{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,15]],"date-time":"2026-05-15T17:06:20Z","timestamp":1778864780123,"version":"3.51.4"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2019,5,21]],"date-time":"2019-05-21T00:00:00Z","timestamp":1558396800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2019,5,21]],"date-time":"2019-05-21T00:00:00Z","timestamp":1558396800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000770","name":"University of Manchester","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100000770","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2019,8]]},"DOI":"10.1007\/s10915-019-00965-8","type":"journal-article","created":{"date-parts":[[2019,5,22]],"date-time":"2019-05-22T21:02:19Z","timestamp":1558558939000},"page":"1058-1082","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["An Adaptive Step Implicit Midpoint Rule for the Time Integration of Newton\u2019s Linearisations of Non-Linear Problems with Applications in Micromagnetics"],"prefix":"10.1007","volume":"80","author":[{"given":"David","family":"Shepherd","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"James","family":"Miles","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Matthias","family":"Heil","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3817-5677","authenticated-orcid":false,"given":"Milan","family":"Mihajlovi\u0107","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,5,21]]},"reference":[{"key":"965_CR1","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971392","volume-title":"Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations","author":"U Ascher","year":"1998","unstructured":"Ascher, U., Petzold, L.: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia (1998)"},{"key":"965_CR2","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971231","volume-title":"Numerical Solution of Boundary Value Problems for Ordinary Differential Equations","author":"U Ascher","year":"1995","unstructured":"Ascher, U., Mattehij, R., Russell, R.: Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. SIAM, Philadelphia (1995)"},{"key":"965_CR3","volume-title":"Incompressible Flow and the Finite Element Method: Advection-Diffusion and Isothermal Laminar Flow","author":"PM Gresho","year":"1998","unstructured":"Gresho, P.M., Sani, R.L.: Incompressible Flow and the Finite Element Method: Advection-Diffusion and Isothermal Laminar Flow, 1st edn. Wiley, Chichester (1998)","edition":"1"},{"issue":"4","key":"965_CR4","doi-asserted-by":"publisher","first-page":"2018","DOI":"10.1137\/070688018","volume":"30","author":"PM Gresho","year":"2008","unstructured":"Gresho, P.M., Griffiths, D.F., Silvester, D.J.: Adaptive time-stepping for incompressible flow part I: scalar advection-diffusion. SIAM J. Sci. Comput. 30(4), 2018\u20132054 (2008)","journal-title":"SIAM J. Sci. Comput."},{"key":"965_CR5","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1137\/080728032","volume":"32","author":"D Kay","year":"2010","unstructured":"Kay, D., Gresho, P., Griffiths, D., Silvester, D.: Adaptive time stepping for incompressible flow; part II: Navier\u2013Stokes equations. SIAM J. Sci. Comput. 32, 111\u2013128 (2010). \n                    https:\/\/doi.org\/10.1137\/070688018","journal-title":"SIAM J. Sci. Comput."},{"key":"965_CR6","doi-asserted-by":"publisher","first-page":"3900","DOI":"10.1016\/j.jcp.2011.02.014","volume":"230","author":"H Elman","year":"2011","unstructured":"Elman, H., Mihajlovi\u0107, M., Silvester, D.: Fast iterative solvers for buoyancy driven flow problems. J. Comput. Phys. 230, 3900\u20133914 (2011). \n                    https:\/\/doi.org\/10.1016\/j.jcp.2011.02.014","journal-title":"J. Comput. Phys."},{"key":"965_CR7","doi-asserted-by":"publisher","first-page":"14","DOI":"10.1016\/j.jcp.2017.06.005","volume":"346","author":"L Exl","year":"2017","unstructured":"Exl, L., Mauser, N., Schrefl, T., Suess, D.: The extrapolated explicit midpoint scheme for variable order and step size controlled integration of the Landau\u2013Lifshitz\u2013Gilbert equation. J. Comput. Phys. 346, 14\u201324 (2017). \n                    https:\/\/doi.org\/10.1016\/j.jcp.2017.06.005","journal-title":"J. Comput. Phys."