{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,5]],"date-time":"2026-05-05T11:48:58Z","timestamp":1777981738828,"version":"3.51.4"},"reference-count":59,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2018,6,5]],"date-time":"2018-06-05T00:00:00Z","timestamp":1528156800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2018,6,5]],"date-time":"2018-06-05T00:00:00Z","timestamp":1528156800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100003347","name":"Fudan University","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100003347","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000086","name":"Directorate for Mathematical and Physical Sciences","doi-asserted-by":"publisher","award":["DMS1715504"],"award-info":[{"award-number":["DMS1715504"]}],"id":[{"id":"10.13039\/100000086","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100011535","name":"Missouri University of Science and Technology","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100011535","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2018,11]]},"DOI":"10.1007\/s10915-018-0748-0","type":"journal-article","created":{"date-parts":[[2018,6,5]],"date-time":"2018-06-05T10:18:26Z","timestamp":1528193906000},"page":"1210-1233","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":45,"title":["A Second Order in Time, Decoupled, Unconditionally Stable Numerical Scheme for the Cahn\u2013Hilliard\u2013Darcy System"],"prefix":"10.1007","volume":"77","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2859-7609","authenticated-orcid":false,"given":"Daozhi","family":"Han","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaoming","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2018,6,5]]},"reference":[{"issue":"2","key":"748_CR1","doi-asserted-by":"publisher","first-page":"492","DOI":"10.1063\/1.1425843","volume":"14","author":"H-G Lee","year":"2002","unstructured":"Lee, H.-G., Lowengrub, J.S., Goodman, J.: Modeling pinchoff and reconnection in a Hele-Shaw cell. I. The models and their calibration. Phys. Fluids 14(2), 492\u2013513 (2002). \n                    https:\/\/doi.org\/10.1063\/1.1425843","journal-title":"Phys. Fluids"},{"issue":"2","key":"748_CR2","doi-asserted-by":"publisher","first-page":"514","DOI":"10.1063\/1.1425844","volume":"14","author":"H-G Lee","year":"2002","unstructured":"Lee, H.-G., Lowengrub, J.S., Goodman, J.: Modeling pinchoff and reconnection in a Hele\u2013Shaw cell. II. Analysis and simulation in the nonlinear regime. Phys. Fluids 14(2), 514\u2013545 (2002). \n                    https:\/\/doi.org\/10.1063\/1.1425844","journal-title":"Phys. Fluids"},{"issue":"1","key":"748_CR3","doi-asserted-by":"publisher","first-page":"38","DOI":"10.1007\/s10915-010-9363-4","volume":"44","author":"SM Wise","year":"2010","unstructured":"Wise, S.M.: Unconditionally stable finite difference, nonlinear multigrid simulation of the Cahn\u2013Hilliard\u2013Hele\u2013Shaw system of equations. J. Sci. Comput. 44(1), 38\u201368 (2010). \n                    https:\/\/doi.org\/10.1007\/s10915-010-9363-4","journal-title":"J. Sci. Comput."},{"key":"748_CR4","volume-title":"Dynamics of Fluids in Porous Media","author":"J Bear","year":"1988","unstructured":"Bear, J.: Dynamics of Fluids in Porous Media. Courier Dover Publications, Mineola (1988)"},{"key":"748_CR5","doi-asserted-by":"publisher","first-page":"546","DOI":"10.1007\/978-1-4757-3033-3","volume-title":"Convection in Porous Media","author":"DA Nield","year":"1999","unstructured":"Nield, D.A., Bejan, A.: Convection in Porous Media, 2nd edn, p. 546. Springer, New York (1999)","edition":"2"},{"issue":"3","key":"748_CR6","doi-asserted-by":"publisher","first-page":"936","DOI":"10.1002\/num.22036","volume":"32","author":"D Han","year":"2016","unstructured":"Han, D., Wang, X.: Decoupled energy-law preserving numerical schemes for the Cahn\u2013Hilliard\u2013Darcy system. Numer. Methods Partial Differ. Equ. 32(3), 936\u2013954 (2016). \n                    https:\/\/doi.org\/10.1002\/num.22036","journal-title":"Numer. Methods Partial Differ. Equ."