{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,16]],"date-time":"2026-07-16T14:17:39Z","timestamp":1784211459936,"version":"3.55.0"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2014,11,26]],"date-time":"2014-11-26T00:00:00Z","timestamp":1416960000000},"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":["Adv Comput Math"],"published-print":{"date-parts":[[2015,12]]},"DOI":"10.1007\/s10444-014-9394-8","type":"journal-article","created":{"date-parts":[[2014,12,2]],"date-time":"2014-12-02T15:06:30Z","timestamp":1417532790000},"page":"955-986","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":50,"title":["Construction and analysis of higher order Galerkin variational integrators"],"prefix":"10.1007","volume":"41","author":[{"given":"Sina","family":"Ober-Bl\u00f6baum","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nils","family":"Saake","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2014,11,26]]},"reference":[{"key":"9394_CR1","volume-title":"Mathematical Methods of Classical Mechanics","author":"VI Arnold","year":"1999","unstructured":"Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, Berlin Heidelberg New York (1999)"},{"key":"9394_CR2","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1093\/imanum\/drn018","volume":"29","author":"N Bou-Rabee","year":"2008","unstructured":"Bou-Rabee, N., Owhadi, H.: Stochastic variational integrators. IMA J. Numer. Anal. 29, 421\u2013443 (2008)","journal-title":"IMA J. Numer. Anal."},{"key":"9394_CR3","unstructured":"Campos, C.M., Junge, O., Ober-Bl\u00f6baum, S.: Higher order variational time discretization of optimal control problems In: 20th International Symposium on Mathematical Theory of Networks and Systems, Melbourne (2012)"},{"issue":"5","key":"9394_CR4","doi-asserted-by":"crossref","first-page":"1365","DOI":"10.1088\/0951-7715\/14\/5\/322","volume":"14","author":"J Cort\u00e9s","year":"2001","unstructured":"Cort\u00e9s, J., Mart\u00ednez, S.: Non-holonomic integrators. Nonlinearity 14(5), 1365\u20131392 (2001)","journal-title":"Nonlinearity"},{"issue":"3","key":"9394_CR5","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1137\/S1111111102406038","volume":"2","author":"RC Fetecau","year":"2003","unstructured":"Fetecau, R.C., Marsden, J.E., Ortiz, M., West, M.: Nonsmooth Lagrangian Mechanics and Variational Collision Integrators. SIAM J. Appl. Dyn. Syst. 2(3), 381\u2013416 (2003)","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"9394_CR6","volume-title":"Springer Series in Computational Mathematics","author":"E Hairer","year":"2010","unstructured":"Hairer, E., Wanner, G.: Solving ordinary differential equations II: Stiff and differential-algebraic problems. Springer Series in Computational Mathematics. Springer, Heidelberg (2010). http:\/\/opac.inria.fr\/record=b1130632"},{"key":"9394_CR7","volume-title":"Springer Series in Computational Mathematics, vol. 31","author":"E Hairer","year":"2002","unstructured":"Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration. Springer Series in Computational Mathematics, vol. 31. Springer, Berlin Heidelberg New York (2002)"},{"key":"9394_CR8","unstructured":"Hall, J., Leok, M.: Spectral variational integrators. (preprint, arXiv: 1211.4534 ) (2012)"},{"issue":"10","key":"9394_CR9","doi-asserted-by":"crossref","first-page":"1295","DOI":"10.1002\/1097-0207(20001210)49:10<1295::AID-NME993>3.0.CO;2-W","volume":"49","author":"C Kane","year":"2000","unstructured":"Kane, C., Marsden, J.E., Ortiz, M., West, M.: Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems. Int. J. Numer. Methods Eng. 49(10), 1295\u20131325 (2000)","journal-title":"Int. J. Numer. Methods Eng."},{"issue":"1","key":"9394_CR10","first-page":"61","volume":"1","author":"M Kobilarov","year":"2010","unstructured":"Kobilarov, M., Marsden, J.E., Sukhatme, G.S.: Geometric discretization of nonholonomic systems with symmetries. Discret Contin Dyn. Syst. - Series S 1(1), 61\u201384 (2010)","journal-title":"Discret Contin Dyn. Syst. - Series S"},{"key":"9394_CR11","doi-asserted-by":"crossref","first-page":"952","DOI":"10.1007\/BF00945133","volume":"39","author":"FM Lasagni","year":"1988","unstructured":"Lasagni, F.M.: Canonical Runge-Kutta methods. Zeitschrift f\u00fcr Angewandte Mathematik und Physik ZAMP 39, 952\u2013953 (1988). doi: 10.1007\/BF00945133","journal-title":"Zeitschrift f\u00fcr Angewandte