{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T03:06:22Z","timestamp":1772852782225,"version":"3.50.1"},"reference-count":24,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,4,11]],"date-time":"2023-04-11T00:00:00Z","timestamp":1681171200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,4,11]],"date-time":"2023-04-11T00:00:00Z","timestamp":1681171200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"name":"BK21 FOUR","award":["17890"],"award-info":[{"award-number":["17890"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2023,5]]},"DOI":"10.1007\/s10915-023-02198-2","type":"journal-article","created":{"date-parts":[[2023,4,11]],"date-time":"2023-04-11T11:02:51Z","timestamp":1681210971000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["A Second-order Time-Accurate Unconditionally Stable Method for a Gradient Flow for the Modica\u2013Mortola Functional"],"prefix":"10.1007","volume":"95","author":[{"given":"Seokjun","family":"Ham","sequence":"first","affiliation":[]},{"given":"Soobin","family":"Kwak","sequence":"additional","affiliation":[]},{"given":"Chaeyoung","family":"Lee","sequence":"additional","affiliation":[]},{"given":"Gyeonggyu","family":"Lee","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0484-9189","authenticated-orcid":false,"given":"Junseok","family":"Kim","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,4,11]]},"reference":[{"key":"2198_CR1","first-page":"425","volume":"21","author":"Q Du","year":"2020","unstructured":"Du, Q., Feng, X.: The phase field method for geometric moving interfaces and their numerical approximations. Handb. Numer. Anal. 21, 425\u2013508 (2020)","journal-title":"Handb. Numer. Anal."},{"issue":"8","key":"2198_CR2","doi-asserted-by":"publisher","first-page":"1600","DOI":"10.1109\/TPAMI.2014.2300478","volume":"36","author":"C Garcia-Cardona","year":"2014","unstructured":"Garcia-Cardona, C., Merkurjev, E., Bertozzi, A.L., Flenner, A., Percus, A.G.: Multiclass data segmentation using diffuse interface methods on graphs. IEEE Trans. Pattern Anal. Mach. Intell. 36(8), 1600\u20131613 (2014)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"issue":"1","key":"2198_CR3","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1515\/acv-2020-0113","volume":"16","author":"Y Giga","year":"2021","unstructured":"Giga, Y., Okamoto, J., Uesaka, M.: A finer singular limit of a single-well Modica-Mortola functional and its applications to the Kobayashi-Warren-Carter energy. Adv. Calc. Var. 16(1), 163\u201382 (2021)","journal-title":"Adv. Calc. Var."},{"key":"2198_CR4","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1016\/j.cpc.2017.05.002","volume":"219","author":"Y Gong","year":"2017","unstructured":"Gong, Y., Zhao, J., Wang, Q.: An energy stable algorithm for a quasi-incompressible hydrodynamic phase-field model of viscous fluid mixtures with variable densities and viscosities. Comput. Phys. Commun. 219, 20\u201334 (2017)","journal-title":"Comput. Phys. Commun."},{"issue":"2","key":"2198_CR5","doi-asserted-by":"publisher","first-page":"B528","DOI":"10.1137\/17M1135451","volume":"40","author":"Y Gong","year":"2018","unstructured":"Gong, Y., Zhao, J., Wang, Q.: Second order fully discrete energy stable methods on staggered grids for hydrodynamic phase field models of binary viscous fluids. SIAM J. Sci. Comput. 40(2), B528\u2013B553 (2018)","journal-title":"SIAM J. Sci. Comput."},{"issue":"2","key":"2198_CR6","doi-asserted-by":"publisher","first-page":"025203","DOI":"10.1063\/5.0081229","volume":"12","author":"S Ham","year":"2022","unstructured":"Ham, S., Hwang, Y., Kwak, S., Kim, J.: Unconditionally stable second-order accurate scheme for a parabolic sine-Gordon equation. AIP Adv. 12(2), 025203 (2022)","journal-title":"AIP Adv."