},{"issue":"2","key":"748_CR7","doi-asserted-by":"publisher","first-page":"531","DOI":"10.1007\/s00021-017-0334-5","volume":"20","author":"Luca Ded\u00e8","year":"2017","unstructured":"Ded\u00e8, L., Garcke, H., Lam, K.F.: A Hele\u2013Shaw\u2013Cahn\u2013Hilliard model for incompressible two-phase flows with different densities. J. Math. Fluid Mech. (2017). \n                    https:\/\/doi.org\/10.1007\/s00021-017-0334-5","journal-title":"Journal of Mathematical Fluid Mechanics"},{"issue":"3","key":"748_CR8","doi-asserted-by":"publisher","first-page":"1320","DOI":"10.1137\/110827119","volume":"50","author":"X Feng","year":"2012","unstructured":"Feng, X., Wise, S.: Analysis of a Darcy\u2013Cahn\u2013Hilliard diffuse interface model for the Hele\u2013Shaw flow and its fully discrete finite element approximation. SIAM J. Numer. Anal. 50(3), 1320\u20131343 (2012). \n                    https:\/\/doi.org\/10.1137\/110827119","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"748_CR9","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1016\/j.anihpc.2012.06.003","volume":"30","author":"X Wang","year":"2013","unstructured":"Wang, X., Zhang, Z.: Well-posedness of the Hele\u2013Shaw\u2013Cahn\u2013Hilliard system. Ann. Inst. H. Poincar\u00e9 Anal. Non Lin\u00e9aire 30(3), 367\u2013384 (2013). \n                    https:\/\/doi.org\/10.1016\/j.anihpc.2012.06.003","journal-title":"Ann. Inst. H. Poincar\u00e9 Anal. Non Lin\u00e9aire"},{"issue":"4","key":"748_CR10","doi-asserted-by":"crossref","first-page":"217","DOI":"10.3233\/ASY-2012-1092","volume":"78","author":"X Wang","year":"2012","unstructured":"Wang, X., Wu, H.: Long-time behavior for the Hele\u2013Shaw\u2013Cahn\u2013Hilliard system. Asymptot. Anal. 78(4), 217\u2013245 (2012)","journal-title":"Asymptot. Anal."},{"issue":"5","key":"748_CR11","doi-asserted-by":"publisher","first-page":"691","DOI":"10.1017\/S0956792513000144","volume":"24","author":"J Lowengrub","year":"2013","unstructured":"Lowengrub, J., Titi, E., Zhao, K.: Analysis of a mixture model of tumor growth. Eur. J. Appl. Math. 24(5), 691\u2013734 (2013). \n                    https:\/\/doi.org\/10.1017\/S0956792513000144","journal-title":"Eur. J. Appl. Math."},{"issue":"7","key":"748_CR12","doi-asserted-by":"publisher","first-page":"3032","DOI":"10.1016\/j.jde.2015.04.009","volume":"259","author":"J Jiang","year":"2015","unstructured":"Jiang, J., Wu, H., Zheng, S.: Well-posedness and long-time behavior of a non-autonomous Cahn\u2013Hilliard\u2013Darcy system with mass source modeling tumor growth. J. Differ. Equ. 259(7), 3032\u20133077 (2015). \n                    https:\/\/doi.org\/10.1016\/j.jde.2015.04.009","journal-title":"J. Differ. Equ."},{"issue":"3","key":"748_CR13","doi-asserted-by":"publisher","first-page":"1102","DOI":"10.1007\/s10915-015-0055-y","volume":"66","author":"Daozhi Han","year":"2015","unstructured":"Han, D.: A decoupled unconditionally stable numerical scheme for the Cahn\u2013Hilliard\u2013Hele\u2013Shaw system. J. Sci. Comput. 1\u201320 (2015). \n                    https:\/\/doi.org\/10.1007\/s10915-015-0055-y","journal-title":"Journal of Scientific Computing"},{"issue":"2","key":"748_CR14","doi-asserted-by":"publisher","first-page":"609","DOI":"10.1016\/j.jde.2013.09.014","volume":"256","author":"D Han","year":"2014","unstructured":"Han, D., Wang, X.: Initial-boundary layer associated with the nonlinear Darcy\u2013Brinkman system. J. Differ. Equ. 256(2), 609\u2013639 (2014). \n                    https:\/\/doi.org\/10.1016\/j.jde.2013.09.014","journal-title":"J. Differ. Equ."