{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,25]],"date-time":"2026-04-25T04:44:34Z","timestamp":1777092274338,"version":"3.51.4"},"reference-count":34,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,10,23]],"date-time":"2022-10-23T00:00:00Z","timestamp":1666483200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NSF of China","award":["11861054"],"award-info":[{"award-number":["11861054"]}]},{"name":"NSF of China","award":["U19A2079"],"award-info":[{"award-number":["U19A2079"]}]},{"name":"NSF of China","award":["11671345"],"award-info":[{"award-number":["11671345"]}]},{"name":"NSF of China","award":["11771348"],"award-info":[{"award-number":["11771348"]}]},{"name":"NSF of China","award":["2020GXNSFAA297223"],"award-info":[{"award-number":["2020GXNSFAA297223"]}]},{"name":"NSF of China","award":["YCSW2022185"],"award-info":[{"award-number":["YCSW2022185"]}]},{"name":"Natural Science Foundation of Guangxi","award":["11861054"],"award-info":[{"award-number":["11861054"]}]},{"name":"Natural Science Foundation of Guangxi","award":["U19A2079"],"award-info":[{"award-number":["U19A2079"]}]},{"name":"Natural Science Foundation of Guangxi","award":["11671345"],"award-info":[{"award-number":["11671345"]}]},{"name":"Natural Science Foundation of Guangxi","award":["11771348"],"award-info":[{"award-number":["11771348"]}]},{"name":"Natural Science Foundation of Guangxi","award":["2020GXNSFAA297223"],"award-info":[{"award-number":["2020GXNSFAA297223"]}]},{"name":"Natural Science Foundation of Guangxi","award":["YCSW2022185"],"award-info":[{"award-number":["YCSW2022185"]}]},{"name":"Innovation Project of Guangxi Graduate Education","award":["11861054"],"award-info":[{"award-number":["11861054"]}]},{"name":"Innovation Project of Guangxi Graduate Education","award":["U19A2079"],"award-info":[{"award-number":["U19A2079"]}]},{"name":"Innovation Project of Guangxi Graduate Education","award":["11671345"],"award-info":[{"award-number":["11671345"]}]},{"name":"Innovation Project of Guangxi Graduate Education","award":["11771348"],"award-info":[{"award-number":["11771348"]}]},{"name":"Innovation Project of Guangxi Graduate Education","award":["2020GXNSFAA297223"],"award-info":[{"award-number":["2020GXNSFAA297223"]}]},{"name":"Innovation Project of Guangxi Graduate Education","award":["YCSW2022185"],"award-info":[{"award-number":["YCSW2022185"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we construct a fully discrete and decoupled Crank\u2013Nicolson Leap-Frog (CNLF) scheme for solving the modified phase field crystal model (MPFC) with long-range interaction. The idea of CNLF is to treat stiff terms implicity with Crank\u2013Nicolson and to treat non-stiff terms explicitly with Leap-Frog. In addition, the scalar auxiliary variable (SAV) method is used to allow explicit treatment of the nonlinear potential, then, these technique combines with CNLF can lead to the highly efficient, fully decoupled and linear numerical scheme with constant coefficients at each time step. Furthermore, the Fourier spectral method is used for the spatial discretization. Finally, we show that the CNLF scheme is fully discrete, second-order decoupled and unconditionally stable. Ample numerical experiments in 2D and 3D are provided to demonstrate the accuracy, efficiency, and stability of the proposed method.<\/jats:p>","DOI":"10.3390\/e24111512","type":"journal-article","created":{"date-parts":[[2022,10,24]],"date-time":"2022-10-24T02:31:03Z","timestamp":1666578663000},"page":"1512","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Second-Order Crank\u2013Nicolson Leap-Frog Scheme for the Modified Phase Field Crystal Model with Long-Range Interaction"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5408-6431","authenticated-orcid":false,"given":"Chunya","family":"Wu","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Guangxi Normal University, Guilin 541006, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xinlong","family":"Feng","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lingzhi","family":"Qian","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Guangxi Normal University, Guilin 541006, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"031609","DOI":"10.1103\/PhysRevE.73.031609","article-title":"Diffusive atomistic dynamics of edge dislocations in two dimensions","volume":"73","author":"Berry","year":"2006","journal-title":"Phys. Rev. E Stat. Nonlinear Soft