{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T20:41:59Z","timestamp":1757623319381,"version":"3.44.0"},"reference-count":32,"publisher":"Springer Science and Business Media LLC","issue":"7","license":[{"start":{"date-parts":[[2025,7,25]],"date-time":"2025-07-25T00:00:00Z","timestamp":1753401600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,7,25]],"date-time":"2025-07-25T00:00:00Z","timestamp":1753401600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001725","name":"Royal Swedish Academy of Sciences","doi-asserted-by":"publisher","award":["VR 2018-03648"],"award-info":[{"award-number":["VR 2018-03648"]}],"id":[{"id":"10.13039\/501100001725","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100003077","name":"Knowledge Foundation","doi-asserted-by":"publisher","award":["KK 2019-0213","KK 2020-0152","KK 2023-0010"],"award-info":[{"award-number":["KK 2019-0213","KK 2020-0152","KK 2023-0010"]}],"id":[{"id":"10.13039\/100003077","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100016777","name":"Karlstad University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100016777","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2025,10]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>This work presents global random walk approximations of solutions to one-dimensional Stefan-type moving-boundary problems. We are particularly interested in the case when the moving boundary is driven by an explicit representation of its speed. This situation is usually referred to in the literature as moving-boundary problem with kinetic condition. As a direct application, we propose a numerical scheme to forecast the penetration of small diffusants into a rubber-based material. To check the quality of our results, we compare the numerical results obtained by global random walks either using the analytical solution to selected benchmark cases or relying on finite element approximations with a priori known convergence rates. It turns out that the global random walk concept can be used to produce good quality approximations of the weak solutions to the target class of problems.<\/jats:p>","DOI":"10.1007\/s40314-025-03334-4","type":"journal-article","created":{"date-parts":[[2025,7,25]],"date-time":"2025-07-25T07:22:22Z","timestamp":1753428142000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Global random walk for one-dimensional one-phase Stefan-type moving-boundary problems: simulation results"],"prefix":"10.1007","volume":"44","author":[{"given":"Nicolae","family":"Suciu","sequence":"first","affiliation":[]},{"given":"Surendra","family":"Nepal","sequence":"additional","affiliation":[]},{"given":"Yosief","family":"Wondmagegne","sequence":"additional","affiliation":[]},{"given":"Magnus","family":"\u00d6gren","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1160-0007","authenticated-orcid":false,"given":"Adrian","family":"Muntean","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,7,25]]},"reference":[{"issue":"1","key":"3334_CR1","first-page":"109","volume":"19","author":"T Aiki","year":"2009","unstructured":"Aiki T, Muntean A (2009) Existence and uniqueness of solutions to a mathematical model predicting service life of concrete structures. Adv Math Sci Appl 19(1):109\u2013129","journal-title":"Adv Math Sci Appl"},{"key":"3334_CR2","doi-asserted-by":"publisher","DOI":"10.1016\/j.advwatres.2020.103558","volume":"138","author":"CD Alecsa","year":"2020","unstructured":"Alecsa CD, Boros I, Frank F, Knabner P, Nechita M, Prechtel A, Rupp A, Suciu N (2020) Numerical benchmark study for flow in heterogeneous aquifers. Adv Water Resour 138:103558. https:\/\/doi.org\/10.1016\/j.advwatres.2020.103558","journal-title":"Adv Water Resour"},{"key":"3334_CR3","doi-asserted-by":"publisher","first-page":"878","DOI":"10.1016\/j.matcom.2022.10.026","volume":"205","author":"M-C Casab\u00e1n","year":"2023","unstructured":"Casab\u00e1n M-C, Company R, J\u00f3dar L (2023) Numerical difference solution of moving boundary random Stefan problems. Math Comput Simulat 205:878\u2013901. https:\/\/doi.org\/10.1016\/j.matcom.2022.10.026","journal-title":"Math Comput