},{"issue":"1","key":"748_CR15","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00205-012-0591-7","volume":"208","author":"N Chemetov","year":"2013","unstructured":"Chemetov, N., Neves, W.: The generalized Buckley\u2013Leverett system: solvability. Arch. Ration. Mech. Anal. 208(1), 1\u201324 (2013). \n                    https:\/\/doi.org\/10.1007\/s00205-012-0591-7","journal-title":"Arch. Ration. Mech. Anal."},{"key":"748_CR16","doi-asserted-by":"crossref","unstructured":"Anderson, D.M., McFadden, G.B., Wheeler, A.A.: Diffuse-interface methods in fluid mechanics. In: Annual Review of Fluid Mechanics, Vol. 30. Annu. Rev. Fluid Mech., vol. 30, pp. 139\u2013165. Annual Reviews, Palo Alto, CA (1998)","DOI":"10.1146\/annurev.fluid.30.1.139"},{"issue":"1978","key":"748_CR17","doi-asserted-by":"publisher","first-page":"2617","DOI":"10.1098\/rspa.1998.0273","volume":"454","author":"J Lowengrub","year":"1998","unstructured":"Lowengrub, J., Truskinovsky, L.: Quasi-incompressible Cahn\u2013Hilliard fluids and topological transitions. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454(1978), 2617\u20132654 (1998). \n                    https:\/\/doi.org\/10.1098\/rspa.1998.0273","journal-title":"R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci."},{"key":"748_CR18","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1017\/jfm.2012.461","volume":"714","author":"F Magaletti","year":"2013","unstructured":"Magaletti, F., Picano, F., Chinappi, M., Marino, L., Casciola, C.M.: The sharp-interface limit of the Cahn\u2013Hilliard\u2013Navier\u2013Stokes model for binary fluids. J. Fluid Mech. 714, 95\u2013126 (2013). \n                    https:\/\/doi.org\/10.1017\/jfm.2012.461","journal-title":"J. Fluid Mech."},{"key":"748_CR19","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1016\/j.jcp.2014.01.037","volume":"264","author":"R Guo","year":"2014","unstructured":"Guo, R., Xia, Y., Xu, Y.: An efficient fully-discrete local discontinuous Galerkin method for the Cahn\u2013Hilliard\u2013Hele\u2013Shaw system. J. Comput. Phys. 264, 23\u201340 (2014). \n                    https:\/\/doi.org\/10.1016\/j.jcp.2014.01.037","journal-title":"J. Comput. Phys."},{"issue":"1","key":"748_CR20","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1137\/130950628","volume":"53","author":"AE Diegel","year":"2015","unstructured":"Diegel, A.E., Feng, X.H., Wise, S.M.: Analysis of a mixed finite element method for a Cahn\u2013Hilliard\u2013Darcy\u2013Stokes system. SIAM J. Numer. Anal. 53(1), 127\u2013152 (2015). \n                    https:\/\/doi.org\/10.1137\/130950628","journal-title":"SIAM J. Numer. Anal."},{"issue":"301","key":"748_CR21","doi-asserted-by":"publisher","first-page":"2231","DOI":"10.1090\/mcom3052","volume":"85","author":"W Chen","year":"2016","unstructured":"Chen, W., Liu, Y., Wang, C., Wise, S.M.: Convergence analysis of a fully discrete finite difference scheme for the Cahn\u2013Hilliard\u2013Hele\u2013Shaw equation. Math. Comp. 85(301), 2231\u20132257 (2016). \n                    https:\/\/doi.org\/10.1090\/mcom3052","journal-title":"Math. Comp."},{"issue":"3","key":"748_CR22","doi-asserted-by":"publisher","first-page":"679","DOI":"10.1007\/s00211-016-0813-2","volume":"135","author":"Y Liu","year":"2017","unstructured":"Liu, Y., Chen, W., Wang, C., Wise, S.M.: Error analysis of a mixed finite element method for a Cahn\u2013Hilliard\u2013Hele\u2013Shaw system. Numer. Math. 135(3), 679\u2013709 (2017). \n                    https:\/\/doi.org\/10.1007\/s00211-016-0813-2","journal-title":"Numer. Math."},{"key":"748_CR23","doi-asserted-by":"crossref","unstructured":"Eyre, D.J.: Unconditionally gradient stable time marching the Cahn-Hilliard equation. In:Computational and Mathematical Models of Microstructural Evolution (San Francisco, CA, 1998). Mater. Res. Soc. Sympos. Proc., vol. 529, pp. 39\u201346. MRS, Warrendale, PA (1998)","DOI":"10.1557\/PROC-529-39"},{"issue":"4","key":"748_CR24","doi-asserted-by":"publisher","first-page":"929","DOI":"10.4208\/cicp.171211.130412a","volume":"13","author":"C Collins","year":"2013","unstructured":"Collins, C., Shen, J., Wise, S.M.: An efficient, energy stable scheme for the Cahn\u2013Hilliard\u2013Brinkman system. Commun. Comput. Phys. 13(4), 929\u2013957 (2013)","journal-title":"Commun. Comput. Phys."