},{"key":"2198_CR7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jocs.2018.02.011","volume":"26","author":"J H\u00f6tzer","year":"2018","unstructured":"H\u00f6tzer, J., Reiter, A., Hierl, H., Steinmetz, P., Selzer, M., Nestler, B.: The parallel multi-physics phase-field framework Pace3D. J. Comput. Sci. 26, 1\u201312 (2018)","journal-title":"J. Comput. Sci."},{"issue":"1","key":"2198_CR8","doi-asserted-by":"publisher","first-page":"014003","DOI":"10.1088\/1361-6420\/ab261e","volume":"36","author":"B Jin","year":"2019","unstructured":"Jin, B., Xu, Y.: Adaptive reconstruction for electrical impedance tomography with a piecewise constant conductivity. Inverse Probl. 36(1), 014003 (2019)","journal-title":"Inverse Probl."},{"key":"2198_CR9","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1016\/j.cam.2018.02.026","volume":"340","author":"D Jeong","year":"2018","unstructured":"Jeong, D., Kim, J.: An explicit hybrid finite difference scheme for the Allen-Cahn equation. J. Comput. Appl. Math. 340, 247\u2013255 (2018)","journal-title":"J. Comput. Appl. Math."},{"issue":"5","key":"2198_CR10","doi-asserted-by":"publisher","first-page":"1213","DOI":"10.1137\/060662708","volume":"67","author":"YM Jung","year":"2007","unstructured":"Jung, Y.M., Kang, S.H., Shen, J.: Multiphase image segmentation via Modica-Mortola phase transition. SIAM J. Appl. Math. 67(5), 1213\u20131232 (2007)","journal-title":"SIAM J. Appl. Math."},{"key":"2198_CR11","doi-asserted-by":"crossref","first-page":"2019","DOI":"10.1016\/j.camwa.2020.08.026","volume":"80","author":"H Kim","year":"2020","unstructured":"Kim, H., Yun, A., Yoon, S., Lee, C., Park, J., Kim, J.: Pattern formation in reaction-diffusion systems on evolving surfaces. Comput. Math. Appl. 80, 2019\u20132028 (2020)","journal-title":"Comput. Math. Appl."},{"issue":"8","key":"2198_CR12","doi-asserted-by":"crossref","first-page":"1885","DOI":"10.1016\/j.camwa.2017.06.053","volume":"74","author":"HG Lee","year":"2017","unstructured":"Lee, H.G.: A semi-analytical Fourier spectral method for the Swift-Hohenberg equation. Comput. Math. Appl. 74(8), 1885\u20131896 (2017)","journal-title":"Comput. Math. Appl."},{"key":"2198_CR13","doi-asserted-by":"publisher","first-page":"108161","DOI":"10.1016\/j.aml.2022.108161","volume":"132","author":"HG Lee","year":"2022","unstructured":"Lee, H.G., Shin, J., Lee, J.Y.: Energy quadratization Runge-Kutta scheme for the conservative Allen-Cah equation with a nonlocal Lagrange multiplier. Appl. Math. Lett. 132, 108161 (2022)","journal-title":"Appl. Math. Lett."},{"key":"2198_CR14","doi-asserted-by":"publisher","first-page":"38","DOI":"10.1016\/j.cam.2018.12.024","volume":"353","author":"C Li","year":"2019","unstructured":"Li, C., Huang, Y., Yi, N.: An unconditionally energy stable second order finite element method for solving the Allen-Cahn equation. J. Comput. Appl. Math. 353, 38\u201348 (2019)","journal-title":"J. Comput. Appl. Math."},{"key":"2198_CR15","doi-asserted-by":"publisher","first-page":"111087","DOI":"10.1016\/j.jcp.2022.111087","volume":"458","author":"D Li","year":"2022","unstructured":"Li, D., Quan, C., Xu, J.: Stability and convergence of strang splitting. Part I: Scalar Allen-Cahn equation. J. Comput. Phys. 458, 111087 (2022)","journal-title":"J. Comput. Phys."},{"key":"2198_CR16","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1155\/2021\/2442071","volume":"2021","author":"Y Li","year":"2021","unstructured":"Li, Y., Yoon, S., Wang, J., Park, J., Kim, S., Lee, C., Kim, J.: Fast and efficient numerical finite difference method for multiphase image segmentation. Math. Probl. Eng. 2021, 1\u201323 (2021)","journal-title":"Math. Probl. Eng."},{"issue":"2","key":"2198_CR17","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1016\/j.camwa.2011.05.054","volume":"62","author":"Y Li","year":"2011","unstructured":"Li, Y., Kim, J.: Multiphase image segmentation using a phase-field model. Comput. Math. Appl. 62(2), 737\u2013745 (2011)","journal-title":"Comput. Math. Appl."