terms. We derive the discretization of the system using a GFD scheme in order to prove and illustrate that the uniform stability behavior/ convergence of the continuous model is also preserved for the discrete model. We prove the convergence of the explicit method and give the conditions of convergence. Extensive numerical experiments are presented to illustrate the accuracy, efficiency and robustness of the GFDM.