{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,11]],"date-time":"2026-02-11T13:03:48Z","timestamp":1770815028071,"version":"3.50.1"},"reference-count":19,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2014,11,6]],"date-time":"2014-11-06T00:00:00Z","timestamp":1415232000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2015,9]]},"DOI":"10.1007\/s11075-014-9935-8","type":"journal-article","created":{"date-parts":[[2014,11,5]],"date-time":"2014-11-05T07:11:56Z","timestamp":1415171516000},"page":"61-91","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Defect-based local error estimators for high-order splitting methods involving three linear operators"],"prefix":"10.1007","volume":"70","author":[{"given":"Winfried","family":"Auzinger","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Othmar","family":"Koch","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mechthild","family":"Thalhammer","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2014,11,6]]},"reference":[{"key":"9935_CR1","doi-asserted-by":"crossref","first-page":"9210","DOI":"10.1039\/b903266e","volume":"11","author":"A Yochelis","year":"2009","unstructured":"Yochelis, A., Sheintuch, M.: Towards nonlinear selection of reaction\u2013diffusion patterns in presence of advection: a spatial dynamics approach. Phys. Chem. Chem. Phys. 11, 9210\u20139223 (2009)","journal-title":"Phys. Chem. Chem. Phys."},{"key":"9935_CR2","first-page":"925","volume":"56","author":"D Fournier","year":"1999","unstructured":"Fournier, D., Sibert, J., Hampton, J., Bills, P.: An advection\u2013diffusion\u2013reaction model for the estimation of fish movement parameters from tagging data, with application to skipjack tuna (katsuwonus pelamis). Can. J. Fish. Aquat. Sci. 56, 925\u2013938 (1999)","journal-title":"Can. J. Fish. Aquat. Sci."},{"key":"9935_CR3","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1088\/0951-7715\/24\/1\/016","volume":"24","author":"Y Du","year":"2011","unstructured":"Du, Y., Mei, L.: On a nonlocal reaction\u2013diffusion\u2013advection equation modelling phytoplankton dynamics. Nonlinearity 24, 319\u2013349 (2011)","journal-title":"Nonlinearity"},{"key":"9935_CR4","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1137\/030600209","volume":"2","author":"W Bao","year":"2004","unstructured":"Bao, W.: Ground states and dynamics of multi-component Bose\u2013Einstein condensates. Multiscale Model. Simul. 2, 210\u2013236 (2004)","journal-title":"Multiscale Model. Simul."},{"key":"9935_CR5","doi-asserted-by":"crossref","first-page":"767","DOI":"10.1142\/S0218202505000534","volume":"15","author":"W Bao","year":"2005","unstructured":"Bao, W., Markowich, P., Schmeiser, C., Weish\u00e4upl, R.: On the Gross\u2013Pitaevskii equation with strongly anisotropic confinement: formal asymptotics and numerical experiments. Math. Models Meth. Appl. Sci. 15, 767\u2013782 (2005)","journal-title":"Math. Models Meth. Appl. Sci."},{"key":"9935_CR6","doi-asserted-by":"crossref","first-page":"612","DOI":"10.1016\/j.jcp.2006.01.020","volume":"217","author":"W Bao","year":"2006","unstructured":"Bao, W., Wang, H.: An efficient and spectrally accurate numerical method for computing dynamics of rotating Bose-Einstein condensates. J. Comput. Phys. 217, 612\u2013626 (2006)","journal-title":"J. Comput. Phys."},{"key":"9935_CR7","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.physd.2007.06.026","volume":"234","author":"Y Zhang","year":"2007","unstructured":"Zhang, Y., Bao, W., Li, H.: Dynamics of rotating two-component Bose\u2013Einstein condensates and its efficient computation. Physica D 234, 49\u201369 (2007)","journal-title":"Physica D"},{"key":"9935_CR8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3934\/krm.2013.6.1","volume":"6","author":"W Bao","year":"2013","unstructured":"Bao, W., Cai, Y.: Mathematical theory and numerical methods for Bose\u2013Einstein condensation. Kinet. Relat. Mod. 6, 1\u2013135 (2013)","journal-title":"Kinet. Relat. Mod."},{"key":"9935_CR9","doi-asserted-by":"crossref","first-page":"243","DOI":"10.7494\/OpMath.2014.34.2.243","volume":"34","author":"W Auzinger","year":"2014","unstructured":"Auzinger, W., Herfort, W.: Local error structures and order conditions in terms of Lie elements for exponential splitting schemes. Opuscula Math. 34, 243\u2013255 (2014)","journal-title":"Opuscula Math."