{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T18:41:42Z","timestamp":1783104102756,"version":"3.54.6"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2020,11,6]],"date-time":"2020-11-06T00:00:00Z","timestamp":1604620800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,11,6]],"date-time":"2020-11-06T00:00:00Z","timestamp":1604620800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11471046 and 11571045"],"award-info":[{"award-number":["11471046 and 11571045"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12001055 and 11971072"],"award-info":[{"award-number":["12001055 and 11971072"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2020,12]]},"DOI":"10.1007\/s10444-020-09822-x","type":"journal-article","created":{"date-parts":[[2020,11,6]],"date-time":"2020-11-06T03:02:41Z","timestamp":1604631761000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["Second-order energy stable schemes for the new model of the Cahn-Hilliard-MHD equations"],"prefix":"10.1007","volume":"46","author":[{"given":"Rui","family":"Chen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hui","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,11,6]]},"reference":[{"key":"9822_CR1","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/S0920-3796(00)00433-6","volume":"54","author":"MA Abdou","year":"2001","unstructured":"Abdou, M.A., et al.: On the exploration of innovative concepts for fusion chamber technology fusion. Fusion Eng. Des. 54, 181\u2013247 (2001)","journal-title":"Fusion Eng. Des."},{"key":"9822_CR2","doi-asserted-by":"crossref","first-page":"258","DOI":"10.1063\/1.1744102","volume":"28","author":"JW Cahn","year":"1958","unstructured":"Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258\u2013267 (1958)","journal-title":"J. Chem. Phys."},{"key":"9822_CR3","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1016\/j.jcp.2015.09.025","volume":"302","author":"R Chen","year":"2015","unstructured":"Chen, R., Ji, G., Yang, X., Zhang, H.: Decoupled energy stable schemes for phase field vesicle membrane model. J. Comput. Phys. 302, 509\u2013523 (2015)","journal-title":"J. Comput. Phys."},{"key":"9822_CR4","doi-asserted-by":"crossref","first-page":"A2808","DOI":"10.1137\/17M1119834","volume":"39","author":"R Chen","year":"2017","unstructured":"Chen, R., Yang, X., Zhang, H.: Second order, linear, and unconditionally energy stable schemes for a hydrodynamic model of smectic-A liquid crystals. SIAM J. Sci. Comput. 39, A2808\u2013A2833 (2017)","journal-title":"SIAM J. Sci. Comput."},{"key":"9822_CR5","doi-asserted-by":"crossref","first-page":"661","DOI":"10.4208\/jcm.1703-m2016-0614","volume":"36","author":"R Chen","year":"2018","unstructured":"Chen, R., Yang, X., Zhang, H.: Decoupled, energy stable scheme for hydrodynamic Allen-Cahn phase field moving contact line model. J. Comput. Math. 36, 661\u2013681 (2018)","journal-title":"J. Comput. Math."},{"key":"9822_CR6","doi-asserted-by":"publisher","unstructured":"Chen, W., Feng, W., Liu, Y., Wang, C., Wise, S.M.: A second order energy stable scheme for the Cahn-Hilliard-Hele-Shaw equations. Discrete Cont. Dyn. Sys. B 24. https:\/\/doi.org\/10.3934\/dcdsb.2018090 (2016)","DOI":"10.3934\/dcdsb.2018090"},{"key":"9822_CR7","doi-asserted-by":"crossref","first-page":"B701","DOI":"10.1137\/12088879X","volume":"35","author":"EC Cyr","year":"2013","unstructured":"Cyr, E.C., Shadid, J.N., Tuminaro, R.S., Pawlowski, R.P., Chac\u00f3n, L.: A new approximate block fractorization preconditioner for two-dimensional incompressible (reduced) resistive MHD. SIAM J. Sci. Comput. 35, B701\u2013B730 (2013)","journal-title":"SIAM J. Sci. Comput."