Simulat"},{"issue":"3","key":"3334_CR4","doi-asserted-by":"publisher","first-page":"1145","DOI":"10.1137\/22M1509722","volume":"62","author":"C Cuchiero","year":"2024","unstructured":"Cuchiero C, Reisinger C, Rigger S (2024) Implicit and fully discrete approximation of the supercooled Stefan problem in the presence of blow-ups. SIAM J Numer Anal 62(3):1145\u20131170. https:\/\/doi.org\/10.1137\/22M1509722","journal-title":"SIAM J Numer Anal"},{"issue":"1","key":"3334_CR5","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1093\/qjmam\/56.1.139","volume":"56","author":"JD Evans","year":"2003","unstructured":"Evans JD, King JR (2003) The Stefan problem with nonlinear kinetic undercooling. Q J Mech Appl 56(1):139\u2013161. https:\/\/doi.org\/10.1093\/qjmam\/56.1.139","journal-title":"Q J Mech Appl"},{"key":"3334_CR6","doi-asserted-by":"publisher","DOI":"10.1016\/C2017-0-02306-6","volume-title":"The classical Stefan problem: basic concepts, modelling and analysis","author":"SC Gupta","year":"2003","unstructured":"Gupta SC (2003) The classical Stefan problem: basic concepts, modelling and analysis. Elsevier, London. https:\/\/doi.org\/10.1016\/C2017-0-02306-6"},{"key":"3334_CR7","doi-asserted-by":"publisher","first-page":"1134","DOI":"10.1039\/TF9555101134","volume":"51","author":"MJ Hayes","year":"1955","unstructured":"Hayes MJ, Park GS (1955) The diffusion of benzene in rubber. Part 1. Low concentrations of benzene. T Faraday Soc 51:1134\u20131142. https:\/\/doi.org\/10.1039\/TF9555101134","journal-title":"T Faraday Soc"},{"key":"3334_CR8","doi-asserted-by":"publisher","first-page":"375","DOI":"10.1002\/pc.25832","volume":"42","author":"AV Kaliyathan","year":"2021","unstructured":"Kaliyathan AV, Rane AV, Jackson S, Thomas S (2021) Analysis of diffusion characteristics for aromatic solvents through carbon black filled natural rubber\/butadiene rubber blends. Polym Compos 42:375\u2013396. https:\/\/doi.org\/10.1002\/pc.25832","journal-title":"Polym Compos"},{"issue":"1","key":"3334_CR9","doi-asserted-by":"publisher","first-page":"27","DOI":"10.4171\/ifb\/431","volume":"22","author":"K Kumazaki","year":"2020","unstructured":"Kumazaki K, Muntean A (2020) Global weak solvability, continuous dependence on data, and large time growth of swelling moving interfaces. Interf Free Bound 22(1):27\u201350. https:\/\/doi.org\/10.4171\/ifb\/431","journal-title":"Interf Free Bound"},{"issue":"1","key":"3334_CR10","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1016\/S0377-0427(97)00034-4","volume":"81","author":"S Kutluay","year":"1997","unstructured":"Kutluay S, Bahadir AR, \u00d6zde\u015f A (1997) The numerical solution of one-phase classical Stefan problem. J Comput Appl Math 81(1):135\u2013144. https:\/\/doi.org\/10.1016\/S0377-0427(97)00034-4","journal-title":"J Comput Appl Math"},{"key":"3334_CR11","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1016\/0142-9418(80)90002-1","volume":"1","author":"PM Lewis","year":"1980","unstructured":"Lewis PM (1980) Laboratory testing of rubber durability. Polom Test 1:167\u2013189. https:\/\/doi.org\/10.1016\/0142-9418(80)90002-1","journal-title":"Polom Test"},{"issue":"2","key":"3334_CR12","doi-asserted-by":"publisher","first-page":"539","DOI":"10.2977\/prims\/1195190728","volume":"12","author":"M Mori","year":"1976","unstructured":"Mori M (1976) Stability and convergence of a finite element method for solving the Stefan problem. Publ Res I Math Sci 12(2):539\u2013563. https:\/\/doi.org\/10.2977\/prims\/1195190728","journal-title":"Publ Res I Math Sci"},{"issue":"3","key":"3334_CR13","doi-asserted-by":"publisher","first-page":"723","DOI":"10.2977\/prims\/1195189605","volume":"13","author":"M Mori","year":"1977","unstructured":"Mori M (1977) A finite element method for solving the two phase Stefan problem in one space dimension. Publ Res I Math Sci 13(3):723\u2013753. https:\/\/doi.org\/10.2977\/prims\/1195189605","journal-title":"Publ Res I Math Sci"},{"key":"3334_CR14","first-page":"61","volume":"5","author":"S Nepal","year":"2021","unstructured":"Nepal S, Meyer R, Kr\u00f6ger NH, Aiki T, Muntean A, Wondmagegne Y, Giese U (2021) A moving boundary approach of capturing diffusants penetration into rubber: FEM approximation and comparison with laboratory