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 twodimensional 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.

Convergence and Numerical Solution of a Model for Tumor Growth

“…We have propounded a meshless numerical scheme for the recently published model (1) in [8]. Convergence of the explicit formulae is obtained under some restrictions on the time step, ∆t, depending on the distribution of the nodes and the parameters of the problem.…”

“…The GFDM provides a simple discretization of the spatial derivatives at some point in terms of the values of the solution at the surrounding ones. These aspects (as well as the small number of points per star) have meant that the method has recently been used widely in [7,8] and is currently expanding [9].…”