Fully decoupled, linear and structure-preserving block-centered finite difference methods for the Keller-Segel chemotaxis system on staggered non-uniform grids

Jie Xu; Hongfei Fu

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arXiv · 2606.12095

Fully decoupled, linear and structure-preserving block-centered finite difference methods for the Keller-Segel chemotaxis system on staggered non-uniform grids

Jie Xu·Hongfei Fu

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Abstract

In this paper, we propose two fully decoupled, linear and structure-preserving block-centered finite difference schemes for the classical Keller-Segel chemotaxis system on staggered non-uniform spatial grids. Both novel schemes are second-order accurate in space; one is first-order accurate in time, while the other achieves second-order temporal accuracy. Moreover, we show that the schemes preserve several inherent physical laws at the discrete level: (i) the positivity of both the cell density and the chemoattractant concentration; (ii) the conservation of total cell mass; and (iii) a discrete energy dissipation property for the first-order scheme. In particular, the temporally first-order scheme unconditionally preserves positivity, mass conservation, and energy dissipation, whereas the second-order scheme ensures positivity under a sufficient (but not necessary) time-step condition. The proposed methods yield more accurate and efficient simulations of chemotactic dynamics, especially in the presence of rapid blow-up phenomena, on specified non-uniform spatial grids. Numerical experiments are conducted to validate the theoretical findings and to illustrate the accuracy and reliability of the proposed schemes.

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Jie Xu, Hongfei Fu. 2026-06-10. Fully decoupled, linear and structure-preserving block-centered finite difference methods for the Keller-Segel chemotaxis system on staggered non-uniform grids. https://arxiv.org/abs/2606.12095

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