{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T00:17:49Z","timestamp":1783037869688,"version":"3.54.6"},"reference-count":21,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2016,10,4]],"date-time":"2016-10-04T00:00:00Z","timestamp":1475539200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund (AT)","doi-asserted-by":"publisher","award":["P24157-N13"],"award-info":[{"award-number":["P24157-N13"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001821","name":"WWTF","doi-asserted-by":"crossref","award":["MA14-002"],"award-info":[{"award-number":["MA14-002"]}],"id":[{"id":"10.13039\/501100001821","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2017,5]]},"DOI":"10.1007\/s11075-016-0206-8","type":"journal-article","created":{"date-parts":[[2016,10,3]],"date-time":"2016-10-03T20:53:29Z","timestamp":1475528009000},"page":"261-283","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Adaptive high-order splitting methods for systems of nonlinear evolution equations with periodic boundary conditions"],"prefix":"10.1007","volume":"75","author":[{"given":"Winfried","family":"Auzinger","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Othmar","family":"Koch","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michael","family":"Quell","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2016,10,4]]},"reference":[{"key":"206_CR1","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1016\/j.cam.2005.03.008","volume":"185","author":"G S\u00f6derlind","year":"2006","unstructured":"S\u00f6derlind, G., Wang, L.: Adaptive time-stepping and computational stability. J. Comput. Appl. Math. 185, 225\u2013243 (2006)","journal-title":"J. Comput. Appl. Math."},{"key":"206_CR2","doi-asserted-by":"publisher","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\u2013Heidelberg\u2013New York (2002)"},{"key":"206_CR3","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1017\/S0962492910000048","volume":"19","author":"M Hochbruck","year":"2010","unstructured":"Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209\u2013286 (2010)","journal-title":"Acta Numer."},{"key":"206_CR4","doi-asserted-by":"publisher","first-page":"413","DOI":"10.1142\/S1793744211000412","volume":"03","author":"S Descombes","year":"2011","unstructured":"Descombes, S., Duarte, M., Dumont, T., Louvet, V., Massot, M.: Adaptive time splitting method for multi-scale evolutionary partial differential equations. Confluentes Math. 03, 413\u2013443 (2011)","journal-title":"Confluentes Math."},{"key":"206_CR5","doi-asserted-by":"publisher","first-page":"735","DOI":"10.1023\/A:1022396519656","volume":"40","author":"T Jahnke","year":"2000","unstructured":"Jahnke, T., Lubich, C.: Error bounds for exponential operator splittings. BIT 40, 735\u2013744 (2000)","journal-title":"BIT"},{"issue":"4","key":"206_CR6","doi-asserted-by":"publisher","first-page":"2022","DOI":"10.1137\/060674636","volume":"46","author":"M Thalhammer","year":"2008","unstructured":"Thalhammer, M.: High-order exponential operator splitting methods for time-dependent Schr\u00f6dinger equations. SIAM J. Numer. Anal. 46(4), 2022\u20132038 (2008)","journal-title":"SIAM J. Numer. Anal."},{"key":"206_CR7","doi-asserted-by":"publisher","first-page":"2141","DOI":"10.1090\/S0025-5718-08-02101-7","volume":"77","author":"C Lubich","year":"2008","unstructured":"Lubich, C.: On splitting methods for Schr\u00f6dinger\u2013Poisson and cubic nonlinear Schr\u00f6dinger equations. Math. Comp. 77, 2141\u20132153 (2008)","journal-title":"Math. Comp."},{"key":"206_CR8","doi-asserted-by":"publisher","first-page":"1265","DOI":"10.1051\/m2an\/2013067","volume":"47","author":"O Koch","year":"2013","unstructured":"Koch, O., Neuhauser, C., Thalhammer, M.: Error analysis of high-order splitting methods for nonlinear evolutionary Schr\u00f6dinger equations and application to the MCTDHF equations in electron dynamics. M2AN Math. Model. Numer. Anal. 47, 1265\u20131284 (2013)","journal-title":"M2AN Math. Model. Numer. Anal."