},{"issue":"3","key":"748_CR25","doi-asserted-by":"publisher","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\u2013Stokes\u2013Cahn\u2013Hilliard diffuse interface model for two-phase fluid flows. SIAM J. Numer. Anal. 44(3), 1049\u20131072 (2006). \n                    https:\/\/doi.org\/10.1137\/050638333","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"748_CR26","doi-asserted-by":"publisher","first-page":"15","DOI":"10.4171\/IFB\/178","volume":"10","author":"D Kay","year":"2008","unstructured":"Kay, D., Styles, V., Welford, R.: Finite element approximation of a Cahn\u2013Hilliard\u2013Navier\u2013Stokes system. Interf. Free Bound. 10(1), 15\u201343 (2008). \n                    https:\/\/doi.org\/10.4171\/IFB\/178","journal-title":"Interf. Free Bound."},{"issue":"3","key":"748_CR27","doi-asserted-by":"publisher","first-page":"1159","DOI":"10.1137\/09075860X","volume":"32","author":"J Shen","year":"2010","unstructured":"Shen, J., Yang, X.: A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscosities. SIAM J. Sci. Comput. 32(3), 1159\u20131179 (2010). \n                    https:\/\/doi.org\/10.1137\/09075860X","journal-title":"SIAM J. Sci. Comput."},{"key":"748_CR28","unstructured":"Shen, J.: Modeling and numerical approximation of two-phase incompressible flows by a phase-field approach. In:Multiscale Modeling and Analysis for Materials Simulation. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 22, pp. 147\u2013195. World Sci. Publ., Hackensack, NJ, (2012)"},{"issue":"6","key":"748_CR29","doi-asserted-by":"publisher","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(6), 3036\u20133061 (2013). \n                    https:\/\/doi.org\/10.1137\/130908208","journal-title":"SIAM J. Numer. Anal."},{"key":"748_CR30","doi-asserted-by":"publisher","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\u2013Hilliard\u2013Navier\u2013Stokes equations for variable density flows with a discrete energy law. J. Comput. Phys. 276, 486\u2013507 (2014). \n                    https:\/\/doi.org\/10.1016\/j.jcp.2014.07.038","journal-title":"J. Comput. Phys."},{"issue":"2","key":"748_CR31","doi-asserted-by":"publisher","first-page":"584","DOI":"10.1002\/num.21721","volume":"29","author":"S Minjeaud","year":"2013","unstructured":"Minjeaud, S.: An unconditionally stable uncoupled scheme for a triphasic Cahn\u2013Hilliard\/Navier\u2013Stokes model. Numer. Methods Partial Differ. Equ. 29(2), 584\u2013618 (2013). \n                    https:\/\/doi.org\/10.1002\/num.21721","journal-title":"Numer. Methods Partial Differ. Equ."},{"issue":"1","key":"748_CR32","doi-asserted-by":"publisher","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(1), 279\u2013296 (2015). \n                    https:\/\/doi.org\/10.1137\/140971154","journal-title":"SIAM J. Numer. Anal."},{"key":"748_CR33","doi-asserted-by":"publisher","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). \n                    https:\/\/doi.org\/10.1016\/j.jcp.2014.12.046","journal-title":"J. Comput. Phys."},{"issue":"8","key":"748_CR34","doi-asserted-by":"publisher","first-page":"2834","DOI":"10.1016\/j.jcp.2008.12.036","volume":"228","author":"J-L Guermond","year":"2009","unstructured":"Guermond, J.-L., Salgado, A.: A splitting method for incompressible flows with variable density based on a pressure Poisson equation. J. Comput. Phys. 228(8), 2834\u20132846 (2009). \n                    https:\/\/doi.org\/10.1016\/j.jcp.2008.12.036","journal-title":"J. Comput. Phys."