},{"key":"2198_CR18","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2019.112781","volume":"362","author":"R Ma","year":"2020","unstructured":"Ma, R., Sun, W.: FFT-based solver for higher-order and multi-phase-field fracture models applied to strongly anisotropic brittle materials. Comput. Meth. Appl. Mech. Eng. 362, 112781 (2020)","journal-title":"Comput. Meth. Appl. Mech. Eng."},{"key":"2198_CR19","doi-asserted-by":"publisher","first-page":"102025","DOI":"10.1016\/j.cagd.2021.102025","volume":"89","author":"J Meny","year":"2021","unstructured":"Meny, J., Rumpf, M., Sassen, J.: A phase-field approach to variational hierarchical surface segmentation. Comput. Aided Geom. Des. 89, 102025 (2021)","journal-title":"Comput. Aided Geom. Des."},{"issue":"1","key":"2198_CR20","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41524-020-00445-w","volume":"6","author":"Q Wang","year":"2020","unstructured":"Wang, Q., Zhang, G., Li, Y., Hong, Z., Wang, D., Shi, S.: Application of phase-field method in rechargeable batteries. Npj Comput. Mater. 6(1), 1\u20138 (2020)","journal-title":"Npj Comput. Mater."},{"issue":"1","key":"2198_CR21","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1007\/s10444-011-9227-y","volume":"38","author":"Z Rong","year":"2013","unstructured":"Rong, Z., Wang, L.L., Tai, X.C.: Adaptive wavelet collocation methods for image segmentation using TV-Allen-Cahn type models. Adv. Comput. Math. 38(1), 101\u2013131 (2013)","journal-title":"Adv. Comput. Math."},{"key":"2198_CR22","doi-asserted-by":"publisher","first-page":"382","DOI":"10.1016\/j.jcp.2019.06.030","volume":"395","author":"X Zhao","year":"2019","unstructured":"Zhao, X., Wang, Q.: A second order fully-discrete linear energy stable scheme for a binary compressible viscous fluid model. J. Comput. Phys. 395, 382\u2013409 (2019)","journal-title":"J. Comput. Phys."},{"issue":"1","key":"2198_CR23","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10915-021-01430-1","volume":"87","author":"C Zhang","year":"2021","unstructured":"Zhang, C., Ouyang, J., Wang, X., Chai, Y., Ma, M.: Analysis of the energy stability for stabilized semi-implicit schemes of the functionalized Cahn-Hilliard Mass-conserving gradient flow equation. J. Sci. Comput. 87(1), 1\u201325 (2021)","journal-title":"J. Sci. Comput."},{"issue":"5","key":"2198_CR24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10444-021-09897-0","volume":"47","author":"N Zheng","year":"2021","unstructured":"Zheng, N., Li, X.: Error analysis of the SAV Fourier-spectral method for the Cahn-Hilliard-Hele-Shaw system. Adv. Comput. Math. 47(5), 1\u201327 (2021)","journal-title":"Adv. Comput. Math."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-023-02198-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-023-02198-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-023-02198-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,12,10]],"date-time":"2023-12-10T14:19:36Z","timestamp":1702217976000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-023-02198-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,11]]},"references-count":24,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,5]]}},"alternative-id":["2198"],"URL":"https:\/\/doi.org\/10.1007\/s10915-023-02198-2","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,11]]},"assertion":[{"value":"14 October 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 February 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 March 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 April 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare that they have no conflict of interest.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"63"}}