},{"key":"965_CR8","unstructured":"Hindmarsh, A.C., Serban, R.: User Documentation for $$\\mathtt {cvode}$$ v2.7.0 (2012)"},{"key":"965_CR9","unstructured":"MathWorks, ode45 (2014). \n                    http:\/\/www.mathworks.co.uk\/help\/matlab\/ref\/ode45.html"},{"key":"965_CR10","volume-title":"Introduction to the Theory of Ferromagnetism","author":"A Aharoni","year":"1996","unstructured":"Aharoni, A.: Introduction to the Theory of Ferromagnetism, 2nd edn. Oxford University Press, Oxford (1996)","edition":"2"},{"key":"965_CR11","doi-asserted-by":"publisher","first-page":"7201304","DOI":"10.1109\/TMAG.2014.2325494","volume":"50","author":"D Shepherd","year":"2014","unstructured":"Shepherd, D., Miles, J., Heil, M., Mihajlovi\u0107, M.: Discretisation induced stiffness in micromagnetic simulations. IEEE Trans. Magn. 50, 7201304 (2014). \n                    https:\/\/doi.org\/10.1109\/TMAG.2014.2325494","journal-title":"IEEE Trans. Magn."},{"issue":"2","key":"965_CR12","doi-asserted-by":"publisher","first-page":"730","DOI":"10.1016\/j.jcp.2005.04.001","volume":"209","author":"M d\u2019Aquino","year":"2005","unstructured":"d\u2019Aquino, M., Serpico, C., Miano, G.: Geometrical integration of Landau\u2013Lifshitz\u2013Gilbert equation based on the mid-point rule. J. Comput. Phys. 209(2), 730\u2013753 (2005). \n                    https:\/\/doi.org\/10.1016\/j.jcp.2005.04.001","journal-title":"J. Comput. Phys."},{"key":"965_CR13","unstructured":"Shepherd, D.: Numerical methods for dynamical micromagnetics, Ph.D. thesis, The University of Manchester (2015)"},{"issue":"15","key":"965_CR14","doi-asserted-by":"publisher","first-page":"R135","DOI":"10.1088\/0022-3727\/33\/15\/201","volume":"33","author":"J Fidler","year":"2000","unstructured":"Fidler, J., Schrefl, T.: Micromagnetic modelling\u2014the current state of the art. J. Phys. D Appl. Phys. 33(15), R135\u2013R156 (2000). \n                    https:\/\/doi.org\/10.1088\/0022-3727\/33\/15\/201","journal-title":"J. Phys. D Appl. Phys."},{"key":"965_CR15","doi-asserted-by":"publisher","first-page":"253","DOI":"10.1016\/j.commatsci.2008.03.019","volume":"44","author":"H Szambolics","year":"2008","unstructured":"Szambolics, H., Buda-Prejbeanu, L., Toussaint, J., Fruchart, O.: A constrained finite element formulation for the Landau\u2013Lifshitz\u2013Gilbert equations. Comput. Mater. Sci. 44, 253\u2013258 (2008). \n                    https:\/\/doi.org\/10.1016\/j.commatsci.2008.03.019","journal-title":"Comput. Mater. Sci."},{"key":"965_CR16","unstructured":"Fischbacher, T., Fangohr, H.: Continuum multi-physics modeling with scripting languages: the Nsim simulation compiler prototype for classical field theory, arXiv (2009) 1\u201350. \n                    arXiv:0907.1587v1"},{"key":"965_CR17","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-09947-6","volume-title":"Solving Ordinary Differential Equations I: Nonstiff Problems","author":"E Hairer","year":"1991","unstructured":"Hairer, E., Morsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd edn. Springer, Berlin (1991)","edition":"2"},{"key":"965_CR18","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511845000","volume-title":"Magnetism and Magnetic Materials","author":"J Coey","year":"2010","unstructured":"Coey, J.: Magnetism and Magnetic Materials, 1st edn. Cambridge University Press, Cambridge (2010)","edition":"1"},{"key":"965_CR19","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780199678792.001.0001","volume-title":"Finite Elements and Fast Iterative Solvers","author":"H Elman","year":"2014","unstructured":"Elman, H., Silvester, D., Wathen, A.: Finite Elements and Fast Iterative Solvers, 2nd edn. Oxford University Press, Oxford (2014)","edition":"2"},{"issue":"4","key":"965_CR20","doi-asserted-by":"publisher","first-page":"1405","DOI":"10.1137\/050631070","volume":"44","author":"S