A decoupled linear, mass-conservative block-centered finite difference method for the Keller-Segel chemotaxis system

TL;DR

This paper introduces a decoupled, mass-conservative finite difference scheme for the Keller-Segel chemotaxis system, achieving second-order accuracy on non-uniform grids and validated through theoretical analysis and numerical experiments.

Contribution

It presents a novel, decoupled finite difference method that conserves mass and achieves second-order convergence for the Keller-Segel system on non-uniform grids.

Findings

The scheme is mass conservative at the discrete level.

Second-order convergence in time and space is rigorously proven.

Numerical experiments confirm robustness and accuracy.

Abstract

As a class of nonlinear partial differential equations, the Keller-Segel system is widely used to model chemotaxis in biology. In this paper, we present the construction and analysis of a decoupled linear, mass-conservative, block-centered finite difference method for the classical Keller-Segel chemotaxis system. We show that the scheme is mass conservative for the cell density at the discrete level. In addition, second-order temporal and spatial convergence for both the cell density and the chemoattractant concentration are rigorously discussed, using the mathematical induction method, the discrete energy method and detailed analysis of the truncation errors. Our scheme is proposed and analyzed on non-uniform spatial grids, which leads to more accurate and efficient modeling results for the chemotaxis system with rapid blow-up phenomenon. Furthermore, the existence and uniqueness of solutions to the Keller-Segel chemotaxis system are also discussed. Numerical experiments are presented to verify the theoretical results and to show the robustness and accuracy of the scheme.