},{"issue":"15","key":"748_CR35","doi-asserted-by":"publisher","first-page":"5323","DOI":"10.1016\/j.jcp.2009.04.020","volume":"228","author":"Z Hu","year":"2009","unstructured":"Hu, Z., Wise, S.M., Wang, C., Lowengrub, J.S.: Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation. J. Comput. Phys. 228(15), 5323\u20135339 (2009). \n                    https:\/\/doi.org\/10.1016\/j.jcp.2009.04.020","journal-title":"J. Comput. Phys."},{"issue":"5","key":"748_CR36","doi-asserted-by":"publisher","first-page":"2851","DOI":"10.1137\/120880677","volume":"51","author":"A Baskaran","year":"2013","unstructured":"Baskaran, A., Lowengrub, J.S., Wang, C., Wise, S.M.: Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation. SIAM J. Numer. Anal. 51(5), 2851\u20132873 (2013). \n                    https:\/\/doi.org\/10.1137\/120880677","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"748_CR37","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1137\/110822839","volume":"50","author":"J Shen","year":"2012","unstructured":"Shen, J., Wang, C., Wang, X., Wise, S.M.: Second-order convex splitting schemes for gradient flows with Ehrlich\u2013Schwoebel type energy: application to thin film epitaxy. SIAM J. Numer. Anal. 50(1), 105\u2013125 (2012). \n                    https:\/\/doi.org\/10.1137\/110822839","journal-title":"SIAM J. Numer. Anal."},{"key":"748_CR38","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1016\/j.jcp.2015.02.046","volume":"290","author":"D Han","year":"2015","unstructured":"Han, D., Wang, X.: A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn\u2013Hilliard\u2013Navier\u2013Stokes equation. J. Comput. Phys. 290, 139\u2013156 (2015). \n                    https:\/\/doi.org\/10.1016\/j.jcp.2015.02.046","journal-title":"J. Comput. Phys."},{"issue":"3","key":"748_CR39","doi-asserted-by":"publisher","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(3), 870\u2013891 (1986). \n                    https:\/\/doi.org\/10.1137\/0907059","journal-title":"SIAM J. Sci. Statist. Comput."},{"issue":"2","key":"748_CR40","doi-asserted-by":"publisher","first-page":"511","DOI":"10.1016\/j.jcp.2003.07.035","volume":"193","author":"J Kim","year":"2004","unstructured":"Kim, J., Kang, K., Lowengrub, J.: Conservative multigrid methods for Cahn\u2013Hilliard fluids. J. Comput. Phys. 193(2), 511\u2013543 (2004). \n                    https:\/\/doi.org\/10.1016\/j.jcp.2003.07.035","journal-title":"J. Comput. Phys."},{"issue":"3","key":"748_CR41","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1007\/s00211-017-0887-5","volume":"137","author":"Amanda E. Diegel","year":"2017","unstructured":"Diegel, A.E., Wang, C., Wang, X., Wise, S.M.: Convergence analysis and error estimates for a second order accurate finite element method for the Cahn\u2013Hilliard\u2013Navier\u2013Stokes system. Numerische Mathematik, 1\u201340 (2017). \n                    https:\/\/doi.org\/10.1007\/s00211-017-0887-5","journal-title":"Numerische Mathematik"},{"issue":"17","key":"748_CR42","doi-asserted-by":"publisher","first-page":"5788","DOI":"10.1016\/j.jcp.2012.04.041","volume":"231","author":"S Dong","year":"2012","unstructured":"Dong, S., Shen, J.: A time-stepping scheme involving constant coefficient matrices for phase-field simulations of two-phase incompressible flows with large density ratios. J. Comput. Phys. 231(17), 5788\u20135804 (2012). \n                    https:\/\/doi.org\/10.1016\/j.jcp.2012.04.041","journal-title":"J. Comput. Phys."},{"key":"748_CR43","doi-asserted-by":"publisher","first-page":"58","DOI":"10.1016\/j.jcp.2013.12.055","volume":"262","author":"S Aland","year":"2014","unstructured":"Aland, S.: Time integration for diffuse interface models for two-phase flow. J. Comput. Phys. 262, 58\u201371 (2014). \n                    https:\/\/doi.org\/10.1016\/j.jcp.2013.12.055","journal-title":"J. Comput. Phys."},{"key":"748_CR44","doi-asserted-by":"publisher","first-page":"140","DOI":"10.1016\/j.jcp.2012.09.020","volume":"234","author":"F Guill\u00e9n-Gonz\u00e1lez","year":"2013","unstructured":"Guill\u00e9n-Gonz\u00e1lez, F., Tierra, G.: On linear schemes for a Cahn\u2013Hilliard diffuse interface model. J. Comput. Phys. 234, 140\u2013171 (2013). \n                    https:\/\/doi.org\/10.1016\/j.jcp.2012.09.020","journal-title":"J. Comput. Phys."