{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T05:11:09Z","timestamp":1771477869639,"version":"3.50.1"},"reference-count":36,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,12,26]],"date-time":"2024-12-26T00:00:00Z","timestamp":1735171200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Education","award":["2021R1A6A1A03044326"],"award-info":[{"award-number":["2021R1A6A1A03044326"]}]},{"name":"Ministry of Education","award":["BK 21"],"award-info":[{"award-number":["BK 21"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>This article presents an efficient and accurate adaptive time-stepping finite difference method (FDM) for solving the Landau\u2013Lifshitz (LL) equation, which is an important mathematical model in understanding magnetic materials and processes. Our proposed algorithm strategically selects an adaptive time step, ensuring that the maximum displacement falls within a predefined tolerance threshold. Furthermore, this adaptive approach allows the utilization of larger time steps near equilibrium states and results in faster computations. For example, we introduce a numerical test where the adaptive time step decreases from 3.05\u00d710\u22127 to 3.52\u00d710\u22129. If a uniform time step is applied, around a 100 times smaller time step must be applied at unnecessary cases. To demonstrate the high performance of our proposed algorithm, we conduct several characteristic benchmark tests. The computational results confirm that the proposed algorithm is efficient and accurate. Overall, our adaptive time-stepping FDM offers a promising solution for accurately and efficiently solving the LL equation and contributes to advancements in the understanding and analysis of magnetic phenomena.<\/jats:p>","DOI":"10.3390\/a18010001","type":"journal-article","created":{"date-parts":[[2024,12,26]],"date-time":"2024-12-26T04:31:16Z","timestamp":1735187476000},"page":"1","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["An Efficient and Accurate Adaptive Time-Stepping Method for the Landau\u2013Lifshitz Equation"],"prefix":"10.3390","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-3986-3777","authenticated-orcid":false,"given":"Hyundong","family":"Kim","sequence":"first","affiliation":[{"name":"Department of Mathematics and Physics, Gangneung-Wonju National University, Gangneung 25457, Republic of Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Soobin","family":"Kwak","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Korea University, Seoul 02841, Republic of Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Moumni","family":"Mohammed","sequence":"additional","affiliation":[{"name":"MAMCS Group, FST Errachidia, Moulay Ismail University of Meknes, Boutalamine, P.O. Box 509, 52000 Errachidia, Morocco"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Seungyoon","family":"Kang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Korea University, Seoul 02841, Republic of Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Seokjun","family":"Ham","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Korea University, Seoul 02841, Republic of Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0484-9189","authenticated-orcid":false,"given":"Junseok","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Korea University, Seoul 02841, Republic of Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,26]]},"reference":[{"key":"ref_1","first-page":"101","article-title":"On the theory of the dispersion of magnetic permeability in ferromagnetic bodies","volume":"8","author":"Landau","year":"1935","journal-title":"Phys. Z. Sowjetunion"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1016\/j.physb.2003.08.064","article-title":"Analytical solutions of Landau\u2013Lifshitz equation for precessional dynamics","volume":"343","author":"Bertotti","year":"2004","journal-title":"Physica B"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"105674","DOI":"10.1016\/j.cnsns.2020.105674","article-title":"Self-organization in the one-dimensional Landau\u2013Lifshitz\u2013Gilbert\u2013Slonczewski equation with non-uniform anisotropy fields","volume":"96","author":"Humire","year":"2021","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1016\/j.cam.2010.01.002","article-title":"A Crank\u2013Nicolson scheme for the Landau\u2013Lifshitz equation without damping","volume":"234","author":"Jeong","year":"2010","journal-title":"J. Comput. Appl. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"476","DOI":"10.1016\/j.cap.2013.12.028","article-title":"An accurate and robust numerical method for micromagnetics simulations","volume":"14","author":"Jeong","year":"2014","journal-title":"Curr. Appl. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"106646","DOI":"10.1016\/j.cnsns.2022.106646","article-title":"Performance assessment of energy-preserving, adaptive time-step variational integrators","volume":"114","author":"Sharma","year":"2022","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/s10665-015-9836-4","article-title":"A finite-difference scheme for a model of magnetization dynamics with inertial effects","volume":"100","author":"Moumni","year":"2016","journal-title":"J. Eng. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"109046","DOI":"10.1016\/j.jcp.2019.109046","article-title":"Two improved Gauss-Seidel projection methods for Landau\u2013Lifshitz\u2013Gilbert equation","volume":"401","author":"Li","year":"2020","journal-title":"J. Comput. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"106073","DOI":"10.1016\/j.cnsns.2021.106073","article-title":"Adaptive numerical solutions of time-fractional advection\u2013diffusion\u2013reaction equations","volume":"105","author":"Jannelli","year":"2022","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1006\/jcph.2001.6793","article-title":"A Gauss\u2013Seidel projection method for micromagnetics simulations","volume":"171","author":"Wang","year":"2001","journal-title":"J. Comput. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"668","DOI":"10.1016\/j.aam.2011.02.007","article-title":"Quilted Gabor frames\u2014A new concept for adaptive time-frequency representation","volume":"47","year":"2011","journal-title":"Adv. Appl. Math."