[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84366-en":3,"doc-seo-84366-105":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},84366,13056703020460,"Valentina","https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923",8,"Research & Report","High-order Complete Flux Schemes for Convection-Diffusion Equations on Arbitrary Subdivisions","Develops a complete-flux finite volume method for convection–diffusion equations on arbitrary 2D and 3D meshes and for discrete spaces of any polynomial degree. Exact normal numerical fluxes are derived from the underlying PDE, splitting into a homogeneous Scharfetter–Gummel part and an inhomogeneous Green’s-function contribution capturing tangential flux and sources. The formulation is exactly equivalent to the continuous model and provides high-order accuracy without correction or stabilization, achieving optimal second-order behavior for linear spaces and optimal L2 convergence for higher-degree spaces.","arXiv :2607 .08422v1 [math .NA] 9 Jul 2026  \nHIGH-ORDER COMPLETE FLUX SCHEMES FOR CONVECTION-DIFFUSION EQUATIONS ON ARBITRARY  \nSUBDIVISIONS  \nPENG YANG∗, WENYU LEI∗ , AND LIWEI XU∗  \nAbstract. We develop a complete flux finite volume method for convection-diffusion equations that works on arbitrary meshes in two and three dimensions and for discrete spaces of any polynomial degree. Unlike standard finite volume discretizations, where the numerical flux is directly approximated from the flux definition, we derive the exact normal flux across each control volume edge/face from the underlying PDE. This exact flux splits naturally into a homogeneous part (the classical Scharfetter-Gummel flux) and an inhomogeneous part expressed via a Green’s function that incorporates the tangential flux and the source term. The resulting formulation is exactly equivalent to the continuous equation and, once the discrete space is chosen, yields high-order schemes without any correction or stabilization.  \nFor piecewise linear spaces, the scheme achieves optimal second-order accuracy in convectiondominated regimes and can preserve positivity on moderately coarse meshes. For quadratic spaces, standard finite volume methods, based on the Lagrange elements or B-splines, fail to attain optimal L2 convergence unless the control volume mesh is specially designed. The proposed complete flux scheme, however, always achieves optimal L2 convergence independently of the control volume mesh. Numerical experiments in two and three dimensions confirm the robustness and optimal accuracy of the approach.  \nKey words. complete flux, finite volume, convection-diffusion, high-order, B-spline  \nMSC codes. 65N08, 65D07, 65L11  \n1. Introduction. Steady-state convection-diffusion equations arise in numerous physical and engineering applications, including semiconductor device simulation [24], groundwater contaminant transport [30], heat and mass transfer in fluid flows [20], and atmospheric pollutant dispersion [23] . When the convection coefficient dominates over diffusion, the problem becomes singularly perturbed and its numerical approximation is notoriously challenging: standard Galerkin methods produce spurious oscillations unless the mesh is excessively refined [3], while simple upwinding introduces excessive numerical diffusion [20] .  \nFinite volume methods are attractive for these problems due to their local conservation property and their natural ability to handle discontinuous coefficients and complex geometries [1, 7, 10, 15, 16] . A particularly influential class of methods for convection-dominated problems originates from the work of Scharfetter and Gummel [22] in semiconductor physics. The Scharfetter–Gummel (SG) scheme solves a local one-dimensional boundary value problem on the interval between two adjacent grid points, assuming constant coefficients and neglecting the source term. The resulting numerical flux, expressed in terms of the Bernoulli function, resolves sharp layers without spurious oscillations even on relatively coarse meshes. Building on the same exponential fitting technique, related methods have been developed in various numerical frameworks, including finite element and finite volume discretizations [21], and recently discontinuous Galerkin methods [14] . A more systematic generalization of this idea is the complete flux (CF) scheme proposed by Thiart [27] and further investigated by Ten Thije Boonkkamp and co-workers [19, 25, 26], who derived the CF formulation by retaining the source term contribution in the Scharfetter–Gummel flux. In multiple spatial dimensions, the tangential flux divergence naturally enters  \n∗ School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China ([pyang@uestc.edu.cn](pyang@uestc.edu.cn), [wenyu.lei@uestc.edu.cn](wenyu.lei@uestc.edu.cn), [xul@uestc.edu.cn](xul@uestc.edu.cn)).  \n1  \nThis manuscript is for review purposes only.  \n2 P. YANG, W. LEI AND L. XU  \nthe local ODE as part o","cbCais7BxshzG41l","https://ap.wps.com/l/cbCais7BxshzG41l","pdf",1718664,4,1,23,"English","en",105,"# Introduction\n## Background and motivation from convection-dominated problems\n## Finite volume methods and Scharfetter–Gummel-type exponential fitting\n## Complete flux methodology and prior extensions\n## Challenges for high-order finite volume schemes on arbitrary meshes","[{\"question\":\"What distinguishes the complete-flux method from standard finite volume discretizations for convection–diffusion equations?\",\"answer\":\"Standard methods approximate the numerical flux directly from a flux definition. The complete-flux approach derives the exact normal flux across each control-volume edge/face from the original PDE, then splits it into a homogeneous Scharfetter–Gummel part and an inhomogeneous Green’s-function part.\"},{\"question\":\"How does the proposed method incorporate source terms and tangential effects?\",\"answer\":\"The inhomogeneous flux term is expressed using a Green’s function that incorporates the tangential flux divergence and the source contribution, embedding these effects into the numerical flux construction.\"},{\"question\":\"Why are high-order schemes difficult for quadratic and higher-degree polynomial spaces, and how does the proposal address it?\",\"answer\":\"Standard finite volume methods fail to achieve optimal L2 convergence on arbitrary dual/control-volume meshes for quadratic (and higher even-degree) spaces. The proposed complete-flux scheme attains optimal L2 convergence independently of how the control-volume mesh is chosen, supported by numerical experiments in 2D and 3D.\"}]",1784195127,58,{"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},"high-order-complete-flux-schemes-for-convection-diffusion-equations-on-arbitrary-subdivisions","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"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":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/high-order-complete-flux-schemes-for-convection-diffusion-equations-on-arbitrary-subdivisions/84366/",{"url":52,"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","2026-07-23","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 distinguishes the complete-flux method from standard finite volume discretizations for convection–diffusion equations?","Question",{"text":75,"@type":76},"Standard methods approximate the numerical flux directly from a flux definition. The complete-flux approach derives the exact normal flux across each control-volume edge/face from the original PDE, then splits it into a homogeneous Scharfetter–Gummel part and an inhomogeneous Green’s-function part.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed method incorporate source terms and tangential effects?",{"text":80,"@type":76},"The inhomogeneous flux term is expressed using a Green’s function that incorporates the tangential flux divergence and the source contribution, embedding these effects into the numerical flux construction.",{"name":82,"@type":73,"acceptedAnswer":83},"Why are high-order schemes difficult for quadratic and higher-degree polynomial spaces, and how does the proposal address it?",{"text":84,"@type":76},"Standard finite volume methods fail to achieve optimal L2 convergence on arbitrary dual/control-volume meshes for quadratic (and higher even-degree) spaces. The proposed complete-flux scheme attains optimal L2 convergence independently of how the control-volume mesh is chosen, supported by numerical experiments in 2D and 3D.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"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":20,"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"]