},{"key":"9822_CR8","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.1137\/050638333","volume":"44","author":"X Feng","year":"2006","unstructured":"Feng, X.: Fully discrete finite element approximations of the Navier-Stokes-Cahn-Hilliard diffuse interface model for two-phase fluid flows. SIAM J. Numer. Anal. 44, 1049\u20131072 (2006)","journal-title":"SIAM J. Numer. Anal."},{"key":"9822_CR9","doi-asserted-by":"crossref","first-page":"3036","DOI":"10.1137\/130908208","volume":"51","author":"G Gr\u00fcn","year":"2013","unstructured":"Gr\u00fcn, G.: On convergent schemes for diffuse interface models for two-phase flow of incompressible fluids with general mass densities. SIAM J. Numer. Anal. 51, 3036\u20133061 (2013)","journal-title":"SIAM J. Numer. Anal."},{"key":"9822_CR10","doi-asserted-by":"crossref","first-page":"486","DOI":"10.1016\/j.jcp.2014.07.038","volume":"276","author":"Z Guo","year":"2014","unstructured":"Guo, Z., Lin, P., Lowengrub, J.S.: A numerical method for the quasi-incompressible Cahn-Hilliard-Navier-Stokes equations for variable density flows with a discrete energy law. J. Comput. Phys. 276, 486\u2013507 (2014)","journal-title":"J. Comput. Phys."},{"key":"9822_CR11","doi-asserted-by":"crossref","DOI":"10.1093\/acprof:oso\/9780198566656.001.0001","volume-title":"Mathematical Methods for the Magnetohydrodynamics of Liquid Metals","author":"JF Gerbeau","year":"2006","unstructured":"Gerbeau, J.F., Le Bris, C., Leli\u00e9vre, T.: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Oxford University Press, Oxford (2006)"},{"key":"9822_CR12","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1090\/S0025-5718-1991-1066834-0","volume":"56","author":"MD Gunzburger","year":"1991","unstructured":"Gunzburger, M.D., Meir, A.J., Peterson, J.S.: On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary incompressible magnetohydrodynamics. Math. Comp. 56, 523\u2013563 (1991)","journal-title":"Math. Comp."},{"key":"9822_CR13","doi-asserted-by":"crossref","first-page":"1210","DOI":"10.1007\/s10915-018-0748-0","volume":"77","author":"D Han","year":"2018","unstructured":"Han, D., Wang, X.: A second order in time, decoupled, unconditionally stable numerical scheme for the Cahn-Hilliard-Darcy system. J. Sci. Comput. 77, 1210\u20131233 (2018)","journal-title":"J. Sci. Comput."},{"key":"9822_CR14","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1142\/S0218202518500173","volume":"24","author":"R Hiptmair","year":"2018","unstructured":"Hiptmair, R., Li, L., Mao, S., Zheng, W.: A fully divergence-free finite element method for magneto-hydrodynamic equations. Math. Mod. Meth. Appl. Sci. 24, 659\u2013695 (2018)","journal-title":"Math. Mod. Meth. Appl. Sci."},{"key":"9822_CR15","doi-asserted-by":"crossref","first-page":"1953","DOI":"10.1090\/S0025-5718-2013-02678-6","volume":"82","author":"R Ingram","year":"2013","unstructured":"Ingram, R.: A new linearly extrapolated Crank-Nicolson time-stepping scheme for the Navier-Stokes equations. Math. Comp. 82, 1953\u20131973 (2013)","journal-title":"Math. Comp."},{"key":"9822_CR16","doi-asserted-by":"crossref","first-page":"870","DOI":"10.1137\/0907059","volume":"7","author":"J van Kan","year":"1986","unstructured":"van Kan, J.: A second-order accurate pressure-correction scheme for viscous incompressible flow. SIAM J. Sci. Statist. Comput. 7, 870\u2013891 (1986)","journal-title":"SIAM J. Sci. Statist. Comput."},{"key":"9822_CR17","doi-asserted-by":"crossref","first-page":"514","DOI":"10.1063\/1.1425844","volume":"14","author":"H Lee","year":"2002","unstructured":"Lee, H., Lowengrub, J.S., Goodman, J.: Modeling pinchoff and reconnection in a Hele-Shaw cell. II. Analysis and simulation in the nonlinear regime. Phys. Fluids. 