[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83583-en":3,"doc-seo-83583-105":30,"detail-sidebar-cat-0-en-105":92},{"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},83583,8796095360427,"Lucas Martin","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",8,"Research & Report","Hybrid Two-Level Transport Method with Solution Decomposition in Macro and Micro Components","This paper develops a hybrid Monte Carlo/deterministic approach for solving the one-group steady-state Boltzmann transport equation via decomposition of the solution into macro and micro components. The macro component represents large-scale behavior through angular moments obtained using a P1-defined approximation and hybrid low-order moment equations with exact closures. The micro component is computed with Monte Carlo simulation. A fixed-point iteration couples both levels, yielding reduced stochastic variance and improved computational efficiency compared with purely stochastic solutions.","arXiv :2607 .01346v1 [math .NA] 1 Jul 2026  \nHybrid Two-Level Transport Method with Solution Decomposition  \nin Macro and Micro Components  \nCaleb A. Shaw and Dmitriy Y. Anistratov  \nDepartment of Nuclear Engineering, North Carolina State University, Raleigh, NC 27695  \n[cashaw4@ncsu.edu](cashaw4@ncsu.edu), [anistratov@ncsu.edu](anistratov@ncsu.edu)  \nAbstract  \nThis paper presents a new hybrid MC/deterministic method for solving the one-group steady-state Boltzmann transport equation based on decomposition of solution in macro and micro components. The macro component captures the large scale structure of the solution. It is represented by angular moments of the high-order transport solution. The P1 approximation is applied to define the macro component. The first two angular moments are obtained as a solution of hybrid low-order moment equations with exact closures. The equation for the micro component is solved using a MC simulation. The hybrid two-level system of equations for macro and micro components is solved by fixed-point iteration scheme. Numerical results are presented to demonstrate variance reduction of stochastic numerical solution and improvement in computational efficiency.  \nKeywords: Boltzmann transport equation, hybrid Monte Carlo methods, low-order equations, Eddington factor, finite volume schemes  \n1. Introduction  \nThe particle transport problem is relevant to many applications, so achieving efficient accurate solutions to this problem is highly desirable. Generally this problem is solved in two ways; with deterministic methods or with stochastic methods. Deterministic methods solve a discrete form of the high-dimensional transport equation on a finite mesh. This introduces discretization error that converges with mesh refinement. Stochastic methods, such as Monte Carlo (MC), introduce only stochastic error which decreases with increasing particles. Various hybrid MC/deterministic projection operator schemes using discretized loworder forms of the transport equation have been shown improve efficiency and reduce variance [1, 2 , 3 , 4 , 5 , 6 , 7] . There exist also computational methods for multiscale problems applying decomposition of solution in microscopic and macroscopic components for variance reduction [8] . This approach has been used to develop hybrid methods for thermal radiative transfer problems [7] .  \nIn this paper, we present a new hybrid MC/deterministic method for one-group steady-state transport problems based on decomposition of solution in macro and micro components. The large scale structure of the solution is captured by the macro component. This component is formed using angular moments that are the solution of hybrid low-order quasidiffusion (VEF) equations [9, 10] . We apply the P1 approximation to define the macro component. The micro component is solved using a MC simulation. The hybrid twolevel system of equations for macro and micro components is solved by an iterative process. We present numerical results demonstrating variance reduction of stochastic numerical solution and improvement in computational efficiency.  \nPreprint  \nC. Shaw & D. Anistratov, Hybrid Transport Method with Macro-Micro Decomposition of Solution 2  \n2. Micro-Macro Transport Method  \nWe consider the steady-state monoenergetic linear transport problem in 1D slab geometry defined by  \nµ ψx (x,µ) + Σt(x)ψ (x,µ) = 12 􀀒 Σs(x) Z−11 ψ (x,µ′)dµ′ + q(x)􀀓 , (1)  \nψ(0,µ) = ψ+in(µ) for µ > 0, ψ(X,µ) = ψ−in(µ) for µ \u003C 0 . (2)  \nHere ψ is the angular flux, x ∈ [0, X] is position, µ ∈ [−1, 1] is the directional cosine. Σ t and Σs are the total and scattering cross sections, respectively. q is the external source. ψ±in are prescribed incoming fluxes at boundaries. The scalar flux of particles is the zeroth angular moment of the angular flux defined by ϕ (x) =R1 ψ (x,µ)dµ . The particle current is the first angular moment of ψ given by J(x) =R1 µψ (x,µ)dµ . We represent the solution of the transport equation (Eq. (1)) as","cbCaidEV3TWTe5On","https://ap.wps.com/l/cbCaidEV3TWTe5On","pdf",549309,6,1,9,"English","en",105,"# Abstract\n# Introduction\n# Micro-Macro Transport Method\n## Problem Formulation\n## Solution Decomposition and Macro Component\n## Micro Component Equation\n## Quasidiffusion (Variable Eddington Factor) Closures","[{\"question\":\"What is the central idea of the proposed hybrid transport method?\",\"answer\":\"The solution of the one-group steady-state Boltzmann transport equation is decomposed into macro and micro components. The macro level is handled with deterministic low-order moment equations, while the micro level is solved using Monte Carlo simulation.\"},{\"question\":\"How is the macro component constructed in the method?\",\"answer\":\"The macro component is defined to preserve the first two angular moments of the angular flux. A P1 approximation is applied to express the macro angular flux in terms of scalar flux and current, which are then obtained from hybrid low-order quasidiffusion equations with exact closures.\"},{\"question\":\"How are the macro and micro equations coupled and solved?\",\"answer\":\"A fixed-point iteration scheme solves the coupled two-level system for macro and micro components. Numerical results show that this coupling reduces variance in the stochastic solution and improves computational efficiency.\"}]",1784188999,23,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"hybrid-two-level-transport-method-with-solution-decomposition-in-macro-and-micro-components","",{"@graph":36,"@context":86},[37,54,69],{"@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":53},"https://docshare.wps.com/document/hybrid-two-level-transport-method-with-solution-decomposition-in-macro-and-micro-components/83583/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-26","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What is the central idea of the proposed hybrid transport method?","Question",{"text":76,"@type":77},"The solution of the one-group steady-state Boltzmann transport equation is decomposed into macro and micro components. The macro level is handled with deterministic low-order moment equations, while the micro level is solved using Monte Carlo simulation.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How is the macro component constructed in the method?",{"text":81,"@type":77},"The macro component is defined to preserve the first two angular moments of the angular flux. A P1 approximation is applied to express the macro angular flux in terms of scalar flux and current, which are then obtained from hybrid low-order quasidiffusion equations with exact closures.",{"name":83,"@type":74,"acceptedAnswer":84},"How are the macro and micro equations coupled and solved?",{"text":85,"@type":77},"A fixed-point iteration scheme solves the coupled two-level system for macro and micro components. Numerical results show that this coupling reduces variance in the stochastic solution and improves computational efficiency.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,111,115,120,123,127,130,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":46,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},"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":22,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":107,"slug":137},19,"General","general"]