Bartels","year":"2006","unstructured":"Bartels, S., Prohl, A.: Convergence of an implicit finite element method for the Landau\u2013Lifshitz\u2013Gilbert equation. SIAM J. Numer. Anal. 44(4), 1405 (2006). \n                    https:\/\/doi.org\/10.1137\/050631070","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"965_CR21","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1007\/s11831-008-9021-2","volume":"15","author":"I Cimr\u00e1k","year":"2008","unstructured":"Cimr\u00e1k, I.: A survey on the numerics and computations for the Landau\u2013Lifshitz equation of micromagnetism. Arch. Comput. Methods Eng. 15(3), 277\u2013309 (2008). \n                    https:\/\/doi.org\/10.1007\/s11831-008-9021-2","journal-title":"Arch. Comput. Methods Eng."},{"issue":"2","key":"965_CR22","doi-asserted-by":"publisher","first-page":"298","DOI":"10.1016\/S0304-8853(02)00341-4","volume":"248","author":"D Suess","year":"2002","unstructured":"Suess, D., Tsiantos, V., Schrefl, T., Fidler, J., Scholtz, W., Forster, H., Dittrich, R., Miles, J.: Time resolved micromagnetics using a preconditioned time integration method. J. Magn. Magn. Mater. 248(2), 298\u2013311 (2002). \n                    https:\/\/doi.org\/10.1016\/S0304-8853(02)00341-4","journal-title":"J. Magn. Magn. Mater."},{"key":"965_CR23","volume-title":"Infinite Element Methods","author":"L Ying","year":"1995","unstructured":"Ying, L.: Infinite Element Methods. Springer, Berlin (1995)"},{"key":"965_CR24","unstructured":"Yang, B.: Numerical studies of dynamical micromagnetics, Ph.D. thesis, University of California (1997)"},{"issue":"2","key":"965_CR25","doi-asserted-by":"publisher","first-page":"415","DOI":"10.1109\/20.106342","volume":"26","author":"D Fredkin","year":"1990","unstructured":"Fredkin, D., Koehler, T.: Hybrid method for computing demagnetizing fields. IEEE Trans. Magn. 26(2), 415\u2013417 (1990)","journal-title":"IEEE Trans. Magn."},{"issue":"11","key":"965_CR26","doi-asserted-by":"publisher","first-page":"3149","DOI":"10.1109\/TMAG.2008.2001666","volume":"44","author":"O Bottauscio","year":"2008","unstructured":"Bottauscio, O., Chiampi, M., Manzin, A.: A finite element procedure for dynamic micromagnetic computations. IEEE Trans. Magn. 44(11), 3149\u20133152 (2008). \n                    https:\/\/doi.org\/10.1109\/TMAG.2008.20016","journal-title":"IEEE Trans. Magn."},{"issue":"7","key":"965_CR27","doi-asserted-by":"publisher","first-page":"07D542","DOI":"10.1063\/1.3072032","volume":"105","author":"A Knittel","year":"2009","unstructured":"Knittel, A., Franchin, M., Bordignon, G., Fischbacher, T., Bending, S., Fangohr, H.: Compression of boundary element matrix in micromagnetic simulations. J. Appl. Phys. 105(7), 07D542 (2009)","journal-title":"J. Appl. Phys."},{"key":"965_CR28","unstructured":"Borm, S., Grasedyck, L.: HLib. \n                    http:\/\/www.hlib.org\/"},{"key":"965_CR29","doi-asserted-by":"publisher","DOI":"10.1002\/9781118164495","volume-title":"Numerical Solution of Ordinary Differential Equations","author":"KE Atkinson","year":"2009","unstructured":"Atkinson, K.E., Han, W., Stewart, D.E.: Numerical Solution of Ordinary Differential Equations, 1st edn. Wiley, Hoboken (2009)","edition":"1"},{"key":"965_CR30","doi-asserted-by":"publisher","first-page":"A725","DOI":"10.1137\/140966915","volume":"37","author":"J Lee","year":"2015","unstructured":"Lee, J., Nam, J., Pasquali, M.: A new stabilization of adaptive step trapezoid rule based on finite difference interrupts. SIAM J. Sci. Comput. 37, A725\u2013A746 (2015). \n                    https:\/\/doi.org\/10.1137\/140966915","journal-title":"SIAM J. Sci. Comput."