},{"key":"9935_CR10","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.apnum.2012.09.002","volume":"63","author":"O Koch","year":"2013","unstructured":"Koch, O., Neuhauser, C., Thalhammer, M.: Embedded split\u2013step formulae for the time integration of nonlinear evolution equations. Appl. Numer. Math. 63, 14\u201324 (2013)","journal-title":"Appl. Numer. Math."},{"key":"9935_CR11","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1016\/j.cam.2014.06.012","volume":"273","author":"W Auzinger","year":"2014","unstructured":"Auzinger, W., Hofst\u00e4tter, H., Koch, O., Thalhammer, M.: Defect-based local error estimators for splitting methods, with application to Schr\u00f6dinger equations, Part III: The nonlinear case. J. Comput. Appl. Math. 273, 182\u2013204 (2014)","journal-title":"J. Comput. Appl. Math."},{"key":"9935_CR12","doi-asserted-by":"crossref","first-page":"2643","DOI":"10.1016\/j.cam.2012.01.001","volume":"236","author":"W Auzinger","year":"2012","unstructured":"Auzinger, W., Koch, O., Thalhammer, M.: Defect-based local error estimators for splitting methods, with application to Schr\u00f6dinger equations, Part I: The linear case. J. Comput. Appl. Math. 236, 2643\u20132659 (2012)","journal-title":"J. Comput. Appl. Math."},{"key":"9935_CR13","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1016\/j.cam.2013.04.043","volume":"255","author":"W Auzinger","year":"2013","unstructured":"Auzinger, W., Koch, O., Thalhammer, M.: Defect-based local error estimators for splitting methods, with application to Schr\u00f6dinger equations, Part II: Higher-order methods for linear problems. J. Comput. Appl. Math. 255, 384\u2013403 (2013)","journal-title":"J. Comput. Appl. Math."},{"key":"9935_CR14","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-05018-7","volume-title":"Geometric Numerical Integration","author":"E Hairer","year":"2002","unstructured":"Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Springer-Verlag, Berlin (2002)"},{"key":"9935_CR15","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1007\/s10543-009-0235-y","volume":"49","author":"F Castella","year":"2009","unstructured":"Castella, F., Chartier, P., Descombes, S., Vilmart, G.: Splitting methods with complex times for parabolic equations. BIT Numer. Math. 49, 487\u2013508 (2009)","journal-title":"BIT Numer. Math."},{"key":"9935_CR16","doi-asserted-by":"crossref","first-page":"454","DOI":"10.1007\/BF02731494","volume":"20","author":"E Gross","year":"1961","unstructured":"Gross, E.: Structure of a quantized vortex in boson systems. Nuovo Cimento 20, 454\u2013457 (1961)","journal-title":"Nuovo Cimento"},{"key":"9935_CR17","first-page":"451","volume":"13","author":"L Pitaevskii","year":"1961","unstructured":"Pitaevskii, L.: Vortex lines in an imperfect Bose gas. Soviet Phys. JETP 13, 451\u2013454 (1961)","journal-title":"Soviet Phys. JETP"},{"key":"9935_CR18","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1088\/0951-7715\/10\/2\/013","volume":"10","author":"A Doelman","year":"1997","unstructured":"Doelman, A., Kaper, T., Zegeling, P.: Pattern formation in the one-dimensional Gray-Scott model. Nonlinearity 10, 523\u2013563 (1997)","journal-title":"Nonlinearity"},{"key":"9935_CR19","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1021\/j100247a009","volume":"89","author":"P Gray","year":"1985","unstructured":"Gray, P., Scott, S.: Sustained oscillations and other exotic patterns of behavior in isothermal reactions. J. Phys. Chem. 89, 22\u201332 (1985)","journal-title":"J. Phys. Chem."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-014-9935-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-014-9935-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-014-9935-8","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,3]],"date-time":"2019-06-03T14:01:13Z","timestamp":1559570473000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-014-9935-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,11,6]]},"references-count":19,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2015,9]]}},"alternative-id":["9935"],"URL":"https:\/\/doi.org\/10.1007\/s11075-014-9935-8","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,11,6]]}}}