Solving a reaction-diffusion system with chemotaxis and non-local terms using Generalized Finite Difference Method. Study of the convergence

Abstract

In this paper a parabolic-parabolic chemotaxis system of PDEs that describes the evolution of a population with non-local terms is studied. We derive the discretization of the system using the meshless method called Generalized Finite Difference Method. We prove the conditional convergence of the solution obtained from the numerical method to the analytical solution in the two dimensional case. Several examples of the application are given to illustrate the accuracy and efficiency of the numerical method. We also present two examples of a parabolic-elliptic model, as generalized by the parabolic-parabolic system addressed in this paper, to show the validity of the discretization of the non-local terms.