{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T01:07:03Z","timestamp":1780448823649,"version":"3.54.1"},"reference-count":35,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T00:00:00Z","timestamp":1780358400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Journal of Computational Physics"],"published-print":{"date-parts":[[2026,10]]},"DOI":"10.1016\/j.jcp.2026.114982","type":"journal-article","created":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T15:47:37Z","timestamp":1777564057000},"page":"114982","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"C","title":["Convex-concave splitting for the Allen-Cahn equation leads to \u03b52-slow movement of interfaces"],"prefix":"10.1016","volume":"562","author":[{"given":"Patrick","family":"Dondl","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2110-8881","authenticated-orcid":false,"given":"Akwum","family":"Onwunta","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ludwig","family":"Striet","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stephan","family":"Wojtowytsch","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"key":"10.1016\/j.jcp.2026.114982_bib0001","first-page":"285","article-title":"Un esempio di \u0393-convergenza","volume":"14","author":"Modica","year":"1977","journal-title":"Boll. Un. Mat. Ital. B"},{"key":"10.1016\/j.jcp.2026.114982_bib0002","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1007\/BF00251230","article-title":"The gradient theory of phase transitions and the minimal interface criterion","volume":"98","author":"Modica","year":"1987","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"2","key":"10.1016\/j.jcp.2026.114982_bib0003","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/0022-0396(91)90147-2","article-title":"Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics","volume":"90","author":"Bronsard","year":"1991","journal-title":"J. Differ. Equ."},{"issue":"2","key":"10.1016\/j.jcp.2026.114982_bib0004","doi-asserted-by":"crossref","first-page":"417","DOI":"10.4310\/jdg\/1214454300","article-title":"Convergence of the Allen-Cahn equation to Brakke\u2019s motion by mean curvature","volume":"38","author":"Ilmanen","year":"1993","journal-title":"J. Differ. Geom."},{"issue":"8","key":"10.1016\/j.jcp.2026.114982_bib0005","doi-asserted-by":"crossref","first-page":"1597","DOI":"10.1002\/cpa.21747","article-title":"Convergence of the Allen-Cahn equation to multiphase mean curvature flow","volume":"71","author":"Laux","year":"2018","journal-title":"Commun. Pure Appl. Math."},{"issue":"6","key":"10.1016\/j.jcp.2026.114982_bib0006","doi-asserted-by":"crossref","first-page":"6222","DOI":"10.1137\/20M1322182","article-title":"Convergence rates of the Allen\u2013Cahn equation to mean curvature flow: a short proof based on relative entropies","volume":"52","author":"Fischer","year":"2020","journal-title":"SIAM J. Math. Anal."},{"issue":"3","key":"10.1016\/j.jcp.2026.114982_bib0007","doi-asserted-by":"crossref","first-page":"1350","DOI":"10.1137\/17M1117835","article-title":"Generalizing diffuse interface methods on graphs: nonsmooth potentials and hypergraphs","volume":"78","author":"Bosch","year":"2018","journal-title":"SIAM J. Appl. Math."},{"issue":"3","key":"10.1016\/j.jcp.2026.114982_bib0008","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1007\/s00332-023-09903-3","article-title":"Stochastic gradient descent with noise of machine learning type Part I: discrete time analysis","volume":"33","author":"Wojtowytsch","year":"2023","journal-title":"J. Nonlinear Sci."},{"key":"10.1016\/j.jcp.2026.114982_bib0009","first-page":"1","article-title":"Acceleration by stepsize hedging: silver stepsize schedule for smooth convex optimization","author":"Altschuler","year":"2024","journal-title":"Math. Program."