A decoupled linear, mass-conservative block-centered finite difference method for the Keller-Segel chemotaxis system
Jie Xu, Hongfei Fu
TL;DR
The study addresses numerical approximation of the classical Keller–Segel chemotaxis system, consisting of $\partial_t\rho = \Delta\rho - \lambda \nabla\cdot(\rho\nabla c)$ and $\partial_t c = \Delta c - c + \rho$ in $\Omega$ with homogeneous Neumann boundary conditions. It develops a decoupled linear, mass-conservative, block-centered finite difference scheme (DeC-MC-BCFD) on general non-uniform grids, using Crank–Nicolson time discretization and extrapolation to achieve a linear, fully-implicit, yet decoupled update between density and chemoattractant. The authors prove discrete mass conservation, derive rigorous second-order error estimates in time and space via discrete energy methods and truncation-error analysis, and establish existence and uniqueness of the discrete solution. Numerical experiments confirm the predicted accuracy, demonstrate robustness on non-uniform grids, and showcase the scheme’s ability to capture blow-up phenomena, with indications that tailored grids improve resolution while maintaining conservation properties.
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.