14, 514\u2013545 (2002)","journal-title":"Phys. Fluids."},{"key":"9822_CR18","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1016\/j.jcp.2017.09.025","volume":"351","author":"L Li","year":"2017","unstructured":"Li, L., Zheng, W.: A robust solver for the finite element approximation of stationary incompressible MHD equations in 3D. J. Comput. Phys. 351, 254\u2013270 (2017)","journal-title":"J. Comput. Phys."},{"key":"9822_CR19","doi-asserted-by":"crossref","first-page":"1815","DOI":"10.1007\/s11425-016-5137-2","volume":"59","author":"X Li","year":"2016","unstructured":"Li, X., Qiao, Z.H., Zhang, H.: An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation. Sci. China Math. 59, 1815\u20131834 (2016)","journal-title":"Sci. China Math."},{"key":"9822_CR20","doi-asserted-by":"crossref","first-page":"693","DOI":"10.4208\/jcm.1611-m2016-0517","volume":"35","author":"X Li","year":"2017","unstructured":"Li, X., Qiao, Z.H., Zhang, H.: A second-order convex-splitting scheme for the Cahn-Hilliard equation with variable interfacial parameters. J. Comput. Math. 35, 693\u2013710 (2017)","journal-title":"J. Comput. Math."},{"key":"9822_CR21","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1007\/s10915-014-9867-4","volume":"62","author":"C Liu","year":"2014","unstructured":"Liu, C., Shen, J., Yang, X.: Decoupled energy stable schemes for a phase-field model of two-phase incompressible flows with variable density. J. Sci. Comput. 62, 601\u2013622 (2014)","journal-title":"J. Sci. Comput."},{"key":"9822_CR22","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1016\/j.jcp.2016.04.019","volume":"316","author":"Y Ma","year":"2016","unstructured":"Ma, Y., Hu, K., Hu, X., Xu, J.: Robust preconditioners for incompressible MHD models. J. Comput. Phys. 316, 721\u2013746 (2016)","journal-title":"J. Comput. Phys."},{"key":"9822_CR23","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-015-7883-7","volume-title":"Magnetohydrodynamics","author":"R Moreau","year":"1990","unstructured":"Moreau, R.: Magnetohydrodynamics. Kluwer Academic Publishers, Dordrecht, Boston (1990)"},{"key":"9822_CR24","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1016\/j.jcp.2007.07.025","volume":"227","author":"M-J Ni","year":"2007","unstructured":"Ni, M.-J., Munipalli, R., Huang, P., Morley, N.B., Abdou, M.A.: A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part I. On a rectangular collocated grid system. J. Comp. Phys. 227, 174\u2013204 (2007)","journal-title":"J. Comp. Phys."},{"key":"9822_CR25","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/j.jcp.2007.07.023","volume":"227","author":"M-J Ni","year":"2007","unstructured":"Ni, M.-J., Munipalli, R., Huang, P., Morley, N.B., Abdou, M.A.: A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part II: on an arbitrary collocated mesh. J. Comp. Phys. 227, 205\u2013228 (2007)","journal-title":"J. Comp. Phys."},{"key":"9822_CR26","doi-asserted-by":"crossref","first-page":"B930","DOI":"10.1137\/140955082","volume":"36","author":"EG Phillips","year":"2014","unstructured":"Phillips, E.G., Elman, H.C., Cyr, E.C., Shadid, J.N., Pawlowski, R.P.: A block preconditioner for an exact penalty formulation for stationary MHD. SIAM J. Sci Comput. 36, B930\u2013B951 (2014)","journal-title":"SIAM J. Sci Comput."},{"key":"9822_CR27","doi-asserted-by":"crossref","first-page":"B1009","DOI":"10.1137\/16M1074084","volume":"38","author":"EG Phillips","year":"2016","unstructured":"Phillips, E.G., Shadid, J.N., Cyr, E.C., Elman, H.C., Pawlowski, R.P.: Block preconditioners for stable mixed nodal and edge finite element representations of incompressible resistive MHD. SIAM J. Sci. Comput. 