},{"issue":"11","key":"748_CR45","doi-asserted-by":"publisher","first-page":"1993","DOI":"10.1142\/S0218202517500373","volume":"27","author":"Xiaofeng Yang","year":"2017","unstructured":"Yang, X., Zhao, J., Wang, Q., Shen, J.: Numerical approximations for a three-components cahnhilliard phase-field model based on the invariant energy quadratization method. Math. Models Methods Appl. Sci. 1\u201338 (2017). \n                    https:\/\/doi.org\/10.1142\/S0218202517500373\n                    \n                  . \n                    http:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218202517500373","journal-title":"Mathematical Models and Methods in Applied Sciences"},{"key":"748_CR46","doi-asserted-by":"publisher","first-page":"44","DOI":"10.1016\/j.jcp.2017.04.010","volume":"341","author":"Q Cheng","year":"2017","unstructured":"Cheng, Q., Yang, X., Shen, J.: Efficient and accurate numerical schemes for a hydro-dynamically coupled phase field diblock copolymer model. J. Comput. Phys. 341, 44\u201360 (2017). \n                    https:\/\/doi.org\/10.1016\/j.jcp.2017.04.010","journal-title":"J. Comput. Phys."},{"key":"748_CR47","doi-asserted-by":"publisher","first-page":"1005","DOI":"10.1016\/j.cma.2017.02.011","volume":"318","author":"X Yang","year":"2017","unstructured":"Yang, X., Ju, L.: Linear and unconditionally energy stable schemes for the binary fluid-surfactant phase field model. Comput. Methods Appl. Mech. Eng. 318, 1005\u20131029 (2017). \n                    https:\/\/doi.org\/10.1016\/j.cma.2017.02.011","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"748_CR48","doi-asserted-by":"publisher","first-page":"803","DOI":"10.1016\/j.cma.2017.01.031","volume":"318","author":"J Zhao","year":"2017","unstructured":"Zhao, J., Yang, X., Gong, Y., Wang, Q.: A novel linear second order unconditionally energy stable scheme for a hydrodynamic Q-tensor model of liquid crystals. Comput. Methods Appl. Mech. Eng. 318, 803\u2013825 (2017). \n                    https:\/\/doi.org\/10.1016\/j.cma.2017.01.031","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"748_CR49","doi-asserted-by":"publisher","first-page":"104","DOI":"10.1016\/j.jcp.2016.12.025","volume":"333","author":"X Yang","year":"2017","unstructured":"Yang, X., Zhao, J., Wang, Q.: Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method. J. Comput. Phys. 333, 104\u2013127 (2017). \n                    https:\/\/doi.org\/10.1016\/j.jcp.2016.12.025","journal-title":"J. Comput. Phys."},{"key":"748_CR50","doi-asserted-by":"publisher","first-page":"1116","DOI":"10.1016\/j.jcp.2016.10.020","volume":"330","author":"X Yang","year":"2017","unstructured":"Yang, X., Han, D.: Linearly first- and second-order, unconditionally energy stable schemes for the phase field crystal equation. J. Comput. Phys. 330, 1116\u20131134 (2017)","journal-title":"J. Comput. Phys."},{"issue":"44\u201347","key":"748_CR51","doi-asserted-by":"publisher","first-page":"6011","DOI":"10.1016\/j.cma.2005.10.010","volume":"195","author":"JL Guermond","year":"2006","unstructured":"Guermond, J.L., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Engrg. 195(44\u201347), 6011\u20136045 (2006). \n                    https:\/\/doi.org\/10.1016\/j.cma.2005.10.010","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"issue":"2","key":"748_CR52","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1007\/s002110050366","volume":"80","author":"J-L Guermond","year":"1998","unstructured":"Guermond, J.-L., Quartapelle, L.: On the approximation of the unsteady Navier-Stokes equations by finite element projection methods. Numer. Math. 80(2), 207\u2013238 (1998). \n                    https:\/\/doi.org\/10.1007\/s002110050366","journal-title":"Numer. Math."