{"id":"https://openalex.org/W2067445564","doi":"https://doi.org/10.1137/080721480","title":"A Novel Multigrid Method for Sn Discretizations of the Mono-Energetic Boltzmann Transport Equation in the Optically Thick and Thin Regimes with Anisotropic Scattering, Part I","display_name":"A Novel Multigrid Method for Sn Discretizations of the Mono-Energetic Boltzmann Transport Equation in the Optically Thick and Thin Regimes with Anisotropic Scattering, Part I","publication_year":2010,"publication_date":"2010-01-01","ids":{"openalex":"https://openalex.org/W2067445564","doi":"https://doi.org/10.1137/080721480","mag":"2067445564"},"language":"en","primary_location":{"id":"doi:10.1137/080721480","is_oa":false,"landing_page_url":"https://doi.org/10.1137/080721480","pdf_url":null,"source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Scientific Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5020207252","display_name":"B. Lee","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"B. Lee","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5020207252"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.3067,"has_fulltext":false,"cited_by_count":17,"citation_normalized_percentile":{"value":0.93245095,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"31","issue":"6","first_page":"4744","last_page":"4773"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10597","display_name":"Nuclear reactor physics and engineering","score":0.9988999962806702,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10597","display_name":"Nuclear reactor physics and engineering","score":0.9988999962806702,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.994700014591217,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9936000108718872,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/multigrid-method","display_name":"Multigrid method","score":0.8396780490875244},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6523521542549133},{"id":"https://openalex.org/keywords/convection\u2013diffusion-equation","display_name":"Convection\u2013diffusion equation","score":0.610855221748352},{"id":"https://openalex.org/keywords/integral-equation","display_name":"Integral equation","score":0.5722618103027344},{"id":"https://openalex.org/keywords/scalar","display_name":"Scalar (mathematics)","score":0.5400375723838806},{"id":"https://openalex.org/keywords/boltzmann-equation","display_name":"Boltzmann equation","score":0.5271793603897095},{"id":"https://openalex.org/keywords/galerkin-method","display_name":"Galerkin method","score":0.4703918695449829},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.46581295132637024},{"id":"https://openalex.org/keywords/preconditioner","display_name":"Preconditioner","score":0.43034443259239197},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3720572590827942},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.275737464427948},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.270630419254303},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.23192718625068665},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.19292095303535461},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.17668703198432922},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.16244205832481384}],"concepts":[{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.8396780490875244},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6523521542549133},{"id":"https://openalex.org/C17456955","wikidata":"https://www.wikidata.org/wiki/Q343279","display_name":"Convection\u2013diffusion equation","level":2,"score":0.610855221748352},{"id":"https://openalex.org/C27016315","wikidata":"https://www.wikidata.org/wiki/Q580101","display_name":"Integral equation","level":2,"score":0.5722618103027344},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.5400375723838806},{"id":"https://openalex.org/C165995430","wikidata":"https://www.wikidata.org/wiki/Q891653","display_name":"Boltzmann equation","level":2,"score":0.5271793603897095},{"id":"https://openalex.org/C186899397","wikidata":"https://www.wikidata.org/wiki/Q1491980","display_name":"Galerkin method","level":3,"score":0.4703918695449829},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.46581295132637024},{"id":"https://openalex.org/C167431342","wikidata":"https://www.wikidata.org/wiki/Q1754327","display_name":"Preconditioner","level":3,"score":0.43034443259239197},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3720572590827942},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.275737464427948},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.270630419254303},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.23192718625068665},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.19292095303535461},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.17668703198432922},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.16244205832481384},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/080721480","is_oa":false,"landing_page_url":"https://doi.org/10.1137/080721480","pdf_url":null,"source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Scientific