Structure-preserving block-centered finite difference methods for the Keller–Segel chemotaxis system on staggered non-uniform grids

报告题目: Structure-preserving block-centered finite difference methods for the Keller–Segel chemotaxis system on staggered non-uniform grids

报告人：付红斐教授中国海洋大学数学科学学院

报告时间：2025年7月10日下午15:00

报告地点：红瓦楼726

报告内容简介：As a class of nonlinear partial differential equations, the Keller–Segel system is widely used to model chemotaxis in biology. The model  upholds three important physical properties：mass conservation, entropy-dissipation, and positivity preserving. In addition, the cell density solution may blow up within a finite time. However, the strong nonlinearity and coupling of the cell (or organism) density and the chemoattractant concentration present a significant challenge for the numerical methods.  In this talk, we present the construction and analysis of decoupled linear, structure-preserving, block-centered finite difference method for the classical Keller–Segel chemotaxis system on staggered non-uniform spatial grids. The proposed schemes achieve second-order accuracy in space and either first- or second-order accuracy in time. We demonstrate that the schemes satisfy the following properties: (i) they preserve the positivity of both the cell density and the chemoattractant concentration; (ii) they conserve the total mass of the cell density; and (iii) the first-order scheme is discretely energy dissipative. By formulating the schemes on non-uniform spatial grids, the methods yield more accurate and efficient simulations of chemotactic dynamics, particularly in the presence of rapid blow-up phenomena. In particular, 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. Furthermore, the existence and uniqueness of solutions to the Keller–Segel chemotaxis system are also discussed.  Numerical experiments are conducted to validate the theoretical properties and to illustrate the robustness and accuracy of the proposed schemes. This is a joint work with Dr. Jie Xu (许洁).

报告人简介：付红斐，中国海洋大学数学科学学院教授、博士生导师，中国工业与应用数学学会油水资源数值方法专委会委员、中国仿真学会仿真算法专业委员会委员、山东数学会理事、山东数学会计算数学专委会委员、美国数学会《数学评论》评论员，山东大学计算数学博士,美国南卡罗莱纳大学访问学者。主要从事偏微分方程数值解法、最优控制问题数值算法和高阶保结构算法等方面的研究和教学工作。

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