[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83525-en":3,"doc-seo-83525-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83525,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","A Linear, Decoupled and Positivity-Preserving Time-Staggered Block-Centered Finite Difference Method for the Multi-Species Keller–Segel Chemotaxis System","A linearly implicit, second-order block-centered finite difference prediction-then-projection scheme is developed for the multi-species Keller–Segel chemotaxis system on nonuniform spatio-temporal grids. A Crank–Nicolson time-marching step combined with an L2 projection enforces positivity and mass conservation. Variable time stepping with time-staggered discretization decouples cell densities from the chemoattractant concentration, enabling linearization and improved efficiency. Jointly, it supports adaptive resolution and local refinement near blow-up. Mathematical induction and energy analysis prove unique solvability, and discrete L2/H1 error estimates establish second-order convergence, validated by numerical experiments.","arXiv :2607 .00713v1 [math .NA] 1 Jul 2026  \nA linear, decoupled and positivity-preserving time-staggered block-centered finite difference method for the multi-species Keller–Segel chemotaxis system  \nAo Zhanga , Bingyin Zhanga , Hongfei Fua,b,∗  \na School of Mathematical Sciences, Ocean University of China, Qingdao 266100, P.R. China b Laboratory of Marine Mathematics, Ocean University of China, Qingdao 266100, P. R. China  \nAbstract  \nIn 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 nonuniform spatio-temporal grids. The proposed scheme integrates a standard Crank-Nicolson timemarching algorithm with an L2 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 timestepping 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 L2 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.  \nKeywords: Keller–Segel chemotaxis system, Block-centered finite difference method, Projection method, Mass conservation, positivity-preserving, Error estimates.  \n1. Introduction  \nIn the 1970s, Keller and Segel [1, 2] established a pioneering mathematical framework for chemotaxis. They formulated a system of nonlinear partial differential equations to represent the essential biological mechanism, in which cellular or organismal movement is directed by chemical cues that can be attractive or repulsive. Mathematically, the multi-species (d-species) Keller–Segel chemotaxis model is to find the cell (or organism) density functions ρi (x, t) (i = 1 ,..., d) and the chemoattractant concentration function c (x, t) such that  \n􀀸􀀾 ∂t ρi = κi ∆ρi − χi ∇ · (ρi ∇c), in Ω × (0, T],  \n􀀾  \n ∂tc = β∆c − αc + dX γiρi , in Ω × (0, T] . (1.1)  \n􀀺 i=1  \nHere Ω ⊂ R2 is assumed to be a two-dimensional convex, bounded and open domain. The parameters κi (i = 1 ,..., d) and β are positive diffusion coefficients, χi > 0 (i = 1 ,..., d) is the chemoattractant  \n∗ Corresponding author.  \nEmail addresses: [zhangao6290@stu.ouc.edu.cn](zhangao6290@stu.ouc.edu.cn) (Ao Zhang), [zhangbingyin@stu.ouc.edu.cn](zhangbingyin@stu.ouc.edu.cn) (Bingyin Zhang),  \n[fhf@ouc.edu.cn](fhf@ouc.edu.cn) (Hongfei Fu)  \nsensitivity constant, α ≥ 0 is the consumption rate of chemoattractant, and γi ≥ 0 (i = 1 ,..., d)  \nrepresents the production rate of chemoattractant.  \nWithout loss of generality, we only consider the two-species Keller–Segel chemotaxis model (1.1), which involves identifying three real functions u = u (x, t), v = v (x, t) and c = c (x, t) such that  \n􀀸􀀾 ∂tu = κ 1 ∆u − χ1 ∇ · (u∇c), in Ω × (0, T],  \n􀀼 ∂tv = κ2 ∆v − χ2 ∇ · (v∇c), in Ω × (0, T], (1.2)  \n􀀾􀀺 ∂tc = β∆c − αc + γ1 u + γ2 v, in Ω × (0, T],  \nsubject to homogeneous Neumann boundary conditions  \n∂nu := ∇u · n = 0, ∂nv = 0, ∂nc = 0 , on ∂Ω × (0, T], (1.3)  \nand initial conditions  \nu (x, 0) = u0 (x), v (x, 0) = v0 (x), c (x, 0) = c0 (x), in Ω , (1 .4)  \nwhere n represents the unit outer normal vector onto the boundary.  \nSignificantly, t","cbCaitHea8Wx95Yl","https://ap.wps.com/l/cbCaitHea8Wx95Yl","pdf",1463599,3,1,26,"English","en",105,"# Introduction\n## Multi-species Keller–Segel model and properties\n## Structure-preserving numerical methods","[{\"question\":\"What numerical scheme is proposed for the multi-species Keller–Segel chemotaxis system?\",\"answer\":\"A linearly implicit, second-order block-centered finite difference prediction-then-projection scheme is proposed, combining Crank–Nicolson time marching with an L2 projection step.\"},{\"question\":\"How does the method ensure the solution preserves physical properties?\",\"answer\":\"The projection step is designed to enforce positivity and mass conservation. The time-staggered discretization and variable time stepping also support stable decoupling of unknowns.\"},{\"question\":\"What convergence and solvability results are proved?\",\"answer\":\"Unique solvability is rigorously proved using mathematical induction and energy analysis. The cell densities achieve second-order convergence in the discrete L2 norm, while the chemoattractant concentration achieves second-order convergence in the discrete H1 norm.\"}]",1784188621,66,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"a-linear-decoupled-and-positivity-preserving-time-staggered-block-centered-finite-difference-method-for-the-multi-species-kellersegel-chemotaxis-system","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/a-linear-decoupled-and-positivity-preserving-time-staggered-block-centered-finite-difference-method-for-the-multi-species-kellersegel-chemotaxis-system/83525/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-26","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What numerical scheme is proposed for the multi-species Keller–Segel chemotaxis system?","Question",{"text":75,"@type":76},"A linearly implicit, second-order block-centered finite difference prediction-then-projection scheme is proposed, combining Crank–Nicolson time marching with an L2 projection step.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the method ensure the solution preserves physical properties?",{"text":80,"@type":76},"The projection step is designed to enforce positivity and mass conservation. The time-staggered discretization and variable time stepping also support stable decoupling of unknowns.",{"name":82,"@type":73,"acceptedAnswer":83},"What convergence and solvability results are proved?",{"text":84,"@type":76},"Unique solvability is rigorously proved using mathematical induction and energy analysis. The cell densities achieve second-order convergence in the discrete L2 norm, while the chemoattractant concentration achieves second-order convergence in the discrete H1 norm.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]