{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T20:12:25Z","timestamp":1773778345491,"version":"3.50.1"},"reference-count":39,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2005,9,1]],"date-time":"2005-09-01T00:00:00Z","timestamp":1125532800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bit Numer Math"],"published-print":{"date-parts":[[2005,9]]},"DOI":"10.1007\/s10543-005-0014-3","type":"journal-article","created":{"date-parts":[[2005,11,25]],"date-time":"2005-11-25T15:37:12Z","timestamp":1132933032000},"page":"517-542","source":"Crossref","is-referenced-by-count":12,"title":["Global Error Estimation and Extrapolated Multistep Methods For Index 1 Differential-Algebraic Systems"],"prefix":"10.1007","volume":"45","author":[{"given":"G. Yu.","family":"Kulikov","sequence":"first","affiliation":[]},{"given":"S. K.","family":"Shindin","sequence":"additional","affiliation":[]}],"member":"297","reference":[{"key":"14_CRAiLe97","doi-asserted-by":"crossref","unstructured":"R. A\u00efd and L. Levacher, Numerical investigations on global error estimation for ordinary differential equations, J. Comput. Appl. Math., 82 (1997), pp. 21\u201339.","DOI":"10.1016\/S0377-0427(97)00079-4"},{"key":"14_CRBaDe83a","doi-asserted-by":"crossref","unstructured":"G. Bader and P. Deuflhard, A semi-implicit mid-point rule for stiff systems of ordinary differential equations, Numer. Math., 41 (1983), pp. 373\u2013398.","DOI":"10.1007\/BF01418331"},{"key":"14_CRBa75","unstructured":"N. S. Bakhvalov, Numerical Methods (in Russian), Nauka, Moscow, 1975."},{"key":"14_CRBeZh62","unstructured":"I. S. Berezin and N. P. Zhidkov, Methods of Computations, Vol. 1 (in Russian), Gos. izd-vo fiz.-mat. lit-ry, Moscow, 1962."},{"key":"14_CRBuSt66","doi-asserted-by":"crossref","unstructured":"R. Bulirsch and J. Stoer, Numerical treatment of ordinary differential equations by extrapolation methods, Numer. Math., 8 (1966), pp. 1\u201313.","DOI":"10.1007\/BF02165234"},{"key":"14_CRCrLi84","doi-asserted-by":"crossref","unstructured":"M. Crozeix and F. J. Lisbona, The convergence of variable-stepsize, variable formula, multistep methods, SIAM J. Numer. Anal., 21 (1984), pp. 512\u2013534.","DOI":"10.1137\/0721037"},{"key":"14_CRDa56","doi-asserted-by":"crossref","unstructured":"G. Dahlquist, Convergence and stability in the numerical integration of ordinary differential equations, Math. Scand., 4 (1956), pp. 33\u201353.","DOI":"10.7146\/math.scand.a-10454"},{"key":"14_CRDe83","doi-asserted-by":"crossref","unstructured":"P. Deuflhard, Order and stepsize control in extrapolation methods, Numer. Math., 41 (1983), pp. 399\u2013422.","DOI":"10.1007\/BF01418332"},{"key":"14_CRDe85","doi-asserted-by":"crossref","unstructured":"P. Deuflhard, Recent progress in extrapolation methods for ordinary differential equations, SIAM Rev., 27 (1985), pp. 505\u2013535.","DOI":"10.1137\/1027140"},{"key":"14_CRDeHa87","doi-asserted-by":"crossref","unstructured":"P. Deuflhard, E. Hairer, and J. Zugck, One-step and extrapolation methods for differential-algebraic systems, Numer. Math., 51 (1987), pp. 501\u2013516.","DOI":"10.1007\/BF01400352"},{"key":"14_CRGeTu74","doi-asserted-by":"crossref","unstructured":"C. W. Gear and K. W. Tu, The effects of variable mesh size on the stability of multistep methods, SIAM J. Numer. Anal., 11 (1974), pp. 1025\u20131043.","DOI":"10.1137\/0711079"},{"key":"14_CRGeWa74","doi-asserted-by":"crossref","unstructured":"C. W. Gear and D. S. Watanabe, Stability and convergence of variable order multistep methods, SIAM J. Numer. Anal., 11 (1974), pp. 1044\u20131058.","DOI":"10.1137\/0711080"},{"key":"14_CRGr64","unstructured":"W. B. Gragg, Repeated Extrapolation to the Limit in the Numerical Solution of Ordinary Differential Equations, PhD. thesis, University of California, 1964."