},{"key":"ref_12","first-page":"187","article-title":"A new finite element scheme for Landau\u2013Lifshitz equations","volume":"1","author":"Alouges","year":"2008","journal-title":"Discrete Contin. Dyn. Syst. Ser. S"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1142\/S0218202506001169","article-title":"Convergence of a finite element discretization for the Landau\u2013Lifshitz equations in micromagnetism","volume":"16","author":"Alouges","year":"2006","journal-title":"Math. Models Methods Appl. Sci."},{"key":"ref_14","first-page":"53","article-title":"A finite element approximation of a current-induced magnetization dynamics model","volume":"10","author":"Mohammed","year":"2022","journal-title":"J. Math. Model."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"100380","DOI":"10.1016\/j.rinam.2023.100380","article-title":"Fourier-spectral method for the Landau\u2013Lifshitz\u2013Gilbert equation in micromagnetism","volume":"19","author":"Moumni","year":"2023","journal-title":"Results Appl. Math."},{"key":"ref_16","first-page":"1647","article-title":"Numerical methods for the Landau\u2013Lifshitz equation","volume":"39","author":"Weinan","year":"2001","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/j.aml.2016.05.014","article-title":"A stable numerical method for space fractional Landau\u2013Lifshitz equations","volume":"61","author":"Yang","year":"2016","journal-title":"Appl. Math. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF03024947","article-title":"A survey on the numerics and computations for the Landau\u2013Lifshitz equation of micromagnetism","volume":"15","year":"2007","journal-title":"Arch. Comput. Methods Eng."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Bastos, J.P.A., and Sadowski, N. (2017). Magnetic Materials and 3D Finite Element Modeling, CRC Press.","DOI":"10.1201\/b15558"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"18952","DOI":"10.1002\/mma.9601","article-title":"Error analysis of a linear numerical scheme for the Landau\u2013Lifshitz equation with large damping parameters","volume":"46","author":"Cai","year":"2023","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.apnum.2021.05.027","article-title":"Convergence analysis of a second-order semi-implicit projection method for Landa\u2013Lifshitz equation","volume":"168","author":"Chen","year":"2021","journal-title":"Appl. Numer. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.apnum.2020.08.014","article-title":"Unconditional optimal error estimates of linearized second-order BDF Galerkin FEMs for the Landau\u2013Lifshitz equation","volume":"159","author":"Yang","year":"2021","journal-title":"Appl. Numer. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1007\/s13160-011-0054-9","article-title":"Finite difference scheme for the Landau\u2013Lifshitz equation","volume":"29","author":"Fuwa","year":"2012","journal-title":"Jpn. J. Ind. Appl. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"111248","DOI":"10.1016\/j.jcp.2022.111248","article-title":"A positivity-preserving scheme for fluctuating hydrodynamics","volume":"463","author":"Magaletti","year":"2022","journal-title":"J. Comput. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Daribayev, B., Mukhanbet, A., Azatbekuly, N., and Imankulov, T. (2024). A quantum approach for exploring the numerical results of the heat equation. Algorithms, 17.","DOI":"10.3390\/a17080327"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Krivovichev, G.V. (2024). Stability optimization of explicit Runge\u2013Kutta methods with higher-order derivatives. Algorithms, 17.","DOI":"10.3390\/a17120535"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1007\/s40314-020-01230-7","article-title":"An efficient and highly accurate spectral method for modeling the propagation of solitary magnetic spin waves in thin films","volume":"39","author":"Christou","year":"2020","journal-title":"Comput. Appl. Math."},{"key":"ref_28","first-page":"2731593","article-title":"An adaptive time-stepping algorithm for the Allen\u2013Cahn equation","volume":"2022","author":"Lee","year":"2022","journal-title":"J. Funct. Spaces"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"A530","DOI":"10.1137\/22M1501143","article-title":"Length preserving numerical schemes for Landau\u2013Lifshitz equation based on Lagrange multiplier approaches","volume":"45","author":"Cheng","year":"2023","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"He, J., Yang, L., and Zhan, J. (2024). Temporal High-Order Accurate Numerical Scheme for the Landau\u2013Lifshitz\u2013Gilbert Equation. Mathematics, 12.","DOI":"10.3390\/math12081179"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1007\/s40324-021-00254-1","article-title":"Recent results for the Landau\u2013Lifshitz equation","volume":"79","year":"2022","journal-title":"SeMA J."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Atkinson, K., Han, W., and Stewart, D.E. (2009). Numerical Solution of Ordinary Differential Equations, John Wiley & Sons. [2nd ed.].","DOI":"10.1002\/9781118164495"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1134\/S0965542508030068","article-title":"Optimal first-to sixth-order accurate Runge-Kutta schemes","volume":"48","author":"Alshina","year":"2008","journal-title":"Comput. Math. Math. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1016\/j.matcom.2023.01.016","article-title":"Stability analysis for a maximum principle preserving explicit scheme of the Allen\u2013Cahn equation","volume":"207","author":"Ham","year":"2023","journal-title":"Math. Comput. Simul."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1207","DOI":"10.1007\/s10614-022-10242-w","article-title":"Accurate and efficient finite difference method for the Black\u2013Scholes model with no far-field boundary conditions","volume":"61","author":"Lee","year":"2023","journal-title":"Comput. Econ."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Dieguez, G., Batistela, C., and Piqueira, J.R.C. (2023). Controlling COVID-19 spreading: A three-level algorithm. Mathematics, 11.","DOI":"10.3390\/math11173766"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/1\/1\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T17:00:27Z","timestamp":1760115627000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/1\/1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,26]]},"references-count":36,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["a18010001"],"URL":"https:\/\/doi.org\/10.3390\/a18010001","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,12,26]]}}}