},{"issue":"10","key":"748_CR53","doi-asserted-by":"publisher","first-page":"3887","DOI":"10.1016\/j.jde.2014.07.013","volume":"257","author":"D Han","year":"2014","unstructured":"Han, D., Wang, X., Wu, H.: Existence and uniqueness of global weak solutions to a Cahn\u2013Hilliard\u2013Stokes\u2013Darcy system for two phase incompressible flows in karstic geometry. J. Differ. Equ. 257(10), 3887\u20133933 (2014). \n                    https:\/\/doi.org\/10.1016\/j.jde.2014.07.013","journal-title":"J. Differ. Equ."},{"issue":"4","key":"748_CR54","doi-asserted-by":"publisher","first-page":"1867","DOI":"10.1093\/imanum\/drv065","volume":"36","author":"AE Diegel","year":"2016","unstructured":"Diegel, A.E., Wang, C., Wise, S.M.: Stability and convergence of a second-order mixed finite element method for the Cahn\u2013Hilliard equation. IMA J. Numer. Anal. 36(4), 1867\u20131897 (2016). \n                    https:\/\/doi.org\/10.1093\/imanum\/drv065","journal-title":"IMA J. Numer. Anal."},{"issue":"3\u20134","key":"748_CR55","first-page":"251","volume":"20","author":"F Hecht","year":"2012","unstructured":"Hecht, F.: New development in freefem++. J. Numer. Math. 20(3\u20134), 251\u2013265 (2012)","journal-title":"J. Numer. Math."},{"key":"748_CR56","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1016\/j.jcp.2013.09.049","volume":"257","author":"Andrew Christlieb","year":"2014","unstructured":"Christlieb, A., Jones, J., Promislow, K., Wetton, B., Willoughby, M.: High accuracy solutions to energy gradient flows from material science models. J. Comput. Phys. 257(part A), 193\u2013215 (2014). \n                    https:\/\/doi.org\/10.1016\/j.jcp.2013.09.049","journal-title":"Journal of Computational Physics"},{"key":"748_CR57","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1016\/j.jcp.2016.03.042","volume":"315","author":"K Glasner","year":"2016","unstructured":"Glasner, K., Orizaga, S.: Improving the accuracy of convexity splitting methods for gradient flow equations. J. Comput. Phys. 315, 52\u201364 (2016). \n                    https:\/\/doi.org\/10.1016\/j.jcp.2016.03.042","journal-title":"J. Comput. Phys."},{"key":"748_CR58","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1017\/S0022112005007998","volume":"550","author":"M Le Bars","year":"2006","unstructured":"Le Bars, M., Worster, M.G.: Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification. J. Fluid Mech. 550, 149\u2013173 (2006). \n                    https:\/\/doi.org\/10.1017\/S0022112005007998","journal-title":"J. Fluid Mech."},{"key":"748_CR59","doi-asserted-by":"publisher","first-page":"312","DOI":"10.1098\/rspa.1958.0085","volume":"245","author":"PG Saffman","year":"1958","unstructured":"Saffman, P.G., Taylor, G.: The penetration of a fluid into a porous medium or Hele\u2013Shaw cell containing a more viscous liquid. Proc. Roy. Soc. Lond. Ser. A 245, 312\u20133292 (1958)","journal-title":"Proc. Roy. Soc. Lond. Ser. A"}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-018-0748-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-018-0748-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-018-0748-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,17]],"date-time":"2020-05-17T09:31:41Z","timestamp":1589707901000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-018-0748-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,6,5]]},"references-count":59,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2018,11]]}},"alternative-id":["748"],"URL":"https:\/\/doi.org\/10.1007\/s10915-018-0748-0","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,6,5]]},"assertion":[{"value":"16 September 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 May 2018","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 May 2018","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 June 2018","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}