},{"key":"14_CRGr65","unstructured":"W. B. Gragg, On extrapolation algorithms for ordinary initial value problems, SIAM J. Numer. Anal., 2 Ser. B (1965), pp. 384\u2013403."},{"key":"14_CRGr83","doi-asserted-by":"crossref","unstructured":"R. D. Grigorieff, Stability of multistep methods on variable grids, Numer. Math., 42 (1983), pp. 359\u2013377.","DOI":"10.1007\/BF01389580"},{"key":"14_CRHaLu84","unstructured":"E. Hairer and Ch. Lubich, Asymptotic expansions of the global error of fixed-stepsize methods, 45 (1984), pp. 345\u2013360."},{"key":"14_CRHaLu89","doi-asserted-by":"crossref","unstructured":"E. Hairer, Ch. Lubich, and M. Roche, The numerical solution of differential-algebraic systems by Runge-Kutta methods, Lect. Notes Math, Vol. 1409, Springer, Berlin, 1989.","DOI":"10.1007\/BFb0093947"},{"key":"14_CRHaNo93","unstructured":"E. Hairer, S. P. N\u00f8rsett, and G. Wanner, Solving ordinary differential equations I: Nonstiff problems, Springer, Berlin, 1993."},{"key":"14_CRHaWa96","doi-asserted-by":"crossref","unstructured":"E. Hairer and G. Wanner, Solving ordinary differential equations II: Stiff and differential-algebraic problems, Springer, Berlin, 1996.","DOI":"10.1007\/978-3-642-05221-7"},{"key":"14_CRKu93Vest","unstructured":"G. Yu. Kulikov, The numerical solution of the autonomous Cauchy problem with an algebraic relation between the phase variables (non-degenerate case) (in Russian), Vestnik Moskov. Univ. Ser. 1 Mat. Mekh. (1993), No. 3, pp. 10\u201314; translation in Moscow Univ. Math. Bull., 48 (1993), No. 3, pp. 8\u201312."},{"key":"14_CRKu97KJCAM","unstructured":"G. Yu. Kulikov, Numerical methods solving the semi-explicit differential-algebraic equations by implicit multistep fixed stepsize methods, Korean J. Comput. Appl. Math., 4 (1997), No. 2, pp. 281\u2013318."},{"key":"14_CRKuSh99KJCAM","unstructured":"G. Yu. Kulikov and S. K. Shindin, A local-global stepsize control for multistep methods applied to semi-explicit index 1 differential-algebraic equations, Korean J. Comput. Appl. Math., 6 (1999), No. 3, pp. 463\u2013492."},{"key":"14_CRKuSh00ZhVM","unstructured":"G. Yu. Kulikov and S. K. Shindin, A technique for controlling the global error in multistep methods (in Russian), Zh. Vychisl. Mat. Mat. Fiz., 40 (2000), No. 9, pp. 1308\u20131329; translation in Comput. Math. Math. Phys., 40 (2000), No. 9, pp. 1255\u20131275."},{"key":"14_CRKuSh00ENUMATH","unstructured":"G. Yu. Kulikov and S. K. Shindin, An advanced local-global stepsize control for multistep methods, in Numerical Mathematics and Advanced Applications, Proceedings of the 3rd European Conference, P. Neittaanm\u00e4ki, T. Tiihonen, P. Tarvainen, eds., World Scientific, Singapore, 2000, pp. 593\u2013600."},{"key":"14_CRKuSh00DAN","unstructured":"G. Yu. Kulikov and S. K. Shindin, On multistep extrapolation methods for ordinary differential equations (in Russian), Dokl. Akad. Nauk, 372 (2000), No. 3, pp. 301\u2013304; translation in Dokl. Math., 61 (2000), No. 3, pp. 357\u2013360."},{"key":"14_CRKu02RJNAMM","unstructured":"G. Yu. Kulikov, On implicit extrapolation methods for ordinary differential equations, Russ. J. Numer. Anal. Math. Model., 17 (2002), No. 1, pp. 41\u201369."},{"key":"14_CRKuSh03ICCS","unstructured":"G. Yu. Kulikov, and S. K. Shindin, Extrapolated multistep methods and local-global step size control, in Computational Science\u2014ICCS 2003, International Conference, Melbourne, Australia and St. Petersburg, Russia, June 2\u20134, 2003, Proceedings, Part I, P. M. A. Sloot et al., eds., Lect. Notes Comp. Sci., Vol. 2657, Springer, Berlin, 2003, pp. 540\u2013549."