Matter Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"046107","DOI":"10.1103\/PhysRevE.80.046107","article-title":"Phase field crystal study of deformation and plasticity in nanocrystalline materials","volume":"80","author":"Stefanovic","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"245701","DOI":"10.1103\/PhysRevLett.88.245701","article-title":"Modeling elasticity in crystal growth","volume":"88","author":"Elder","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"051605","DOI":"10.1103\/PhysRevE.70.051605","article-title":"Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals","volume":"70","author":"Elder","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"225504","DOI":"10.1103\/PhysRevLett.96.225504","article-title":"Phase-field crystals with elastic interactions","volume":"96","author":"Stefanovic","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2621","DOI":"10.1021\/ma00164a028","article-title":"Equilibrium morphology of block copolymer melts","volume":"19","author":"Ohta","year":"1986","journal-title":"Macromolecules"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1007\/s10444-020-09789-9","article-title":"Stability and error estimates of the SAV Fourier-spectral method for the phase field crystal equation","volume":"46","author":"Li","year":"2020","journal-title":"Adv. Comput. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/s10444-021-09897-0","article-title":"Error analysis of the SAV Fourier-spectral method for the Cahn-Hilliard-Hele-Shaw system","volume":"47","author":"Zheng","year":"2021","journal-title":"Adv. Comput. Math."},{"key":"ref_9","first-page":"33","article-title":"Numerical analysis for a nonlocal Allen-Cahn equation","volume":"6","author":"Bates","year":"2009","journal-title":"Int. J. Numer. Anal. Mod."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"4564","DOI":"10.1002\/mma.2869","article-title":"Stabilized semi-implicit spectral deferred correction methods for Allen-Cahn and Cahn-Hilliard equations","volume":"38","author":"Liu","year":"2015","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"616","DOI":"10.1016\/j.apnum.2006.07.026","article-title":"On large time-stepping methods for the Cahn-Hilliard equation","volume":"57","author":"He","year":"2007","journal-title":"Appl. Numer. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.jcp.2018.02.023","article-title":"Stabilized linear semi-implicit schemes for the nonlocal Cahn-Hilliard equation","volume":"363","author":"Du","year":"2018","journal-title":"J. Comput. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1016\/j.cma.2015.09.018","article-title":"The numerical simulation of the phase field crystal (PFC) and modified phase field crystal (MPFC) models via global and local mesh less methods","volume":"298","author":"Dehghan","year":"2016","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1016\/j.apnum.2019.10.019","article-title":"Two fast and efficient linear semi-implicit approaches with unconditional energy stability for nonlocal phase field crystal equation","volume":"150","author":"Liu","year":"2020","journal-title":"Appl. Numer. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"572","DOI":"10.4208\/cicp.OA-2016-0197","article-title":"A second-order energy stable BDF numerical scheme for the Cahn-Hilliard equation","volume":"23","author":"Yan","year":"2018","journal-title":"Commun. Comput. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"294","DOI":"10.1016\/j.jcp.2016.09.029","article-title":"Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends","volume":"327","author":"Yang","year":"2016","journal-title":"J. Comput. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"113335","DOI":"10.1016\/j.cam.2020.113335","article-title":"Efficient second-order unconditionally stable numerical schemes for the modified phase field crystal model with long-range interaction","volume":"389","author":"Li","year":"2021","journal-title":"J. Comput. Appl. Math."},{"key":"ref_18","first-page":"753","article-title":"The Unstable Mode in the Crank-Nicolson Leap-Frog Method is Stable","volume":"13","author":"Hurl","year":"2016","journal-title":"Int. J. Numer. Anal. Mod."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1515\/cmam-2015-0010","article-title":"Analysis of a stabilized CNLF method with fast slow wave splittings for flow problems","volume":"15","author":"Jiang","year":"2015","journal-title":"Comput. Methods Appl. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/j.cam.2014.09.026","article-title":"A Crank-Nicolson Leapfrog stabilization: Unconditional stability and two applications","volume":"281","author":"Jiang","year":"2015","journal-title":"J. Comput. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1016\/j.apnum.2011.10.006","article-title":"Stability of two IMEX methods, CNLF and BDF2-AB2, for uncoupling systems of evolution equations","volume":"62","author":"Layton","year":"2012","journal-title":"Appl. Numer. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1192","DOI":"10.1002\/num.21751","article-title":"Uncoupling evolutionary groundwater-surface water flows using the Crank-Nicolson Leapfrog method","volume":"29","author":"Kubacki","year":"2013","journal-title":"Numer. Methods Partial. Differ. Equ."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1009","DOI":"10.1007\/s10543-014-0493-1","article-title":"Stability analysis of the Crank-Nicolson-Leapfrog method with the Robert-Asselin-Williams time filter","volume":"54","author":"Hurl","year":"2014","journal-title":"BIT Numer. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2851","DOI":"10.1137\/120880677","article-title":"Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation","volume":"51","author":"Baskaran","year":"2013","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1016\/j.jcp.2016.09.053","article-title":"First and second order numerical methods based on a new convex splitting for phase-field crystal equation","volume":"327","author":"Shin","year":"2016","journal-title":"J. Comput. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1116","DOI":"10.1016\/j.jcp.2016.10.020","article-title":"Linearly first- and second-order, unconditionally energy stable schemes for the phase field crystal model","volume":"330","author":"Yang","year":"2017","journal-title":"J. Comput. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/j.cma.2019.03.030","article-title":"Fast, provably unconditionally energy stable, and second-order accurate algorithms for the anisotropic Cahn-Hilliard model","volume":"351","author":"Chen","year":"2019","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1016\/j.jcp.2017.10.021","article-title":"The scalar auxiliary variable (SAV) approach for gradient flows","volume":"353","author":"Shen","year":"2018","journal-title":"J. Comput. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2047","DOI":"10.1090\/mcom\/3428","article-title":"Energy stability and convergence of sav block-centered finite difference method for gradient flows","volume":"88","author":"Li","year":"2019","journal-title":"Math. Comput."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"474","DOI":"10.1137\/17M1150153","article-title":"A new class of efficient and robust energy stable schemes for gradient flows","volume":"61","author":"Shen","year":"2019","journal-title":"SIAM. Rev."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1103\/PhysRevA.15.319","article-title":"Hydrodynamics fluctuations at the convective instability","volume":"15","author":"Swift","year":"1977","journal-title":"Phys. Rev. A"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1712","DOI":"10.1137\/080728809","article-title":"On the phase diagram for microphase separation of diblock copolymers: An approach via a nonlocal Cahn-Hilliard functional","volume":"69","author":"Choksi","year":"2009","journal-title":"SIAM J. Appl. Math."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"C1551","DOI":"10.1007\/s10444-019-09678-w","article-title":"Efficient numerical schemes with unconditional energy stabilities for the modified phase field crystal equation","volume":"45","author":"Li","year":"2019","journal-title":"Adv. Comput. Math."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"C310","DOI":"10.1016\/j.commatsci.2015.09.038","article-title":"Three dimensional structures predicted by the modified phase field crystal equation","volume":"111","author":"Bueno","year":"2016","journal-title":"Comput. Mater. Sci."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/11\/1512\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:01:10Z","timestamp":1760144470000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/11\/1512"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,23]]},"references-count":34,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["e24111512"],"URL":"https:\/\/doi.org\/10.3390\/e24111512","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,23]]}}}