38, B1009\u2013B1031 (2016)","journal-title":"SIAM J. Sci. Comput."},{"key":"9822_CR28","doi-asserted-by":"crossref","first-page":"2977","DOI":"10.1016\/j.jcp.2010.12.046","volume":"230","author":"R Planas","year":"2011","unstructured":"Planas, R., Badia, S., Codina, R.: Approximation of the inductionless MHD problem using a stabilized finite element method. J. Comput. Phys. 230, 2977\u20132996 (2011)","journal-title":"J. Comput. Phys."},{"key":"9822_CR29","doi-asserted-by":"crossref","first-page":"1065","DOI":"10.1051\/m2an:2008034","volume":"42","author":"A Prohl","year":"2008","unstructured":"Prohl, A.: Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamics system. ESAIM Math. Model Num. Anal. 42, 1065\u20131087 (2008)","journal-title":"ESAIM Math. Model Num. Anal."},{"key":"9822_CR30","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1137\/120902410","volume":"53","author":"ZH Qiao","year":"2015","unstructured":"Qiao, Z.H., Tang, T., Xie, H.: Error analysis of a mixed finite element method for molecular beam epitaxy model. SIAM J. Numer. Anal. 53, 184\u2013205 (2015)","journal-title":"SIAM J. Numer. Anal."},{"key":"9822_CR31","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1007\/s00211-003-0487-4","volume":"96","author":"D Sch\u00f6tzau","year":"2004","unstructured":"Sch\u00f6tzau, D.: Mixed finite element methods for stationary incompressible magneto Chydrodynamics. Numer. Math. 96, 771\u2013800 (2004)","journal-title":"Numer. Math."},{"key":"9822_CR32","doi-asserted-by":"crossref","first-page":"1039","DOI":"10.1090\/S0025-5718-96-00750-8","volume":"65","author":"J Shen","year":"1996","unstructured":"Shen, J.: On error estimates of the projection methods for the Navier-Stokes equations: second-order schemes. Math. Comp. 65, 1039\u20131065 (1996)","journal-title":"Math. Comp."},{"key":"9822_CR33","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1137\/130921593","volume":"36","author":"J Shen","year":"2014","unstructured":"Shen, J., Yang, X.: Decoupled energy stable schemes for phase-field models of two-phase complex fluids. SIAM J. Sci. Comput. 36, 122\u2013145 (2014)","journal-title":"SIAM J. Sci. Comput."},{"key":"9822_CR34","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1137\/140971154","volume":"53","author":"J Shen","year":"2015","unstructured":"Shen, J., Yang, X.: Decoupled energy stable schemes for phase-field models of two-phase incompressible flows. SIAM J. Numer. Anal. 53, 279\u2013296 (2015)","journal-title":"SIAM J. Numer. Anal."},{"key":"9822_CR35","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1016\/j.jcp.2014.12.046","volume":"284","author":"J Shen","year":"2015","unstructured":"Shen, J., Yang, X., Yu, H.: Efficient energy stable numerical schemes for a phase field moving contact line model. J. Comput. Phys. 284, 617\u2013630 (2015)","journal-title":"J. Comput. Phys."},{"key":"9822_CR36","doi-asserted-by":"crossref","first-page":"1669","DOI":"10.3934\/dcds.2010.28.1669","volume":"28","author":"J Shen","year":"2010","unstructured":"Shen, J., Yang, X.F.: Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discete Cont. Dyn. Sys.-A 28, 1669\u20131691 (2010)","journal-title":"Discete Cont. Dyn. Sys.-A"},{"key":"9822_CR37","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1016\/j.jcp.2017.10.021","volume":"353","author":"J Shen","year":"2018","unstructured":"Shen, J., Xu, J., Yang, J.: The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353, 407\u2013416 (2018)","journal-title":"J. Comput. Phys."