{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:22:41Z","timestamp":1776784961143,"version":"3.51.2"},"reference-count":11,"publisher":"American Mathematical Society (AMS)","issue":"254","license":[{"start":{"date-parts":[[2006,12,19]],"date-time":"2006-12-19T00:00:00Z","timestamp":1166486400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We consider the numerical solution of systems of semi-explicit index\n \n \n \n 2<\/mml:mn>\n 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n differential-algebraic equations (DAEs) by methods based on Runge-Kutta (RK) coefficients. For nonstiffly accurate RK coefficients, such as Gauss and Radau IA coefficients, the standard application of implicit RK methods is generally not superconvergent. To reestablish superconvergence projected RK methods and partitioned RK methods have been proposed. In this paper we propose a simple alternative which does not require any extra projection step and does not use any additional internal stage. Moreover, symmetry of Gauss methods is preserved. The main idea is to replace the satisfaction of the constraints at the internal stages in the standard definition by enforcing specific linear combinations of the constraints at the numerical solution and at the internal stages to vanish. We call these methods\n \n specialized Runge-Kutta methods for index\n \n \n \n 2<\/mml:mn>\n 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n DAEs (SRK-DAE\u00a0\n \n \n \n 2<\/mml:mn>\n 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n )\n <\/italic>\n .\n <\/p>","DOI":"10.1090\/s0025-5718-05-01809-0","type":"journal-article","created":{"date-parts":[[2006,2,15]],"date-time":"2006-02-15T11:05:20Z","timestamp":1140001520000},"page":"641-654","source":"Crossref","is-referenced-by-count":9,"title":["Specialized Runge-Kutta methods for index 2 differential-algebraic equations"],"prefix":"10.1090","volume":"75","author":[{"given":"Laurent","family":"Jay","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2005,12,19]]},"reference":[{"issue":"4","key":"1","doi-asserted-by":"publisher","first-page":"1097","DOI":"10.1137\/0728059","article-title":"Projected implicit Runge-Kutta methods for differential-algebraic equations","volume":"28","author":"Ascher, Uri M.","year":"1991","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"2","series-title":"Classics in Applied Mathematics","isbn-type":"print","volume-title":"Numerical solution of initial-value problems in differential-algebraic equations","volume":"14","author":"Brenan, K. E.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0898713536"},{"issue":"1-3","key":"3","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1016\/S0168-9274(01)00143-X","article-title":"Post-projected Runge-Kutta methods for index-2 differential-algebraic equations","volume":"42","author":"Chan, R. P. K.","year":"2002","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"issue":"3","key":"4","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1007\/s00211-003-0499-0","article-title":"Reversible methods of Runge-Kutta type for index-2 DAEs","volume":"97","author":"Chan, R. P. K.","year":"2004","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"5","isbn-type":"print","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1142\/9789812798886_0017","article-title":"Implicit Runge-Kutta methods for higher index differential-algebraic systems","author":"Hairer, E.","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/9810214375"},{"key":"6","series-title":"Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0093947","volume-title":"The numerical solution of differential-algebraic systems by Runge-Kutta methods","volume":"1409","author":"Hairer, Ernst","year":"1989","ISBN":"https:\/\/id.crossref.org\/isbn\/3540518606"},{"key":"7","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-05221-7","volume-title":"Solving ordinary differential equations. II","volume":"14","author":"Hairer, E.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/3540604529","edition":"2"},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1007\/BF01990349","article-title":"Convergence of a class of Runge-Kutta methods for differential-algebraic systems of index 2","volume":"33","author":"Jay, Laurent","year":"1993","journal-title":"BIT","ISSN":"https:\/\/id.crossref.org\/issn\/0006-3835","issn-type":"print"},{"issue":"1","key":"9","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1023\/A:1023696822355","article-title":"Solution of index 2 implicit differential-algebraic equations by Lobatto Runge-Kutta methods","volume":"43","author":"Jay, L. O.","year":"2003","journal-title":"BIT","ISSN":"https:\/\/id.crossref.org\/issn\/0006-3835","issn-type":"print"},{"issue":"3","key":"10","doi-asserted-by":"publisher","first-page":"545","DOI":"10.1007\/BF01933267","article-title":"On projected Runge-Kutta methods for differential-algebraic equations","volume":"31","author":"Lubich, Ch.","year":"1991","journal-title":"BIT","ISSN":"https:\/\/id.crossref.org\/issn\/0006-3835","issn-type":"print"},{"key":"11","doi-asserted-by":"crossref","unstructured":"A. Murua, Partitioned Runge-Kutta methods for semi-explicit differential-algebraic systems of index 2, Tech. Report EHU-KZAA-IKT-196, Univ. of the Basque country, 1996.","DOI":"10.1007\/BF02684403"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2006-75-254\/S0025-5718-05-01809-0\/S0025-5718-05-01809-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-254\/S0025-5718-05-01809-0\/S0025-5718-05-01809-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:34:22Z","timestamp":1776782062000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-254\/S0025-5718-05-01809-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,12,19]]},"references-count":11,"journal-issue":{"issue":"254","published-print":{"date-parts":[[2006,4]]}},"alternative-id":["S0025-5718-05-01809-0"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-05-01809-0","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2005,12,19]]}}}