},{"issue":"2","key":"10.1016\/j.jcp.2026.114982_bib0010","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3708502","article-title":"Acceleration by stepsize hedging: multi-step descent and the silver stepsize schedule","volume":"72","author":"Altschuler","year":"2025","journal-title":"J. ACM"},{"key":"10.1016\/j.jcp.2026.114982_bib0011","doi-asserted-by":"crossref","DOI":"10.1287\/ijoo.2024.0057","article-title":"Accelerated objective gap and gradient norm convergence for gradient descent via long steps","author":"Grimmer","year":"2025","journal-title":"INFORMS J. Optim."},{"key":"10.1016\/j.jcp.2026.114982_bib0012","series-title":"Numerical Methods For Nonlinear Partial Differential Equations","volume":"volume 47","author":"Bartels","year":"2015"},{"key":"10.1016\/j.jcp.2026.114982_bib0013","unstructured":"O. Akande, P. Dondl, K. Gupta, A. Onwunta, S. Wojtowytsch, Momentum-based minimization of the Ginzburg-Landau functional on Euclidean spaces and graphs, (2024). arXiv: 2501.00389."},{"key":"10.1016\/j.jcp.2026.114982_bib0014","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1512\/iumj.2011.60.3949","article-title":"Convergence of perturbed Allen-Cahn equations to forced mean curvature flow","volume":"60","author":"Mugnai","year":"2011","journal-title":"Ind. Univ. Math. J."},{"issue":"5","key":"10.1016\/j.jcp.2026.114982_bib0015","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1002\/cpa.3160420502","article-title":"Metastable patterns in solutions of ut=\u03b52uxx\u2212f(u)","volume":"42","author":"Carr","year":"1989","journal-title":"Commun. Pure Appl. Math."},{"key":"10.1016\/j.jcp.2026.114982_bib0016","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/BF01048791","article-title":"Slow-motion manifolds, dormant instability, and singular perturbations","volume":"1","author":"Fusco","year":"1989","journal-title":"J. Dyn. Differ. Equ."},{"key":"10.1016\/j.jcp.2026.114982_bib0017","article-title":"On the slowness of phase boundary motion in one space dimension","volume":"91","author":"Bronsard","year":"1990","journal-title":"NASA STI\/Recon Tech. Rep. N"},{"issue":"6","key":"10.1016\/j.jcp.2026.114982_bib0018","doi-asserted-by":"crossref","first-page":"1544","DOI":"10.1137\/S0036141094275361","article-title":"Traveling waves as limits of solutions on bounded domains","volume":"27","author":"Fusco","year":"1996","journal-title":"SIAM J. Math. Anal."},{"key":"10.1016\/j.jcp.2026.114982_bib0019","series-title":"Diffusion Generated Motion by Mean Curvature","author":"Merriman","year":"1992"},{"issue":"5","key":"10.1016\/j.jcp.2026.114982_bib0020","doi-asserted-by":"crossref","first-page":"808","DOI":"10.1002\/cpa.21527","article-title":"Threshold dynamics for networks with arbitrary surface tensions","volume":"68","author":"Esedoglu","year":"2015","journal-title":"Commun. Pure Appl. Math."},{"issue":"5","key":"10.1016\/j.jcp.2026.114982_bib0021","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/s00526-016-1053-0","article-title":"Convergence of the thresholding scheme for multi-phase mean-curvature flow","volume":"55","author":"Laux","year":"2016","journal-title":"Calc. Var. Partial Differ. Equ."},{"key":"10.1016\/j.jcp.2026.114982_bib0022","series-title":"The Role of Metrics in the Theory of Partial Differential Equations","first-page":"63","article-title":"The thresholding scheme for mean curvature flow and de giorgi\u2019s ideas for minimizing movements","volume":"volume 85","author":"Laux","year":"2020"},{"key":"10.1016\/j.jcp.2026.114982_bib0023","first-page":"111","article-title":"A generalization of the bence, merriman and osher algorithm for motion by mean curvature","author":"Ishii","year":"1995","journal-title":"Curvature Flows Relat. Top."},{"issue":"2","key":"10.1016\/j.jcp.2026.114982_bib0024","doi-asserted-by":"crossref","first-page":"267","DOI":"10.2969\/jmsj\/05120267","article-title":"Threshold dynamics type approximation schemes for propagating fronts","volume":"51","author":"Ishii","year":"1999","journal-title":"J. Math. Soc. Jpn."