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

Abstract

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.

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

“…Let us callũ n i := u n i − U n i (similarly forṽ n i ) and take the maximum ofũ n i among all nodes of the star, that is,ũ n := max i=0,...,s |ũ n i |. Then, after some straightforward computations (the reader may see [2] for further details of the computations) together with the Mean Value Theorem applied to the functions f (λ) = (λ) δ , δ = m, α, we arrive to the following…”

“…The above result covers only the parabolic-parabolic case with d = 0. The conditional convergence of the GFD explicit scheme for the parabolic-elliptic case, i.e., d = 0, has been obtained in [2] by the authors.…”

“…This meshless method has been recently proved to solve with accuracy highly nonlinear PDEs, see [10,11]. In [2], the authors obtained conditional convergence for the parabolic-elliptic case of system (1) for d = a 2 = 0, a 0 = a 1 and m = α = γ = 1. In the cited paper, the conditional convergence of the numerical method is proved in a very different way than in the present paper.…”

Convergence and Numerical Solution of a Model for Tumor Growth

“…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].…”

Complex Ginzburg–Landau Equation with Generalized Finite Differences

“…Thanks to its potential to solve highly nonlinear PDEs systems over irregular domains, several authors have recently used the GFDM. In [13] Wang, Gu and Liu applied the method to perform stress analysis in elastic materials in 3D and in [14][15][16] the authors have applied an explicit GFD scheme for tumor growth invasion, chemotaxis models and the Telegraph equation, respectively. We split the solution U(x, y, t) into its real and imaginary parts and obtain a coupled system of two parabolic PDEs.…”