},{"key":"14_CRKuSh04aZhVM","unstructured":"G. Yu. Kulikov and S. K. Shindin, On effective computation of asymptotically correct estimates of the local and global errors for multistep methods with fixed coefficients (in Russian), Zh. Vychisl. Mat. Mat. Fiz., 44 (2004), No. 5, pp. 847\u2013868; translation in Comput. Math. Math. Phys., 44 (2004), No. 5, pp. 794\u2013814."},{"key":"14_CRKuSh04bZhVM","unstructured":"G. Yu. Kulikov and S. K. Shindin, On interpolation-type multistep methods with automatic global error control (in Russian), Zh. Vychisl. Mat. Mat. Fiz., 44 (2004), No. 8, pp. 1400\u20131421; translation in Comput. Math. Math. Phys., 44 (2004), No. 8, pp. 1314\u20131333."},{"key":"14_CRKu04RJNAMM","unstructured":"G. Yu. Kulikov, One-step methods and implicit extrapolation technique for index 1 differential-algebraic systems, Russ. J. Numer. Anal. Math. Model., 19 (2004), No. 6, pp. 527\u2013553."},{"key":"14_CRKuShCMAM","unstructured":"G. Yu. Kulikov and S. K. Shindin, One-leg integration of ordinary differential equations with global error control, Comput. Methods Appl. Math., 5 (2005), No. 1, pp. 86\u201396."},{"key":"14_CRKuShSIAMJNA","unstructured":"G. Yu. Kulikov and S. K. Shindin, Variable-coefficient one-leg formulas for ordinary differential equations and local-global step size control, in preparation."},{"key":"14_CRKuSh","unstructured":"G. Yu. Kulikov and S. K. Shindin, Local and global error estimation in Nordsieck methods, in preparation."},{"key":"14_CRLa69","unstructured":"P. Lancaster, Theory of Matrices, Academic Press, New York and London, 1969."},{"key":"14_CRSk81","unstructured":"R. D. Skeel, A theoretical framework for proving accuracy results for deferred corrections, SIAM J. Numer. Anal., 19 (1981), No. 1, pp. 171\u2013196."},{"key":"14_CRSk86","doi-asserted-by":"crossref","unstructured":"R. D. Skeel, Thirteen ways to estimate global error, Numer. Math., 48 (1986), pp. 1\u201320.","DOI":"10.1007\/BF01389440"},{"key":"14_CRSk89","doi-asserted-by":"crossref","unstructured":"R. D. Skeel, Global error estimation and the backward differention formulas, Appl. Math. Comput., 5 (1989), pp. 197\u2013208.","DOI":"10.1016\/0096-3003(89)90119-7"},{"key":"14_CRSt78","doi-asserted-by":"crossref","unstructured":"H. J. Stetter, The defect correction principle and discretization methods, Numer. Math., 29 (1978), pp. 425\u2013443.","DOI":"10.1007\/BF01432879"},{"key":"14_CRSt82","doi-asserted-by":"crossref","unstructured":"H. J. Stetter, Global Error Estimation in ODE-Solvers, in Numerical of differential equations and large linear systems, Proceedings, Bielefeld 1980, J. Hinze, ed., Lect. Notes Math., Vol. 968, Springer, Berlin, 1982, pp. 269\u2013279.","DOI":"10.1007\/BFb0064894"}],"container-title":["BIT Numerical Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10543-005-0014-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10543-005-0014-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10543-005-0014-3","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,4,11]],"date-time":"2020-04-11T05:58:50Z","timestamp":1586584730000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10543-005-0014-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,9]]},"references-count":39,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2005,9]]}},"alternative-id":["14"],"URL":"https:\/\/doi.org\/10.1007\/s10543-005-0014-3","relation":{},"ISSN":["0006-3835","1572-9125"],"issn-type":[{"value":"0006-3835","type":"print"},{"value":"1572-9125","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,9]]}}}