},{"key":"965_CR31","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1016\/0045-7825(94)90042-6","volume":"111","author":"J Simo","year":"1994","unstructured":"Simo, J., Armero, F.: Unconditional stability and long-term behaviour of transient algorithms for the incompressible Navier\u2013Stokes and Euler equations. Comput. Methods Appl. Mech. Eng. 111, 111\u2013154 (1994). \n                    https:\/\/doi.org\/10.1016\/0045-7825(94)90042-6","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"965_CR32","unstructured":"SymPy: Python library for symbolic mathematics. \n                    http:\/\/www.sympy.org\/"},{"key":"965_CR33","unstructured":"Shepherd, D.: sym-bdf3.py. \n                    https:\/\/github.com\/davidshepherd7\/Landau-Lifshitz-Gilbert-ODE-model\/blob\/master\/algebra\/sym-bdf3.py"},{"key":"965_CR34","volume-title":"A First Course in the Numerical Analysis of Differential Equations","author":"A Iserles","year":"2009","unstructured":"Iserles, A.: A First Course in the Numerical Analysis of Differential Equations, 2nd edn. Cambridge University Press, Cambridge (2009)","edition":"2"},{"issue":"6","key":"965_CR35","doi-asserted-by":"publisher","first-page":"3443","DOI":"10.1109\/TMAG.2004.836740","volume":"40","author":"T Gilbert","year":"2004","unstructured":"Gilbert, T.: A phenomenological theory of damping in ferromagnetic materials. IEEE Trans. Magn. 40(6), 3443\u20133449 (2004). \n                    https:\/\/doi.org\/10.1109\/TMAG.2004.836740","journal-title":"IEEE Trans. Magn."},{"key":"965_CR36","doi-asserted-by":"crossref","unstructured":"Heil, M., Hazel, A.L.: oomph-lib\u2014An object-oriented multi-physics finite-element library. In: Fluid-Structure Interaction, Lecture Notes in Computational Science and Engineering, vol. 53, pp. 19\u201349. Springer, Berlin (2006)","DOI":"10.1007\/3-540-34596-5_2"},{"issue":"4","key":"965_CR37","doi-asserted-by":"publisher","first-page":"1976","DOI":"10.1109\/20.875251","volume":"36","author":"J Mallinson","year":"2000","unstructured":"Mallinson, J.: Damped gyromagnetic switching. IEEE Trans. Magn. 36(4), 1976\u20131981 (2000)","journal-title":"IEEE Trans. Magn."},{"issue":"3","key":"965_CR38","doi-asserted-by":"publisher","first-page":"476","DOI":"10.1016\/j.cap.2013.12.028","volume":"14","author":"D Jeong","year":"2014","unstructured":"Jeong, D., Kim, J.: An accurate and robust numerical method for micromagnetics simulations. Curr. Appl. Phys. 14(3), 476\u2013483 (2014). \n                    https:\/\/doi.org\/10.1016\/j.cap.2013.12.028","journal-title":"Curr. Appl. Phys."},{"key":"965_CR39","first-page":"107","volume":"29","author":"A Fuwa","year":"2006","unstructured":"Fuwa, A., Ishiwata, T., Tsutsumi, M.: Finite difference scheme for the Landau\u2013Lifshitz equation. Proc. Czech-Jpn. Semin. Appl. Math. 29, 107\u2013113 (2006)","journal-title":"Proc. Czech-Jpn. Semin. Appl. Math."},{"key":"965_CR40","unstructured":"McMichael, B.: $$\\mu $$MAG \u2013 Micromagnetic modelling activity group. \n                    http:\/\/www.ctcms.nist.gov\/rdm\/mumag.org.html"},{"issue":"2","key":"965_CR41","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1002\/fld.1650010203","volume":"1","author":"K Cliffe","year":"1981","unstructured":"Cliffe, K.: On conservative finite element formulations of the inviscid Boussinesq equations. Int. J. Numer. Methods Fluids 1(2), 117\u2013127 (1981)","journal-title":"Int. J. Numer. Methods Fluids"}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-019-00965-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-019-00965-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-019-00965-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,19]],"date-time":"2020-05-19T23:13:23Z","timestamp":1589930003000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-019-00965-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,21]]},"references-count":41,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,8]]}},"alternative-id":["965"],"URL":"https:\/\/doi.org\/10.1007\/s10915-019-00965-8","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,5,21]]},"assertion":[{"value":"20 June 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 December 2018","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 April 2019","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 May 2019","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}