Mathematik und Physik ZAMP"},{"key":"9394_CR12","volume-title":"Simulating Hamiltonian dynamics","author":"B Leimkuhler","year":"2004","unstructured":"Leimkuhler, B., Reich, S.: Simulating Hamiltonian dynamics. Cambridge University Press, Cambridge (2004)"},{"key":"9394_CR13","unstructured":"Leok, M., Shingel, T.: General Techniques for Constructing Variational Integrators. Frontiers of Mathematics in China arXiv: 1102.2685 (2011)"},{"key":"9394_CR14","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1007\/s00205-002-0212-y","volume":"167","author":"A Lew","year":"2003","unstructured":"Lew, A., Marsden, J.E., Ortiz, M., West, M.: Asynchronous variational integrators. Arch. Ration. Mech. Anal. 167, 85\u2013146 (2003)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"9394_CR15","unstructured":"Lew, A., Marsden, J.E., Ortiz, M., West, M.: An overview of variational integrators. In: Franca, L.P., Tezduyar, T.E., Masud, A. (eds.) Finite element methods: 1970\u2019s and beyond, CIMNE, pp. 98\u2013115 (2004a)"},{"issue":"1","key":"9394_CR16","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1002\/nme.958","volume":"60","author":"A Lew","year":"2004","unstructured":"Lew, A., Marsden, J.E., Ortiz, M., West, M.: Variational time integrators. Int. J. Numer. Methods Eng. 60(1), 153\u2013212 (2004b)","journal-title":"Int. J. Numer. Methods Eng."},{"key":"9394_CR17","doi-asserted-by":"crossref","unstructured":"Leyendecker, S., Ober-Bl\u00f6baum, S.: A variational approach to multirate integration for constrained systems. In: Samin, J.C., Fisette, P. (eds.) Multibody dynamics, computational methods in applied sciences, vol. 28, pp. 97\u2013121. Springer Netherlands (2013), doi: 10.1007\/978-94-007-5404-1_5","DOI":"10.1007\/978-94-007-5404-1_5"},{"issue":"9","key":"9394_CR18","first-page":"677","volume":"88","author":"S Leyendecker","year":"2008","unstructured":"Leyendecker, S., Marsden, J.E., Ortiz, M.: Variational integrators for constrained dynamical systems. J. Appl. Math. Mech. 88 (9), 677\u2013708 (2008)","journal-title":"J. Appl. Math. Mech."},{"issue":"6","key":"9394_CR19","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1002\/oca.912","volume":"31","author":"S Leyendecker","year":"2010","unstructured":"Leyendecker, S., Ober-Bl\u00f6baum, S., Marsden, J.E., Ortiz, M.: Discrete mechanics and optimal control for constrained systems. Optim. Control, Appl. Methods 31(6), 505\u2013528 (2010)","journal-title":"Optim. Control, Appl. Methods"},{"key":"9394_CR20","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1017\/S096249290100006X","volume":"10","author":"JE Marsden","year":"2001","unstructured":"Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numerica 10, 357\u2013514 (2001)","journal-title":"Acta Numerica"},{"key":"9394_CR21","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1007\/s002200050505","volume":"199","author":"JE Marsden","year":"1998","unstructured":"Marsden, J.E., Patrick, G.W., Shkoller, S.: Multisymplectic geometry, variational integrators, and nonlinear PDEs. Commun. Math. Sci. 199, 351\u2013395 (1998)","journal-title":"Commun. Math. Sci."},{"key":"9394_CR22","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1017\/S0962492902000053","volume":"11","author":"RI McLachlan","year":"2002","unstructured":"McLachlan, R.I., Quispel, G.R.W.: Splitting methods. Acta Numerica 11, 341\u2013434 (2002). doi: 10.1017\/S0962492902000053","journal-title":"Acta Numerica"},{"key":"9394_CR23","unstructured":"Ober-Bl\u00f6baum, S.: Galerkin variational integrators and modified symplectic Runge-Kutta methods. submitted (2014)"},{"issue":"2","key":"9394_CR24","doi-asserted-by":"crossref","first-page":"322","DOI":"10.1051\/cocv\/2010012","volume":"17","author":"S Ober-Bl\u00f6baum","year":"2011","unstructured":"Ober-Bl\u00f6baum, S., Junge, O., Marsden, J.E.: Discrete mechanics and optimal control: an analysis. Control. Optimisation Calc. Var. 17(2), 322\u2013352 (2011)","journal-title":"Control. Optimisation Calc. Var."},{"key":"9394_CR25","doi-asserted-by":"crossref","first-page":"498","DOI":"10.1016\/j.jcp.2013.02.006","volume":"242","author":"S Ober-Bl\u00f6baum","year":"2013","unstructured":"Ober-Bl\u00f6baum, S., Tao, M., Cheng, M., Owhadi, H., Marsden, J.E.: Variational integrators for electric circuits. J. Comput. Phys. 242, 498\u2013530 (2013). doi: 10.1016\/j.jcp.2013.02.006","journal-title":"J. Comput. Phys."