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

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 nonnegative 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.

Automated summary

This paper focuses on analyzing a discrete version of a nonlinear reaction-diffusion system, showing the convergence of numerical solutions and preserving the asymptotic behavior of continuous solutions in two-dimensional space. The study illustrates the efficiency of the developed numerical algorithms in terms of convergence in space and time through various functions and long-time simulations.