[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84700-en":3,"doc-seo-84700-105":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},84700,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","Second Moment Method for Transport Problems with Anisotropic Scattering","A nonlinear two-level acceleration method solves the particle transport equation under anisotropic scattering. The formulation uses the projection operator approach, deriving low-order equations for angular moments with second-moment closures and defining high-order transport via a nonlinear prolongation operator acting on the scattering term. The coupled nonlinear high-order/low-order system is shown to be equivalent to the original Boltzmann transport equation. Spatial discretization employs the lumped linear-discontinuous Galerkin method, with numerical results validating accuracy and performance.","arXiv :2607 .03967v1 [math .NA] 4 Jul 2026  \nSecond-Moment Method for Transport Problems with Anisotropic  \nScattering  \nIndia J. Allan and Dmitriy Y. Anistratov  \nDepartment of Nuclear Engineering, North Carolina State University, Raleigh, NC 27695  \n[ijallan@ncsu.edu](ijallan@ncsu.edu), [anistratov@ncsu.edu](anistratov@ncsu.edu)  \nAbstract  \nThis paper presents a new nonlinear two-level acceleration method for solving the particle transport equation with anisotropic scattering. The method is formulated with the projection operator approach. The low-order equations are defined for the angular moments using projection operators and closures of the second-moment method. A nonlinear prolongation operator is applied to the scattering term to derive the high-order transport equation. The nonlinear system of high-order and low-order equations is equivalent to the original transport equation. The equations are approximated in space by the lumped lineardiscontinuous Galerkin method. Numerical results are presented to demonstrate the performance of the proposed numerical method.  \nKeywords: Boltzmann transport equation, particle transport, anisotropic scattering, low-order equations, Eddington factor, Galerkin method  \n1. Introduction  \nParticle transport problems with highly anisotropic scattering arise in various applications such as radiation shielding, medical physics, atmospheric sciences, plasma physics, and astrophysics. To solve this type of transport problem, acceleration methods have been developed and theoretically analyzed [1, 2 , 3 , 4 , 5 , 6 , 7] . In this paper, we present a new transport acceleration method for the Boltzmann transport equation (BTE) with anisotropic scattering. We apply a projection operator approach to formulate a two-level nonlinear iteration method. The low-order equations for the angular moments are derived using projection operators and closures of the second-moment (SM) method [8] . We formulate the high-order transport equation by applying a nonlinear prolongation operator [1, 4] . The resulting nonlinear system of high-order and low-order equations is equivalent to the original BTE. The system of equations is discretized by the lumped linear-discontinuous (LLD) Galerkin method. We derive the discrete form of the nonlinear prolongation operator for the high-order transport equation approximated with the LLD scheme. Numerical results are presented to demonstrate the performance of the proposed computational method.  \n2. Nonlinear Second-Moment Method  \nThe steady-state one-group Boltzmann transport equation with anisotropic scattering in 1D slab geometry is given by  \nµ ψ (x,µ) + σt(x)ψ (x,µ) = σ4sπ Z02π dγ′ Z−11 dµ′fs(µ0 )ψ (x,µ′) + 41πq (x) , (1)  \nx ∈ [0, X], µ ∈ [−1, 1],  \nψ(0,µ) = ψ+in(µ) for µ > 0, ψ(X,µ) = ψ−in(µ) for µ \u003C 0 . (2)  \nPreprint  \nI. Allan & D. Anistratov, Second-Moment Method for Transport Problems with Anisotropic Scattering 2  \nHere ψ is the angular flux; x is the spatial position; µ is the directional cosine of particle motion; γ is the azimuthal angle; σt and σs are the total and scattering cross sections, respectively; fs(µ0 ) is the scattering probability density function (aka the phase function); µ0 is the cosine of the scattering angle; q is the isotropic source.  \nThe low-order SM (LOSM) equations are derived by operating on the transport equation with 2πR1 (·)dµ and 2π R1 (·)µdµ as well as defining the closure for the highest angular moment in the projected equations given by [8]  \nP (x) = 2π3 Z−11 (1 − 3µ2 )ψ (x,µ)dµ . (3)  \nThe LOSM for the scalar flux, ϕ = 2π R1 ψdµ and current, J = 2π R1 µψdµ are defined by  \ndJdx (x) +􀀀σt(x) − σs,0(x)􀀁ϕ(x) = q(x) , (4a)  \n1 dϕ  dP  \n(x) +􀀀σt(x) − σs,1(x)􀀁 J (x) = (x) , (4b)  \n3 dx dx  \nJ(0) = − ~~ ~~12ϕ(0) + 2Ji+n + B0 , (4c)  \n1  \nJ (X) = 2 ϕ (X) + 2J−in − BX , (4d)  \nwhere σs,ℓ = σsfs,ℓ and  \nfs,ℓ = 2π Z−11 fs(µ0 )Pℓ(µ0 )dµ0 (5)  \nare the coefficients of expansion of the phase function in the Legendre polynomials Pℓ(µ0 ),  \nJ","cbCaibxsqQbrKInC","https://ap.wps.com/l/cbCaibxsqQbrKInC","pdf",487313,1,7,"English","en",105,"# Introduction\n# Nonlinear Second-Moment Method","[{\"question\":\"What problem does the paper address?\",\"answer\":\"The paper targets particle transport problems governed by the Boltzmann transport equation when scattering is highly anisotropic.\"},{\"question\":\"How is the proposed acceleration method structured?\",\"answer\":\"It is a two-level nonlinear iteration scheme using a projection operator approach to build low-order angular-moment equations and a nonlinear prolongation operator to derive the high-order transport equation.\"},{\"question\":\"How are the equations solved numerically?\",\"answer\":\"The resulting nonlinear coupled system is discretized in space using the lumped linear-discontinuous Galerkin method, and the nonlinear system is solved using fixed-point iteration.\"}]",1784197727,18,{"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},"second-moment-method-for-transport-problems-with-anisotropic-scattering","",{"@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/second-moment-method-for-transport-problems-with-anisotropic-scattering/84700/",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 problem does the paper address?","Question",{"text":75,"@type":76},"The paper targets particle transport problems governed by the Boltzmann transport equation when scattering is highly anisotropic.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the proposed acceleration method structured?",{"text":80,"@type":76},"It is a two-level nonlinear iteration scheme using a projection operator approach to build low-order angular-moment equations and a nonlinear prolongation operator to derive the high-order transport equation.",{"name":82,"@type":73,"acceptedAnswer":83},"How are the equations solved numerically?",{"text":84,"@type":76},"The resulting nonlinear coupled system is discretized in space using the lumped linear-discontinuous Galerkin method, and the nonlinear system is solved using fixed-point iteration.","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 & 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