},{"key":"10.1016\/j.jcp.2026.114982_bib0025","unstructured":"J. Budd, Y. van Gennip, Graph MBO as a semi-discrete implicit Euler scheme for graph Allen\u2013Cahn, (2019). arXiv: 1907.10774."},{"issue":"1","key":"10.1016\/j.jcp.2026.114982_bib0026","doi-asserted-by":"crossref","DOI":"10.1002\/gamm.202100004","article-title":"Classification and image processing with a semi-discrete scheme for fidelity forced Allen\u2013Cahn on graphs","volume":"44","author":"Budd","year":"2021","journal-title":"GAMM-Mitt."},{"key":"10.1016\/j.jcp.2026.114982_bib0027","series-title":"Angewandte Funktionalanalysis: Funktionalanalysis, Sobolev-R\u00e4ume und Elliptische Differentialgleichungen","author":"Dobrowolski","year":"2010"},{"key":"10.1016\/j.jcp.2026.114982_bib0028","series-title":"An Introduction to Variational Inequalities and Their Applications","author":"Kinderlehrer","year":"2000"},{"issue":"1","key":"10.1016\/j.jcp.2026.114982_bib0029","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1007\/BF00281746","article-title":"Global C1,1-regularity for solutions of quasilinear variational inequalities","volume":"89","author":"Gerhardt","year":"1985","journal-title":"Arch. Ration. Mech. Anal."},{"key":"10.1016\/j.jcp.2026.114982_bib0030","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00526-019-1543-y","article-title":"The regularity theory for the double obstacle problem","volume":"58","author":"Lee","year":"2019","journal-title":"Calc. Var. Partial Differ. Equ."},{"key":"10.1016\/j.jcp.2026.114982_bib0031","series-title":"Seminario di Analisi Matematica","first-page":"87","article-title":"Approssimazione variazionale di funzionali con curvatura","author":"Bellettini","year":"1993"},{"key":"10.1016\/j.jcp.2026.114982_bib0032","doi-asserted-by":"crossref","first-page":"675","DOI":"10.1007\/s00209-006-0002-6","article-title":"On a modified conjecture of de giorgi","volume":"254","author":"R\u00f6ger","year":"2006","journal-title":"Math. Z."},{"issue":"4","key":"10.1016\/j.jcp.2026.114982_bib0033","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1007\/s00526-017-1178-9","article-title":"Uniform regularity and convergence of phase-fields for Willmore\u2019s energy","volume":"56","author":"Dondl","year":"2017","journal-title":"Calc. Var. Partial Differ. Equ."},{"issue":"2","key":"10.1016\/j.jcp.2026.114982_bib0034","doi-asserted-by":"crossref","first-page":"693","DOI":"10.1007\/s00205-016-1043-6","article-title":"Phase field models for thin elastic structures with topological constraint","volume":"223","author":"Dondl","year":"2017","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"4","key":"10.1016\/j.jcp.2026.114982_bib0035","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1007\/s12532-020-00179-2","article-title":"OSQP: an operator splitting solver for quadratic programs","volume":"12","author":"Stellato","year":"2020","journal-title":"Math. Program. Comput."}],"container-title":["Journal of Computational Physics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0021999126003359?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0021999126003359?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T00:43:59Z","timestamp":1780447439000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0021999126003359"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,10]]},"references-count":35,"alternative-id":["S0021999126003359"],"URL":"https:\/\/doi.org\/10.1016\/j.jcp.2026.114982","relation":{},"ISSN":["0021-9991"],"issn-type":[{"value":"0021-9991","type":"print"}],"subject":[],"published":{"date-parts":[[2026,10]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Convex-concave splitting for the Allen-Cahn equation leads to \u03b52-slow movement of interfaces","name":"articletitle","label":"Article Title"},{"value":"Journal of Computational Physics","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.jcp.2026.114982","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 The Authors. Published by Elsevier Inc.","name":"copyright","label":"Copyright"}],"article-number":"114982"}}