{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T03:05:57Z","timestamp":1772852757945,"version":"3.50.1"},"reference-count":34,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,2,1]],"date-time":"2021-02-01T00:00:00Z","timestamp":1612137600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper introduces the Fourier spectral method combined with the strongly stable exponential time difference method as an attractive and easy-to-implement alternative for the integration of the multi-dimensional Allen\u2013Cahn equation with no-flux boundary conditions. The main advantages of the proposed method are that it utilizes the discrete fast Fourier transform, which ensures efficiency, allows an extension to two and three spatial dimensions in a similar fashion as one-dimensional problems, and deals with various boundary conditions. Several numerical experiments are carried out on multi-dimensional Allen\u2013Cahn equations including a two-dimensional Allen\u2013Cahn equation with a radially symmetric circular interface initial condition to demonstrate the fourth-order temporal accuracy and stability of the method. The numerical results show that the proposed method is fourth-order accurate in the time direction and is able to satisfy the discrete energy law.<\/jats:p>","DOI":"10.3390\/sym13020245","type":"journal-article","created":{"date-parts":[[2021,2,1]],"date-time":"2021-02-01T11:40:48Z","timestamp":1612179648000},"page":"245","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Fourier Spectral High-Order Time-Stepping Method for Numerical Simulation of the Multi-Dimensional Allen\u2013Cahn Equations"],"prefix":"10.3390","volume":"13","author":[{"given":"Harish","family":"Bhatt","sequence":"first","affiliation":[{"name":"Department of Mathematics, Utah Valley University, Orem, UT 84058, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Janak","family":"Joshi","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Cameron University, Lawton, OK 73503, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ioannis","family":"Argyros","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Cameron University, Lawton, OK 73503, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,2,1]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"Existence and nonexistence of solutions for sublinear problems with prescribed number of zeros on exterior domains","volume":"133","author":"Joshi","year":"2017","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_2","first-page":"1","article-title":"Existence of solutions for semilinear problems with prescribed number of zeros on exterior domains","volume":"112","author":"Joshi","year":"2016","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_3","first-page":"1","article-title":"Infinitely many solutions for a semilinear problem on exterior domains with nonlinear boundary condition","volume":"108","author":"Joshi","year":"2018","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1085","DOI":"10.1016\/0001-6160(79)90196-2","article-title":"A microscopic theory for antiphase boundary motion and its application to antipahse domain coarsening","volume":"27","author":"Allen","year":"1979","journal-title":"Acta Metall."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"450","DOI":"10.1016\/j.jcp.2004.01.029","article-title":"A phase field approach in the numerical study of the elastic bending energy for vesicle membranes","volume":"198","author":"Du","year":"2004","journal-title":"J. Comput. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1097","DOI":"10.1002\/cpa.3160450903","article-title":"Phase transitions and generalized motion by mean curvature","volume":"45","author":"Evans","year":"1992","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_7","first-page":"635","article-title":"Motion of level sets by mean curvature","volume":"33","author":"Evans","year":"1991","journal-title":"I. J. Differ. Geom."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/s00211-002-0413-1","article-title":"Numerical analysis of the Allen\u2013Cahn equation and approximation for mean curvature flows","volume":"94","author":"Feng","year":"2003","journal-title":"Numer. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/S0167-2789(03)00030-7","article-title":"A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method","volume":"179","author":"Liu","year":"2003","journal-title":"Physica D"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1016\/j.jcp.2006.02.021","article-title":"Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method","volume":"218","author":"Yang","year":"2006","journal-title":"J. Comput. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/j.jcp.2006.03.016","article-title":"Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing","volume":"219","author":"Yue","year":"2006","journal-title":"J. Comput. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1016\/j.jnnfm.2005.07.002","article-title":"Diffuse-interface simulations of drop coalescence and retraction in viscoelastic fluids","volume":"129","author":"Yue","year":"2005","journal-title":"J. Non-Newtonian Fluid Mech."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1791","DOI":"10.1016\/j.physa.2009.01.026","article-title":"An unconditionally gradient stable numerical method for solving the Allen\u2013Cahn equation","volume":"388","author":"Choi","year":"2009","journal-title":"Physica A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1016\/j.camwa.2014.05.015","article-title":"A semi-analytical Fourier spectral method for the Allen\u2013Cahn equation","volume":"68","author":"Lee","year":"2014","journal-title":"Comput. Math. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1016\/j.physa.2015.03.012","article-title":"A second order operator splitting method for Allen\u2013Cahn type equations with nonlinear source terms","volume":"432","author":"Lee","year":"2015","journal-title":"Physica A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1591","DOI":"10.1016\/j.camwa.2010.06.041","article-title":"An unconditionally stable hybrid numerical method for solving the Allen\u2013Cahn equation","volume":"60","author":"Li","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1137\/0103003","article-title":"The numerical solution of parabolic and elliptic differential equations","volume":"3","author":"Peaceman","year":"1955","journal-title":"J. Soc. Ind. Appl. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1517","DOI":"10.4310\/CMS.2016.v14.n6.a3","article-title":"On the maximum principle preserving schemes for the generalized Allen\u2013Cahn equation","volume":"14","author":"Shen","year":"2016","journal-title":"Commun. Math. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"451","DOI":"10.4208\/jcm.1603-m2014-0017","article-title":"Implicit-explicit scheme for the Allen\u2013Cahn equation preserve the maximum principle","volume":"34","author":"Tang","year":"2016","journal-title":"J. Comput. Math."},{"key":"ref_20","first-page":"1057","article-title":"Error analysis of stabilized semi-implicit method of Allen\u2013Cahn equation","volume":"11","author":"Yang","year":"2009","journal-title":"Discret. Contin. Dyn. B"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3042","DOI":"10.1137\/080738398","article-title":"Numerical studies of discrete approximations to the Allen\u2013Cahn equation in the sharp interface limit","volume":"31","author":"Zhang","year":"2009","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_22","unstructured":"Argyros, I.K. (2007). Computational Theory of Iterative Methods, Elsevier Publication Company."},{"key":"ref_23","unstructured":"Argyros, I.K., and Magrenan, A.A. (2017). Iterative Methods and Their Dynamics with Applications, CRC Press, Taylor and Francis."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"262","DOI":"10.30538\/oms2019.0069","article-title":"Comparative analysis of numerical methods for the multidimensional Brusselator system","volume":"3","author":"Bhatt","year":"2019","journal-title":"Open J. Math. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1007\/s10915-012-9621-8","article-title":"Stability and convergence analysis of fully discrete Fourier collocation spectral method for 3-D viscous Burgers\u2019 equation","volume":"53","author":"Gottlieb","year":"2012","journal-title":"J. Sci. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3124","DOI":"10.1137\/16M1061588","article-title":"Long time stability of high order multi-step numerical schemes for two-dimensional incompressible Navier-Stokes equations","volume":"54","author":"Cheng","year":"2016","journal-title":"Siam J. Numer. Anal."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1007\/s40687-020-00212-9","article-title":"Energy stable higher-order linear ETD multi-step methods for gradient flows: Application to thin film epitaxy","volume":"7","author":"Chen","year":"2020","journal-title":"Res. Math Sci."},{"key":"ref_28","first-page":"572","article-title":"A second-order energy stable BDF numerical scheme for the Cahn-Hilliard Equation","volume":"23","author":"Yen","year":"2018","journal-title":"Commun. Comput. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"937","DOI":"10.1007\/s10543-014-0484-2","article-title":"Fourier spectral methods for fractional-in-space reaction-diffusion equations","volume":"54","author":"Kay","year":"2014","journal-title":"BIT"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1016\/j.cnsns.2016.04.020","article-title":"Fourier spectral method for higher order space fractional reaction\u2013diffusion equations","volume":"40","author":"Pindza","year":"2016","journal-title":"Comm. Nonlinear Sci. Numer. Simulat."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"430","DOI":"10.1006\/jcph.2002.6995","article-title":"Exponential time differencing for stiff systems","volume":"176","author":"Cox","year":"2002","journal-title":"J. Comput. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"176","DOI":"10.1016\/j.cam.2015.11.046","article-title":"A compact fourth-order L-stable scheme for reaction-diffusion systems with nonsmooth data","volume":"299","author":"Bhatt","year":"2016","journal-title":"J. Comput. Appl. Math."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"125202-1","DOI":"10.1063\/1.5126651","article-title":"Comparison of operator splitting schemes for numerical solutions of the Allen\u2013Cahn equation","volume":"9","author":"Ayub","year":"2019","journal-title":"AIP Adv."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2449","DOI":"10.1016\/j.cpc.2014.05.017","article-title":"Numerical simulation of the three dimensional Allen\u2013Cahn equation by the high-order compact ADI method","volume":"185","author":"Zhai","year":"2014","journal-title":"Comput. Phys. Commun."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/2\/245\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:18:21Z","timestamp":1760159901000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/2\/245"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,2,1]]},"references-count":34,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2021,2]]}},"alternative-id":["sym13020245"],"URL":"https:\/\/doi.org\/10.3390\/sym13020245","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,2,1]]}}}