[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85767-en":3,"doc-seo-85767-105":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},85767,2336464648322,"Aria","https://ap-avatar.wpscdn.com/avatar/2200025388227c56fec?_k=1778556882303663488",8,"Research & Report","Error Analysis for 3D Navier–Stokes Equations with Additive Noise","The work studies three-dimensional stochastic Navier–Stokes equations driven by additive stochastic forcing, focusing on temporal discretization. It analyzes semi-implicit Euler and Crank–Nicolson schemes and proves local-in-time convergence: the Euler method attains rate 1, while the Crank–Nicolson method attains rate 3/2. The theoretical findings are validated through extensive numerical simulations, including simulations for a lid-driven cavity in 3D under varying noise intensities.","arXiv :2607 .09926v1 [math .NA] 10 Jul 2026  \nERROR ANALYSIS FOR 3D NAVIER–STOKES EQUATIONS WITH ADDITIVE  \nNOISE  \nSOUMYA RANJAN BEHERA, DOMINIC BREIT, AND ABHISHEK CHAUDHARY  \nAbstract. We consider the three-dimensional stochastic Navier–Stokes equations with additive stochastic forcing. We study temporal discretizations based on a semi-implicit Euler as well as a Crank-Nicolson scheme. We prove that locally in time the former converges with rate 1 while the latter converges with rate 3/2 . These theoretical results are confirmed by extensive numerical simulations. To the best of our knowledge this is the first time that numerical simulations for the three-dimensional stochastic Navier–Stokes equations have been performed.  \n1. Introduction  \nLet (Ω , F,(Ft)t≥0 , P) be a stochastic basis with a complete, right-continuous filtration. Weare interested in the numerical approximation of the following three-dimensional stochastic Navier–Stokes equations  \n􀀸  \n(1) 􀀼  \n􀀺  \ndu = ν∆udt − (u · ∇)udt − ∇pdt + ΦdW in OT ,  \ndiv u = 0 in OT ,  \nu(0) = u0 in T3 ,  \nP-a.s. in OT := (0, T) ×T3 equipped with periodic boundary conditions. Here T3 is the threedimensional, T > 0, ν > 0 is the viscosity and u0 is a given initial datum. The momentum equation is driven by a cylindrical Wiener process W and the diffusion coefficient Φ takes values in the space of Hilbert-Schmidt operators; see Section 2.2 for details. There are various physical motivations to add stochastic components to the equations of fluid mechanics. The most important one is probably the mathematical description of phenomena of turbulence [5, 6 , 25] . From an analytical point of view most deterministic results have found their stochastic counterpart, see [20, 29] for an overview. Also, some remarkable regularization effects have been obtained by a carefully chosen noise [19] .  \nThe primary objective of this work is to develop and analyse the semi-implicit Euler and Crank–Nicolson discretisations schemes for (1) where the noise is additive in nature. Thereby enabling reliable and efficient simulations of stochastic fluid flows and facilitating a quantitative investigation of the influence of stochastic forcing on deterministic flow structures (see, Figures 1 & 2 for the lid-driven cavity problem in 3D) . A major challenge in the numerical approximation of (1) arises from the driving Wiener process, whose sample paths are only Ho¨lder continuous in time. Consequently, the limited temporal regularity of the noise restricts the attainable convergence order and necessitates sufficiently small time-step sizes to achieve accurate numerical simulations.  \n1.1. The 2D problem. For the two-dimensional stochastic Navier–Stokes equations there exists also a growing literature on the numerical approximation. For a (semi)-implicit Euler  \nDate: July 14, 2026 .  \n2010 Mathematics Subject Classification. 65M15, 65C30, 60H15, 60H35 .  \nKey words and phrases. Stochastic Navier–Stokes equations and local strong solutions and error analysis and temporal discretization and convergence rates.  \n2 SOUMYA RANJAN BEHERA, DOMINIC BREIT, AND ABHISHEK CHAUDHARY  \nFigure 1 . Lid-driven cavity in 3D (see Section 5.2): Streamlines and vorticity (on the XYplane z=0 .5 in columns 1-2, XZ-plane y=0 .5 in columns 3-4, YZ-plane x=0 .5 in columns 5-6) of the deterministic solution (µ = 0) in row 1, streamlines and vorticity of the expected value of the solution with smaller noise (µ = 10) in row 2, and larger noise (µ = 40) in row 3 at T = 20 for semi-implicit Euler Scheme.  \nFigure 2 . Lid-driven cavity in 3D: Streamlines and vorticity of solution at T = 20 for Crank-Nicolson scheme  \ndiscretization in time it is shown in [7] and [13] that for any ξ > 0, and m ∈ {1,..., M}  \n(2) P 􀀔 ∥u(tm) − um ∥2L2x + τ∥∇u(tn) − ∇un ∥2L2x > ξ τ 2α􀀕 → 0  \nSTRONG RATE OF CONVERGENCE FOR 3D STOCHASTIC NSES 3  \nas τ → 0, where α \u003C ~~1~~2. This means we have  \n(3) convergence of order (almost) 1/2 for the Euler scheme  \nfor the co","cbCaifAVmti8AcbD","https://ap.wps.com/l/cbCaifAVmti8AcbD","pdf",31825576,1,42,"English","en",105,"# Introduction\n## The 2D problem\n# Numerical method and convergence background","[{\"question\":\"What type of stochastic Navier–Stokes system is studied?\",\"answer\":\"The document studies the three-dimensional stochastic Navier–Stokes equations with additive stochastic forcing represented through a cylindrical Wiener process.\"},{\"question\":\"Which time discretization schemes are analyzed, and what convergence rates are proved?\",\"answer\":\"It analyzes a semi-implicit Euler scheme and a Crank–Nicolson scheme; locally in time, Euler converges with rate 1 and Crank–Nicolson with rate 3/2.\"},{\"question\":\"How are the theoretical results supported?\",\"answer\":\"The convergence rates are confirmed by extensive numerical simulations, including lid-driven cavity simulations in 3D under different noise levels.\"}]",1784206122,106,{"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},"error-analysis-for-3d-navierstokes-equations-with-additive-noise","",{"@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/error-analysis-for-3d-navierstokes-equations-with-additive-noise/85767/",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 stochastic Navier–Stokes system is studied?","Question",{"text":75,"@type":76},"The document studies the three-dimensional stochastic Navier–Stokes equations with additive stochastic forcing represented through a cylindrical Wiener process.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which time discretization schemes are analyzed, and what convergence rates are proved?",{"text":80,"@type":76},"It analyzes a semi-implicit Euler scheme and a Crank–Nicolson scheme; locally in time, Euler converges with rate 1 and Crank–Nicolson with rate 3/2.",{"name":82,"@type":73,"acceptedAnswer":83},"How are the theoretical results supported?",{"text":84,"@type":76},"The convergence rates are confirmed by extensive numerical simulations, including lid-driven cavity simulations in 3D under different noise levels.","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},[93,97,101,105,110,115,120,123,128,131,135],{"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":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]