measurements. KGK-Kaut Gumi Kunst 5:61\u201369","journal-title":"KGK-Kaut Gumi Kunst"},{"key":"3334_CR15","first-page":"101","volume":"19","author":"S Nepal","year":"2022","unstructured":"Nepal S, Wondmagegne Y, Muntean A (2022) Error estimates for semi-discrete finite element approximations for a moving boundary problem capturing the penetration of diffusants into rubber. Int J Numer Anal Mod 19:101\u2013125 (https:\/\/global-sci.org\/intro\/article_detail\/ijnam\/20351.html)","journal-title":"Int J Numer Anal Mod"},{"key":"3334_CR16","doi-asserted-by":"publisher","unstructured":"Nepal S, \u00d6gren M, Wondmagegne Y, Muntean A (2023a) Random walks and moving boundaries: estimating the penetration of diffusants into dense rubbers. Probabilist Eng Mech 74:103546. https:\/\/doi.org\/10.1016\/j.probengmech.2023.103546","DOI":"10.1016\/j.probengmech.2023.103546"},{"key":"3334_CR17","doi-asserted-by":"publisher","unstructured":"Nepal S, Wondmagegne Y, Muntean A (2023b) Analysis of a fully discrete approximation to a moving-boundary problem describing rubber exposed to diffusants. Appl Math Comput 442:127733. https:\/\/doi.org\/10.1016\/j.amc.2022.127733","DOI":"10.1016\/j.amc.2022.127733"},{"key":"3334_CR18","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-031-17820-7_29","volume-title":"Stochastic processes, statistical methods, and engineering mathematics. SPAS 2019, Springer proceedings in mathematics & statistics","author":"M \u00d6gren","year":"2022","unstructured":"\u00d6gren M (2022) Stochastic solutions of Stefan problems with general time-dependent boundary conditions. In: Malyarenko A, Ni Y, Ran\u010di\u00e9 M, Silvestrov S (eds) Stochastic processes, statistical methods, and engineering mathematics. SPAS 2019, Springer proceedings in mathematics & statistics, vol 408. Springer, Cham. https:\/\/doi.org\/10.1007\/978-3-031-17820-7_29"},{"issue":"2","key":"3334_CR19","doi-asserted-by":"publisher","first-page":"320","DOI":"10.1002\/num.20436","volume":"26","author":"FA Radu","year":"2009","unstructured":"Radu FA, Pop IS, Attinger S (2009) Analysis of an Euler implicit-mixed finite element scheme for reactive solute transport in porous media. Numer Meth Part D E 26(2):320\u2013344. https:\/\/doi.org\/10.1002\/num.20436","journal-title":"Numer Meth Part D E"},{"issue":"1","key":"3334_CR20","doi-asserted-by":"publisher","first-page":"4","DOI":"10.1115\/1.1436090","volume":"124","author":"PJ Roache","year":"2002","unstructured":"Roache PJ (2002) Code verification by the method of manufactured solutions. J Fluids Eng 124(1):4\u201310. https:\/\/doi.org\/10.1115\/1.1436090","journal-title":"J Fluids Eng"},{"key":"3334_CR21","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1016\/j.jcp.2004.10.036","volume":"205","author":"CJ Roy","year":"2005","unstructured":"Roy CJ (2005) Review of code and solution verification procedures for computational simulation. J Comput Phys 205:131\u2013156. https:\/\/doi.org\/10.1016\/j.jcp.2004.10.036","journal-title":"J Comput Phys"},{"issue":"15","key":"3334_CR22","doi-asserted-by":"publisher","first-page":"2911","DOI":"10.1016\/S0017-9310(03)00050-4","volume":"46","author":"S Savovi\u0107","year":"2003","unstructured":"Savovi\u0107 S, Caldwell J (2003) Finite difference solution of one-dimensional Stefan problem with periodic boundary conditions. Int J Heat Mass Transf 46(15):2911\u20132916. https:\/\/doi.org\/10.1016\/S0017-9310(03)00050-4","journal-title":"Int J Heat Mass Transf"},{"key":"3334_CR23","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1002\/andp.18912780206","volume":"42","author":"J Stefan","year":"1891","unstructured":"Stefan J (1891) \u00dcber die Theorie der Eisbildung, insbesondere \u00fcber die Eisbildung im Polarmeere. Ann Physik Chemie 42:269\u2013286. https:\/\/doi.org\/10.1002\/andp.18912780206","journal-title":"Ann Physik Chemie"},{"key":"3334_CR24","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898717938","volume-title":"Finite difference schemes and partial differential equations","author":"JC Strikwerda","year":"2004","unstructured":"Strikwerda JC (2004) Finite difference schemes and partial differential