Computing","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/7","display_name":"Affordable and clean energy","score":0.5099999904632568}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":12,"referenced_works":["https://openalex.org/W31320333","https://openalex.org/W169461701","https://openalex.org/W1969600280","https://openalex.org/W1985189338","https://openalex.org/W1995397164","https://openalex.org/W1998461805","https://openalex.org/W2001478694","https://openalex.org/W2013015072","https://openalex.org/W2039024786","https://openalex.org/W2095092117","https://openalex.org/W2114784631","https://openalex.org/W2152342464"],"related_works":["https://openalex.org/W4389317411","https://openalex.org/W1492522302","https://openalex.org/W2096454229","https://openalex.org/W2042199958","https://openalex.org/W2120298265","https://openalex.org/W2103149966","https://openalex.org/W2177482563","https://openalex.org/W2016008514","https://openalex.org/W2464766188","https://openalex.org/W1988875562"],"abstract_inverted_index":{"This":[0,239,301],"paper":[1,184],"presents":[2],"a":[3,71,83,155,209,262,271,339],"new":[4,240,257],"multigrid":[5,51,72,84,131,180,302,336],"method":[6,52,81,132,204,241,258],"applied":[7,54],"to":[8,56,74,86,244,261,268,282],"the":[9,21,27,47,61,66,75,87,136,140,147,168,172,179,195,212,216,224,230,312],"most":[10,305],"common":[11],"Sn":[12,265],"discretizations":[13],"(Petrov\u2013Galerkin,":[14],"diamond-differenced,":[15],"corner-balanced,":[16],"and":[17,30,33,108,112,122,189,276,279,317,360],"discontinuous":[18],"Galerkin)":[19],"of":[20,129,135,142,160,167,194,202,264,273,289],"mono-energetic":[22],"Boltzmann":[23],"transport":[24,148,156],"equation":[25,59,173,218],"in":[26,100,106,110,116,120,124,175,182,186,307,364],"optically":[28,274],"thick":[29,275],"thin":[31,277],"regimes,":[32,278],"with":[34,284,333,355],"strong":[35],"anisotropic":[36,285],"scattering.":[37],"Unlike":[38],"methods":[39,68],"that":[40,69],"use":[41],"scalar":[42,62,76],"DSA":[43,77,213,217,235,253,291],"diffusion":[44,78],"preconditioners":[45],"for":[46,60,96,165,211,232,353],"source":[48,89],"iteration,":[49],"this":[50,80,130,143,151,183,203,256,324,335],"is":[53,93,174,205,228,237,259,304,351],"directly":[55],"an":[57,94,161,348],"integral":[58,88,137,144,152,197],"flux.":[63],"Thus,":[64],"unlike":[65],"former":[67],"apply":[70],"strategy":[73,85],"operator,":[79,91],"applies":[82],"iteration":[90],"which":[92,290,362],"operator":[95,145,153],"5":[97],"independent":[98,114],"variables":[99,115],"spatial":[101,117,266,318],"three":[102],"dimensions":[103],"(3-d),":[104],"3":[105],"space":[107,121],"2":[109,123],"angle,":[111],"4":[113],"2-d":[118],"(2":[119],"angle).":[125],"The":[126],"core":[127],"smoother":[128],"involves":[133,146],"applications":[134,193],"operator.":[138],"Since":[139],"kernel":[141],"sweeps,":[149],"applying":[150],"requires":[154,338],"sweep":[157],"(an":[158],"inversion":[159],"upper":[162],"triangular":[163],"matrix)":[164],"each":[166,200],"angles":[169],"used.":[170],"As":[171],"5-space":[176],"or":[177,295],"4-space,":[178],"approach":[181,303],"coarsens":[185],"both":[187],"angle":[188,349],"space,":[190],"effecting":[191],"efficient":[192],"coarse":[196],"operators.":[198],"Although":[199],"V-cycle":[201,210],"more":[206,246,280],"expensive":[207],"than":[208,252],"preconditioner,":[214],"since":[215],"does":[219],"not":[220,297],"have":[221],"angular":[222],"dependence,":[223],"overall":[225],"computational":[226],"efficiency":[227],"about":[229],"same":[231],"problems":[233,269,283,332,354],"where":[234,311],"preconditioning":[236],"effective.":[238],"also":[242],"appears":[243],"be":[245,328],"robust":[247],"over":[248],"all":[249,288],"parameter":[250],"regimes":[251],"approaches.":[254],"Moreover,":[255],"applicable":[260,298],"variety":[263],"discretizations,":[267],"involving":[270],"combination":[272],"importantly,":[281],"scattering":[286,309],"cross-sections,":[287],"approaches":[292],"perform":[293],"poorly":[294],"are":[296],"at":[299],"all.":[300],"effective":[306],"neutron":[308],"applications,":[310],"total":[313],"cross-section":[314],"coefficient":[315],"$\\sigma_{t}$":[316],"meshsize":[319],"h":[320],"satisfy":[321],"$\\sigma_{t}h\\approx1$.":[322],"For":[323,331],"case,":[325],"coarsening":[326],"can":[327],"done":[329],"aggressively.":[330],"$\\sigma_{t}h\\approx10$,":[334],"scheme":[337],"moderately":[340],"coarsened":[341],"multiple-coarsening":[342],"scheme.":[343],"An":[344],"even":[345],"slow":[346],"coarsening,":[347],"semicoarsening,":[350],"required":[352],"$\\sigma_{t}h$":[356],"ranges":[357],"between":[358],"100":[359],"1000,":[361],"occur":[363],"high-energy":[365],"photon":[366],"applications.":[367]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}