[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84576-en":3,"doc-seo-84576-105":29,"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84576,8796095360427,"Lucas Martin","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",8,"Research & Report","A Nonstandard Finite Difference Scheme for a Nonlinear Parabolic Equation with p-Laplacian-Type Diffusion","A nonstandard finite difference (NSFD) scheme is developed and analyzed for nonlinear parabolic equations with a p-Laplacian-type diffusion operator in one- and two-dimensional spatial domains. Using Mickens’ design principles, the discretization combines a nonlinear denominator function with a nonlocal approximation of the nonlinear diffusion term to produce a structure-preserving discrete model. The continuous problem’s well-posedness is established, while consistency, convergence, and local truncation error are derived. Numerical experiments confirm stability, positivity, and boundedness, avoiding spurious oscillations and nonphysical negative solutions that standard explicit FDM may produce.","arXiv :2607 .00489v1 [math .NA] 1 Jul 2026  \nA Nonstandard Finite Difference Scheme for a Nonlinear Parabolic Equation with p-Laplacian-Type Diffusion  \nAchraf Zinihia,b , Matthias Ehrhardta,∗, Moulay Rchid Sidi Ammib  \na University of Wuppertal, Applied and Computational Mathematics,  \nGaußstrasse 20, 42119 Wuppertal, Germany  \nb Department of Mathematics, AMNEA Group, Faculty of Sciences and Techniques,  \nMoulay Ismail University of Meknes, Errachidia 52000, Morocco  \nAbstract  \nWe propose and analyze a nonstandard finite difference (NSFD) scheme for nonlinear parabolic equations involving a p-Laplacian-type diffusion operator in one-and two-dimensional spatial domains. Following Mickens’ design principles, the proposed discretization employs anonlinear denominator function φ(·) together with a nonlocal approximation of the nonlinear diffusion term ∆p , yielding a structure-preserving discrete model. The scheme is designed to retain key qualitative properties of the continuous problem, including positivity, boundedness, and stability, which may be lost by standard finite difference methods (FDMs) . We establish the well-posedness of the continuous model, derive the NSFD scheme, and investigate its consistency, convergence, and local truncation error. Numerical experiments confirm the theoretical results and demonstrate that, unlike the standard explicit FDM, the proposed NSFD scheme avoids spurious oscillations and nonphysical negative solutions even for relatively large time-step sizes.  \nKeywords: Nonstandard finite difference method, p-Laplacian operator, Parabolic PDE, Nonlinear diffusion.  \n2020 Mathematics Subject Classification: 35K55, 32W50, 65J15 .  \n1. Introduction  \nNumerical simulations have become an indispensable tool for investigating nonlinear parabolic partial differential equations. In many applications, analytical solutions are unavailable, making numerical methods the primary means for exploring the qualitative and quantitative behavior of these systems. This challenge is particularly pronounced for nonlinear parabolic partial differential equations posed in one-or two-dimensional spatial domains, where nonlinear diffusion mechanisms, intricate spatial interactions, and geometric effectssignificantly increase computational complexity.  \n∗ Corresponding author  \nEmail addresses: [a.zinihi@edu.umi.ac.ma](a.zinihi@edu.umi.ac.ma) (Achraf Zinihi), [ehrhardt@uni-wuppertal.de](ehrhardt@uni-wuppertal.de) (Matthias  \nEhrhardt), [rachidsidiammi@yahoo.fr](rachidsidiammi@yahoo.fr) (Moulay Rchid Sidi Ammi)  \nLet Ω ⊂ Rn be a bounded domain with smooth boundary ∂Ω, where n = 1 or 2, and let f be a sufficiently smooth function. We define Φp (y) = yp−2 , p > 1, and λ > 0. Nonlinear parabolic equations of the form  \n􀀸 ∂u  \n􀀾 = λ div(Φp (|∇u|)∇u) + f(u) , in U = [0, T ] × Ω ,  \n􀀼 tu · ~n = 0 , on Σ = (0, T ) × ∂Ω, (1)  \n􀀾  \n􀀾  \n􀀺 u(0 , ·) = u0 (·) , in Ω ,  \narise in a broad range of applications including image processing [1], mathematical epidemiology [2], porous medium flows [3], and nonlinear heat conduction [4] . The case p = 2 recovers the classical linear diffusion equation, while p  2 introduces strong nonlinearity through the so-called p-Laplacian operator ∆pu = div(|∇u|p−2∇u) . A fundamental requirement in many of these applications is that the numerical solution remain non-negative and uniformly bounded, reflecting the physical or biological meaning of the quantity u (e.g., a concentration or density) . Standard explicit finite difference methods (FDMs) generally fail to preserve these properties unless stringent step-size restrictions are imposed, and they may produce spurious oscillations or nonphysical negative values for moderate time steps.  \nTo address these shortcomings, we propose a nonstandard finite difference (NSFD) scheme in the spirit of [5, 6] . NSFD methods replace the standard discrete derivative (um+1 −um )k −1 by a generalized counterpart involving a denominator function φ(k), satisfying φ(k) = k+O(k2 ) , and treat nonline","cbCaiqcUfPYtqs83","https://ap.wps.com/l/cbCaiqcUfPYtqs83","pdf",4563085,1,16,"English","en",105,"# Introduction\n## Problem setup and motivations\n## NSFD methodology and denominator functions\n## Grid and discretization of the explicit scheme","[{\"question\":\"What type of PDE does the proposed method address?\",\"answer\":\"The method targets nonlinear parabolic equations with a p-Laplacian-type diffusion operator in one- and two-dimensional spatial domains.\"},{\"question\":\"How does the NSFD scheme differ from a standard explicit finite difference method?\",\"answer\":\"It uses Mickens’ design principles with a nonlinear denominator function and nonlocal approximations for the nonlinear diffusion term, which helps preserve key qualitative properties.\"},{\"question\":\"Which qualitative properties are emphasized, and what issues are avoided?\",\"answer\":\"The scheme is designed to retain positivity, boundedness, and stability, preventing spurious oscillations and nonphysical negative solutions even for relatively large time-step sizes.\"}]",1784196893,40,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"a-nonstandard-finite-difference-scheme-for-a-nonlinear-parabolic-equation-with-p-laplacian-type-diffusion","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/a-nonstandard-finite-difference-scheme-for-a-nonlinear-parabolic-equation-with-p-laplacian-type-diffusion/84576/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 type of PDE does the proposed method address?","Question",{"text":75,"@type":76},"The method targets nonlinear parabolic equations with a p-Laplacian-type diffusion operator in one- and two-dimensional spatial domains.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the NSFD scheme differ from a standard explicit finite difference method?",{"text":80,"@type":76},"It uses Mickens’ design principles with a nonlinear denominator function and nonlocal approximations for the nonlinear diffusion term, which helps preserve key qualitative properties.",{"name":82,"@type":73,"acceptedAnswer":83},"Which qualitative properties are emphasized, and what issues are avoided?",{"text":84,"@type":76},"The scheme is designed to retain positivity, boundedness, and stability, preventing spurious oscillations and nonphysical negative solutions even for relatively large time-step sizes.","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":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":28,"slug":118},7,"Healthcare","healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]