equations. SIAM, London. https:\/\/doi.org\/10.1137\/1.9780898717938"},{"key":"3334_CR25","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-15081-5","volume-title":"Diffusion in random fields. Applications to transport in groundwater","author":"N Suciu","year":"2019","unstructured":"Suciu N (2019) Diffusion in random fields. Applications to transport in groundwater. Birkh\u00e4user, Cham. https:\/\/doi.org\/10.1007\/978-3-030-15081-5"},{"key":"3334_CR26","doi-asserted-by":"publisher","DOI":"10.1016\/j.advwatres.2022.104268","volume":"166","author":"N Suciu","year":"2022","unstructured":"Suciu N, Radu FA (2022) Global random walk solvers for reactive transport and biodegradation processes in heterogeneous porous media. Adv Water Resour 166:104268. https:\/\/doi.org\/10.1016\/j.advwatres.2022.104268","journal-title":"Adv Water Resour"},{"key":"3334_CR27","doi-asserted-by":"publisher","DOI":"10.1016\/j.advwatres.2021.103935","volume":"152","author":"N Suciu","year":"2021","unstructured":"Suciu N, Illiano D, Prechtel A, Radu FA (2021) Global random walk solvers for fully coupled flow and transport in saturated\/unsaturated porous media. Adv Water Resour 152:103935. https:\/\/doi.org\/10.1016\/j.advwatres.2021.103935","journal-title":"Adv Water Resour"},{"issue":"1","key":"3334_CR28","doi-asserted-by":"publisher","first-page":"158","DOI":"10.33993\/jnaat531-1429","volume":"53","author":"N Suciu","year":"2024","unstructured":"Suciu N, Radu FA, C\u0103tina\u015f E (2024) Iterative schemes for coupled flow and transport in porous media\u2014convergence and truncation errors. Numer Anal Approx Theory 53(1):158\u2013183. https:\/\/doi.org\/10.33993\/jnaat531-1429","journal-title":"Numer Anal Approx Theory"},{"issue":"1","key":"3334_CR29","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1090\/qam\/1433750","volume":"55","author":"DA Tarzia","year":"1997","unstructured":"Tarzia DA, Turner CV (1997) The one-phase supercooled Stefan problem with a convective boundary condition. Q Appl Math 55(1):41\u201350. https:\/\/doi.org\/10.1090\/qam\/1433750","journal-title":"Q Appl Math"},{"key":"3334_CR30","doi-asserted-by":"publisher","first-page":"86","DOI":"10.1007\/BF01560134","volume":"24","author":"DA Tarzia","year":"1989","unstructured":"Tarzia DA, Villa LT (1989) On the free boundary problem in the Wen-Langmuir shrinking core model for noncatalytic gas-solid reactions. Meccanica 24:86\u201392. https:\/\/doi.org\/10.1007\/BF01560134","journal-title":"Meccanica"},{"key":"3334_CR31","first-page":"263","volume":"21","author":"K Tsunoda","year":"2015","unstructured":"Tsunoda K (2015) Derivation of Stefan problem from a one-dimensional exclusion process with speed change. Markov Process Relat 21:263\u2013273","journal-title":"Markov Process Relat"},{"key":"3334_CR32","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4078-5","volume-title":"Models of phase transition","author":"A Visintin","year":"1996","unstructured":"Visintin A (1996) Models of phase transition. Birkh\u00e4user, London"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03334-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-025-03334-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-025-03334-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,9]],"date-time":"2025-09-09T05:06:48Z","timestamp":1757394408000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-025-03334-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,25]]},"references-count":32,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2025,10]]}},"alternative-id":["3334"],"URL":"https:\/\/doi.org\/10.1007\/s40314-025-03334-4","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"type":"print","value":"2238-3603"},{"type":"electronic","value":"1807-0302"}],"subject":[],"published":{"date-parts":[[2025,7,25]]},"assertion":[{"value":"2 November 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 November 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 July 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 July 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have any financial or non-financial interests that are directly or indirectly related to this work.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"377"}}