},{"key":"9394_CR26","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1007\/s00211-009-0245-3","volume":"113","author":"GW Patrick","year":"2009","unstructured":"Patrick, G.W., Cuell, C.: Error analysis of variational integrators of unconstrained Lagrangian systems. Numer. Math. 113, 243\u2013264 (2009). doi: 10.1007\/S00211-009-0245-3","journal-title":"Numer. Math."},{"issue":"4","key":"9394_CR27","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1016\/0167-2789(94)90046-9","volume":"76","author":"S Reich","year":"1994","unstructured":"Reich, S.: Momentum conserving symplectic integrations. Phys. D 76(4), 375\u2013383 (1994)","journal-title":"Phys. D"},{"key":"9394_CR28","unstructured":"Saake, N.: Konstruktion und Analyse variationeller Integratoren h\u00f6herer Ordnung, diploma thesis, Paderborn (2012)"},{"key":"9394_CR29","doi-asserted-by":"crossref","first-page":"877","DOI":"10.1007\/BF01954907","volume":"28","author":"JM Sanz-Serna","year":"1988","unstructured":"Sanz-Serna, J.M.: Runge-Kutta schemes for Hamiltonian systems. BIT Numer. Math. 28, 877\u2013883 (1988). doi: 10.1007\/BF01954907","journal-title":"BIT Numer. Math."},{"key":"9394_CR30","doi-asserted-by":"crossref","first-page":"1779","DOI":"10.1137\/080732936","volume":"7","author":"A Stern","year":"2009","unstructured":"Stern, A., Grinspun, E.: Implicit-explicit variational integration of highly oscillatory problems. SIAM Multiscale Model. Simul. 7, 1779\u20131794 (2009)","journal-title":"SIAM Multiscale Model. Simul."},{"issue":"4","key":"9394_CR31","first-page":"365","volume":"11","author":"G Sun","year":"1993","unstructured":"Sun, G.: Symplectic partitioned Runge-Kutta methods. J. Comput. Math. 11(4), 365\u2013372 (1993) http:\/\/www.jcm.ac.cn\/EN\/abstract\/article_8879.shtml","journal-title":"J. Comput. Math."},{"issue":"1","key":"9394_CR32","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1016\/0041-5553(89)90058-X","volume":"29","author":"YB Suris","year":"1989","unstructured":"Suris, Y.B.: The canonicity of mappings generated by Runge-Kutta type methods when integrating the systems x \u0308 = \u2212 \u2202u \/ \u2202x $\\ddot {x}=-\\partial u\/\\partial x$ . USSR Comput. Math. Math. Phys. 29(1), 138\u2013144 (1989) doi: 10.1016\/0041-5553(89)90058-X","journal-title":"USSR Comput. Math. Math. Phys."},{"key":"9394_CR33","first-page":"78","volume":"2","author":"YB Suris","year":"1990","unstructured":"Suris, Y.B.: Hamiltonian methods of Runge-Kutta type and their variational interpretation. Math. Model. 2, 78\u201387 (1990)","journal-title":"Math. Model."},{"issue":"4","key":"9394_CR34","doi-asserted-by":"crossref","first-page":"1269","DOI":"10.1137\/090771648","volume":"8","author":"M Tao","year":"2010","unstructured":"Tao, M., Owhadi, H., Marsden, J.E.: Nonintrusive and structure preserving multiscale integration of stiff ODEs, SDEs, and Hamiltonian systems with hidden slow dynamics via flow averaging. Multiscale Model. Simul. 8(4), 1269\u20131324 (2010)","journal-title":"Multiscale Model. Simul."},{"key":"9394_CR35","doi-asserted-by":"crossref","first-page":"262","DOI":"10.1016\/0375-9601(90)90092-3","volume":"150","author":"H Yoshida","year":"1990","unstructured":"Yoshida, H.: Construction of higher order symplectic integrators. Phys. Lett. A 150, 262\u2013268 (1990). doi: 10.1016\/0375-9601(90)90092-3","journal-title":"Phys. Lett. A"}],"updated-by":[{"DOI":"10.1007\/s10444-016-9464-1","type":"erratum","label":"Erratum","source":"publisher","updated":{"date-parts":[[2016,7,12]],"date-time":"2016-07-12T00:00:00Z","timestamp":1468281600000}}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-014-9394-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10444-014-9394-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-014-9394-8","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T17:08:20Z","timestamp":1559149700000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10444-014-9394-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,11,26]]},"references-count":35,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2015,12]]}},"alternative-id":["9394"],"URL":"https:\/\/doi.org\/10.1007\/s10444-014-9394-8","relation":{"erratum":[{"id-type":"doi","id":"10.1007\/s10444-016-9464-1","asserted-by":"object"}]},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,11,26]]}}}