On the convergence of the generalized finite difference method for solving a chemotaxis system with no chemical diffusio
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On the convergence of the generalized finite difference method for solving a chemotaxis system with no chemical diffusion J. J. Benito1 · A. García1 · L. Gavete2 · M. Negreanu3 · F. Ureña1 · A. M. Vargas3 Received: 16 April 2020 / Revised: 6 July 2020 / Accepted: 3 September 2020 © OWZ 2020
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 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. Keywords Chemotaxis systems · Generalized finite difference · Meshless method · Asymptotic stability
1 Introduction
periodic asymptotic behavior in the sense: f is a bounded given function fulfilling
In this paper, we use a meshless method called generalized finite difference method (GFDM) to study the discretization of the nonlinear system of differential equations ∂U = ΔU − div(χU ∇V ) ∂t + μU (1 + f (x, t) − U ), x ∈ Ω, t > 0, ∂V = h(U , V ), x ∈ Ω, t > 0, ∂t U (x, 0) = U0 (x), V (x, 0) = V0 (x), x ∈ Ω, ∂V ∂U = = 0, x ∈ ∂Ω, t > 0 ∂n ∂n
(1)
in a smooth bounded domain Ω ⊂ R2 with positive constant chemotaxis χ and μ > 0. The logistic-like source limits the growth of the biological species and it presents a time-
B
A. M. Vargas [email protected]
1
UNED, ETSII, Madrid, Spain
2
UPM, ETSIM, Madrid, Spain
3
Instituto de Matemática Interdisciplinar, Depto. de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
f (x, t) − f ∗ (t) L ∞ (Ω) → 0, as t → ∞, with f ∗ (t) being a time-periodic function independent of the space variable “x”. The parabolic equation describes the behavior of a biological species “U ” and the ordinary differential equation patterns the concentration of a chemical substance “V ”. The regular function h increases as “U ” increases and states the production of the chemical species. In the recently published article [16], the authors prove that for all sufficiently smooth initial data U (x, 0) = U0 (x), V (x, 0) = V0 (x), x ∈ Ω, the problem possesses a unique global-in-time classical solution that is bounded in Ω × (0, ∞), with Ω ⊂ Rn , for n ≥ 1. We prove that the convergence in space and in time of the classical solution is maintained for the discrete model. The system arises in chemotaxis, the phenomenon whereby liv
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