On the convergence of the generalized finite difference method for solving a chemotaxis system with no chemical diffusion

Abstract

This paper focuses on the numerical analysis of a discrete version of a nonlinear reaction-diffusion system consisting of an ordinary equation coupled to a quasilinear parabolic PDE with a chemotactic term. The parabolic equation of the system describes the behavior of a biological species, while the ordinary equation defines the concentration of a chemical substance. The system also includes a logistic-like source, which limits the growth of the biological species and presents a time-periodic asymptotic behavior. We study the convergence of the explicit discrete scheme obtained by means of the Generalized Finite Difference method and prove that the non-negative numerical solutions in two dimensional space preserve the asymptotic behavior of the continuous ones. Using different functions and long-time simulations, we illustrate the efficiency of the developed numerical algorithms in the sense of the convergence in space and in time.

Benchmark Problems for the Numerical Discretization of the Cahn–Hilliard Equation with a Source Term

“…Khain and Sander [18] studied the logistic growth model of the CH equation. In recent years, many studies for tumor growth simulation using the finite difference method have been actively conducted [19][20][21][22][23].…”