A linear, decoupled and positivity-preserving time-staggered block-centered finite difference method for the multi-species Keller-Segel chemotaxis system

In this paper, we present a linearly implicit, second-order block-centered finite difference (BCFD) prediction-then-projection scheme for the multi-species Keller-Segel chemotaxis system on non-uniform spatio-temporal grids.

The proposed scheme integrates a standard Crank-Nicolson time-marching algorithm with an $L^2$ projection step to enforce positivity and mass conservation.

The use of variable time stepsize and time-staggered discretization fully decouples the solutions of the multi-species cell density variables and the chemoattractant concentration variable while facilitating linearization, thereby greatly enhancing computational efficiency.

Notably, the variable time-stepping algorithm and non-uniform grid BCFD discretization jointly enable adaptive resolution and local refinement near blow-up, thereby improving efficiency and accuracy without compromising the desired physical property-preserving in the simulation.

Furthermore, using the mathematical induction method and the energy analysis approach, the unique solvability of the proposed scheme is rigorously proved, and we show that cell densities achieve second-order convergence in both time and space in the discrete $L^2$ norm, while the chemoattractant concentration achieves second-order convergence in the discrete $H^1$ norm.

Representative numerical experiments are presented to validate the theoretical findings and demonstrate the reliability of the proposed scheme in simulating the blow-up phenomenon.