},{"key":"9822_CR38","doi-asserted-by":"crossref","first-page":"2895","DOI":"10.1137\/17M1159968","volume":"56","author":"J Shen","year":"2018","unstructured":"Shen, J., Xu, J.: Convergence and error analysis for the scalar auxiliary variable (SAV) scheme to gradient flows. SIAM J. Num. Anal. 56, 2895\u20132912 (2018)","journal-title":"SIAM J. Num. Anal."},{"key":"9822_CR39","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1016\/j.cam.2018.06.031","volume":"346","author":"Z Xu","year":"2019","unstructured":"Xu, Z., Zhang, H.: Stabilized semi-implicit numerical scheme for the Cahn-Hilliard with variable interfacial parameters. J. Comput. Appl. Math. 346, 307\u2013322 (2019)","journal-title":"J. Comput. Appl. Math."},{"key":"9822_CR40","doi-asserted-by":"publisher","unstructured":"Xu, Z., Yang, X.F., Zhang, H., Xie, Z.Q.: Efficient and Linear Schemes for Anisotropic Cahn-Hilliard Equations Using the Stabilized Invariant Energy Quadratization (S-IEQ) Approach, Comm. Comput. Phys. Online Publishing. https:\/\/doi.org\/10.1016\/j.cpc.2018.12.019 (2019)","DOI":"10.1016\/j.cpc.2018.12.019"},{"key":"9822_CR41","doi-asserted-by":"crossref","first-page":"294","DOI":"10.1016\/j.jcp.2016.09.029","volume":"327","author":"X Yang","year":"2016","unstructured":"Yang, X.: Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends. J. Comput. Phys. 327, 294\u2013316 (2016)","journal-title":"J. Comput. Phys."},{"key":"9822_CR42","doi-asserted-by":"crossref","first-page":"1993","DOI":"10.1142\/S0218202517500373","volume":"27","author":"X Yang","year":"2017","unstructured":"Yang, X., Zhao, J., Wang, Q., Shen, J.: Numerical approximations for a three components Cahn-Hilliard phase-field model based on the invariant energy quadratization method. Math. Mod. Meth. Appl. Sci. 27, 1993\u20132030 (2017)","journal-title":"Math. Mod. Meth. Appl. Sci."},{"key":"9822_CR43","doi-asserted-by":"publisher","unstructured":"Zhang, J., Han, T.Y., Yang, J.C., Ni, M.J.: On the spreading of impacting drops under the influence of a vertical magnetic field. J. Fluid Mech. 809. https:\/\/doi.org\/10.1017\/jfm.2016.725 (2016)","DOI":"10.1017\/jfm.2016.725"},{"key":"9822_CR44","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1017\/jfm.2017.514","volume":"828","author":"J Zhang","year":"2017","unstructured":"Zhang, J., Ni, M.J.: What happens to the vortex structures when the rising bubble transits from zigzag to spiral. J. Fluid Mech. 828, 353\u2013373 (2017)","journal-title":"J. Fluid Mech."},{"key":"9822_CR45","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1016\/j.jcp.2018.09.001","volume":"375","author":"J Zhang","year":"2018","unstructured":"Zhang, J., Ni, M.J.: Direct numerical simulations of incompressible multiphase magnetohydrodynamics with phase change. J. Comput. Phys. 375, 717\u2013746 (2018)","journal-title":"J. Comput. Phys."}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-020-09822-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10444-020-09822-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-020-09822-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,12,15]],"date-time":"2020-12-15T13:10:43Z","timestamp":1608037843000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10444-020-09822-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,6]]},"references-count":45,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2020,12]]}},"alternative-id":["9822"],"URL":"https:\/\/doi.org\/10.1007\/s10444-020-09822-x","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,11,6]]},"assertion":[{"value":"3 February 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 October 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 November 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"79"}}