},{"key":"206_CR9","doi-asserted-by":"publisher","first-page":"1559","DOI":"10.1090\/S0025-5718-2012-02657-3","volume":"82","author":"S Blanes","year":"2013","unstructured":"Blanes, S., Casas, F., Chartier, P., Murua, A.: Optimized high-order splitting methods for some classes of parabolic equations. Math. Comp. 82, 1559\u20131576 (2013)","journal-title":"Math. Comp."},{"key":"206_CR10","doi-asserted-by":"publisher","first-page":"1485","DOI":"10.1090\/S0025-5718-09-02213-3","volume":"78","author":"E Hansen","year":"2009","unstructured":"Hansen, E., Ostermann, A.: Exponential splitting for unbounded operators. Math. Comp. 78, 1485\u20131496 (2009)","journal-title":"Math. Comp."},{"key":"206_CR11","doi-asserted-by":"publisher","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":"206_CR12","doi-asserted-by":"publisher","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-step formulae for the time integration of nonlinear evolution equations. Appl. Numer. Math. 63, 14\u201324 (2013)","journal-title":"Appl. Numer. Math."},{"key":"206_CR13","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1007\/s11075-014-9935-8","volume":"70","author":"W Auzinger","year":"2015","unstructured":"Auzinger, W., Koch, O., Thalhammer, M.: Defect-based local error estimators for high-order splitting methods involving three linear operators. Numer. Algorithms 70, 61\u201391 (2015)","journal-title":"Numer. Algorithms"},{"key":"206_CR14","doi-asserted-by":"crossref","unstructured":"Auzinger, W., Hofst\u00e4tter, H., Ketcheson, D., Koch, O.: Practical splitting methods for the adaptive integration of nonlinear evolution equations. Part I: construction of optimized schemes and pairs of schemes, BIT Numer. Math., published online 28 July 2016","DOI":"10.1007\/s10543-016-0626-9"},{"key":"206_CR15","unstructured":"Auzinger, W., Koch, O.: Coefficients of various splitting methods, http:\/\/www.asc.tuwien.ac.at\/~winfried\/splitting\/"},{"key":"206_CR16","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-12607-3","volume-title":"Solving ordinary differential equations I","author":"E Hairer","year":"1987","unstructured":"Hairer, E., N\u00f8rsett, S., Wanner, G.: Solving ordinary differential equations I. Springer-Verlag, Berlin\u2013Heidelberg\u2013New York (1987)"},{"key":"206_CR17","volume-title":"Numerical recipes in C \u2014 the art of scientific computing","author":"W Press","year":"1988","unstructured":"Press, W., Flannery, B., Teukolsky, S., Vetterling, W.: Numerical recipes in C \u2014 the art of scientific computing. Cambridge University Press, Cambridge (1988)"},{"key":"206_CR18","volume-title":"Chemical waves and instabilities","author":"P Gray","year":"1990","unstructured":"Gray, P., Scott, S.: Chemical waves and instabilities. Clarendon, Oxford (1990)"},{"key":"206_CR19","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-010-0732-0","volume-title":"Infinite-dimensional dynamical systems","author":"J Robinson","year":"2001","unstructured":"Robinson, J.: Infinite-dimensional dynamical systems. Cambridge University Press, Cambridge (2001)"},{"key":"206_CR20","volume-title":"An introduction to harmonic analysis","author":"Y Katznelson","year":"1968","unstructured":"Katznelson, Y.: An introduction to harmonic analysis. Dover Publications, Inc., New York (1968)"},{"key":"206_CR21","unstructured":"Rudin, W.: Real and complex analysis, 3rd edn. McGraw-Hill (1987)"}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-016-0206-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-016-0206-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-016-0206-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,9,14]],"date-time":"2019-09-14T02:54:15Z","timestamp":1568429655000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-016-0206-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,10,4]]},"references-count":21,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2017,5]]}},"alternative-id":["206"],"URL":"https:\/\/doi.org\/10.1007\/s11075-016-0206-8","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,10,4]]}}}