{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T10:02:48Z","timestamp":1790676168159,"version":"4.1.0"},"reference-count":43,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>In this article we consider the application of high-order\/$hp$-version adaptive discontinuous Galerkin finite element methods (DGFEMs) for the discretization of the $k_{eff}$-eigenvalue problem associated with the neutron transport equation. To this end, we exploit the dual weighted residual approach to derive a reliable and efficient a posteriori error estimate for the computed critical value of $k_{eff}$. Moreover, by exploiting the underlying block structure of the $hp$-version DGFEM, we propose and implement an efficient numerical solver based on exploiting Tarjan's strongly connected components algorithm to compute the inverse of the underlying transport operator; this is then utilized as an efficient preconditioner for the $k_{eff}$-eigenvalue problem. Finally, on the basis of the derived a posteriori error estimator we propose an $hp$-adaptive refinement algorithm which automatically refines both the angular and spatial domains. The performance of this adaptive strategy is demonstrated on a series of multienergetic industrial benchmark problems. In particular, we highlight the computational advantages of employing $hp$-refinement for neutron transport criticality problems in comparison with standard low-order $h$-refinement techniques.<\/jats:p>","DOI":"10.1137\/16m1079944","type":"journal-article","created":{"date-parts":[[2017,9,28]],"date-time":"2017-09-28T14:35:38Z","timestamp":1506609338000},"page":"B916-B942","source":"Crossref","is-referenced-by-count":18,"title":["$hp$-Adaptive Discontinuous Galerkin Methods for Neutron Transport Criticality Problems"],"prefix":"10.1137","volume":"39","author":[{"given":"Edward","family":"Hall","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Paul","family":"Houston","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Steven","family":"Murphy","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,9,28]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/S0149-1970(01)00023-3"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/S0306-4549(01)00082-2"},{"key":"atypb3","volume-title":"8th International Conference on Nuclear Criticality Safety, (ICNC 2007","author":"Armishaw M.J.","year":"2003"},{"key":"atypb4","unstructured":"D.J. Baker,\n                      Characteristic-based Methods for Modelling Neutron Transport\n                      , Ph.D. thesis, University of Nottingham, Nottingham, UK, 2012."},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-7605-6"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492901000010"},{"key":"atypb7","volume-title":"Nuclear Reactor Theory","author":"Bell G.I.","year":"1970"},{"key":"atypb8","volume-title":"Technical report, LA-1891","author":"Carlson B.G.","year":"1955"},{"key":"atypb9","volume-title":"Radiative transfer","author":"Chandrasekhar S.","year":"1960"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2013.09.024"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1137\/080731918"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.4208\/cicp.290709.120210a"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-010-9453-3"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-011-9545-8"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1145\/355780.355785"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1007\/s00466-006-0107-0"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1007\/BF02684345"},{"key":"atypb18","volume-title":"Technical report, NEA\/DB\/DOC (2008) 1","author":"Ganapol B.D.","year":"2008"},{"key":"atypb19","first-page":"141","volume":"3","author":"Hartmann R.","year":"2006","journal-title":"Int. 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Murphy,\n                      Methods for Solving Discontinuous-Galerkin Finite Element Equations with Application to Neutron Transport\n                      , Ph.D. thesis, University of Nottingham, Nottingham, UK, 2015."},{"key":"atypb38","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1093\/oso\/9780198500148.003.0017","volume-title":"State of the Art in Numerical Analysis","author":"S\u00fcli E.","year":"1997"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1080\/18811248.1991.9731408"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1137\/0201010"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-03490-3"},{"key":"atypb42","unstructured":"Y. Wang,\n                      Adaptive mesh refinement solution techniques for the multigroup ${S}_{N}$ Transport Equation using a Higher-Order Discontinuous Finite Element Method\n                      , Ph.D. thesis, Texas A&M University, College Station, TX, 2009."},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.13182\/NSE04-1"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/16M1079944","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:08:07Z","timestamp":1787328487000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/16M1079944"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1]]},"references-count":43,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["10.1137\/16M1079944"],"URL":"https:\/\/doi.org\/10.1137\/16m1079944","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1]]}}}