Chemotaxis systems play a crucial role in modeling the dynamics of bacterial and cellular behaviors, including propagation, aggregation, and pattern formation, all under the influence of chemical signals. One notable characteristic of these systems is their ability to simulate concentration phenomena, where cell density undergoes rapid growth near specific concentration points or along certain curves. Such growth can result in singular, spiky structures and lead to finite-time blowups.
Our investigation focuses on the dynamics of the Patlak-Keller-Segel chemotaxis system and its two-species extensions. In the latter case, different species may exhibit distinct chemotactic sensitivities, giving rise to very different rates of cell density growth. Such a situation may be extremely challenging for numerical methods as they may fail to accurately capture the blowup of the slower-growing species mainly due to excessive numerical dissipation.
In this paper, we propose a hybrid finite-difference-particle (FDP) method, in which a particle method is used to solve the chemotaxis equation(s), while finite difference schemes are employed to solve the chemoattractant equation. Thanks to the low-dissipation nature of the particle method, the proposed hybrid scheme is particularly adept at capturing the blowup behaviors in both one- and two-species cases. The proposed hybrid FDP methods are tested on a series of challenging examples, and the obtained numerical results demonstrate that our hybrid method can provide sharp resolution of the singular structures even with a relatively small number of particles. Moreover, in the two-species case, our method adeptly captures the blowing-up solution for the component with lower chemotactic sensitivity, a feature not observed in other works.
A Hybrid Physics-Informed Neural Network Framework for Subcritical and Supercritical Dynamics in Multi-Species Chemotaxis
African Institute for Mathematical Sciences, Research and Innovation Centre（非洲数学科学研究所研究与创新中心）；Technical University Berlin（柏林工业大学）；German Aerospace Center (DLR)（德国航空航天中心）
Fully decoupled, linear and structure-preserving block-centered finite difference methods for the Keller-Segel chemotaxis system on staggered non-uniform grids
Ocean University of China（中国海洋大学）；Laboratory of Marine Mathematics, Ocean University of China（中国海洋大学海洋数学实验室）
A linear, decoupled and positivity-preserving time-staggered block-centered finite difference method for the multi-species Keller-Segel chemotaxis system
Ocean University of China（中国海洋大学）；Laboratory of Marine Mathematics, Ocean University of China（中国海洋大学海洋数学实验室）
The gradient discretisation method for optimal control problems, with super-convergence for non-conforming finite elements and mixed-hybrid mimetic finite differences
Monash University（莫纳什大学）；Indian Institute of Technology Bombay（印度理工学院孟买分校）
An efficient neural-network and finite-difference hybrid method for elliptic interface problems with applications
National Central University（国立中央大学）；National Center for Theoretical Sciences, National Taiwan University（国立台湾大学国家理论科学研究中心）；National Yang Ming Chiao Tung University（国立阳明交通大学）；National University of Kaohsiung（国立高雄大学）
