{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T01:59:28Z","timestamp":1777687168762,"version":"3.51.4"},"reference-count":22,"publisher":"American Mathematical Society (AMS)","issue":"232","license":[{"start":{"date-parts":[[2000,8,17]],"date-time":"2000-08-17T00:00:00Z","timestamp":966470400000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    This paper continues earlier work by the same author concerning the stability and B-convergence properties of multistep Runge-Kutta methods for the numerical solution of nonlinear stiff initial-value problems in a Hilbert space. A series of sufficient conditions and necessary conditions for a multistep Runge-Kutta method to be algebraically stable, diagonally stable,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper B\">\n                        <mml:semantics>\n                          <mml:mi>B<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">B<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    - or optimally\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper B\">\n                        <mml:semantics>\n                          <mml:mi>B<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">B<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -convergent are established, by means of which six classes of high order algebraically stable and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper B\">\n                        <mml:semantics>\n                          <mml:mi>B<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">B<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -convergent multistep Runge-Kutta methods are constructed in a unified pattern. These methods include the class constructed by Burrage in 1987 as special case, and most of them can be regarded as extension of the Gauss, RadauIA, RadauIIA and LobattoIIIC Runge-Kutta methods. We find that the classes of multistep Runge-Kutta methods constructed in the present paper are superior in many respects to the corresponding existing one-step Runge-Kutta schemes.\n                  <\/p>","DOI":"10.1090\/s0025-5718-99-01159-x","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:17:31Z","timestamp":1027707451000},"page":"1481-1504","source":"Crossref","is-referenced-by-count":17,"title":["Stability and \ud835\udc35-convergence properties of multistep Runge-Kutta methods"],"prefix":"10.1090","volume":"69","author":[{"given":"Shoufu","family":"Li","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[1999,8,17]]},"reference":[{"issue":"3","key":"1","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1007\/BF02262216","article-title":"A note on convergence concepts for stiff problems","volume":"44","author":"Auzinger, W.","year":"1990","journal-title":"Computing","ISSN":"https:\/\/id.crossref.org\/issn\/0010-485X","issn-type":"print"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1016\/0168-9274(92)90008-2","article-title":"An extension of \ud835\udc35-convergence for Runge-Kutta methods","volume":"9","author":"Auzinger, W.","year":"1992","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"issue":"1-2","key":"3","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1016\/0377-0427(93)90260-I","article-title":"Modern convergence theory for stiff initial value problems","volume":"45","author":"Auzinger, W.","year":"1993","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"4","key":"4","doi-asserted-by":"publisher","first-page":"635","DOI":"10.1007\/BF01733784","article-title":"Extending convergence theory for nonlinear stiff problems. I","volume":"36","author":"Auzinger, W.","year":"1996","journal-title":"BIT","ISSN":"https:\/\/id.crossref.org\/issn\/0006-3835","issn-type":"print"},{"issue":"1","key":"5","doi-asserted-by":"publisher","first-page":"106","DOI":"10.1137\/0724009","article-title":"High order algebraically stable multistep Runge-Kutta methods","volume":"24","author":"Burrage, K.","year":"1987","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"2","key":"6","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1007\/BF01933191","article-title":"Nonlinear stability of a general class of differential equation methods","volume":"20","author":"Burrage, Kevin","year":"1980","journal-title":"BIT","ISSN":"https:\/\/id.crossref.org\/issn\/0006-3835","issn-type":"print"},{"key":"7","doi-asserted-by":"crossref","unstructured":"J.C. Butcher, A stability property of implicit Runge-Kutta methods, BIT 15(1975), 358\u2013361.","DOI":"10.1007\/BF01931672"},{"issue":"3","key":"8","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1007\/bf01932265","article-title":"On the implementation of implicit Runge-Kutta methods","volume":"16","author":"Butcher, J. C.","year":"1976","journal-title":"Nordisk Tidskr. Informationsbehandling (BIT)","ISSN":"https:\/\/id.crossref.org\/issn\/0901-246X","issn-type":"print"},{"key":"9","first-page":"60","article-title":"Error analysis for a class of methods for stiff non-linear initial value problems","author":"Dahlquist, Germund","year":"1976"},{"issue":"5","key":"10","doi-asserted-by":"publisher","first-page":"753","DOI":"10.1137\/0718051","article-title":"The concept of \ud835\udc35-convergence","volume":"18","author":"Frank, Reinhard","year":"1981","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"11","doi-asserted-by":"publisher","first-page":"497","DOI":"10.1137\/0722030","article-title":"Stability properties of implicit Runge-Kutta methods","volume":"22","author":"Frank, Reinhard","year":"1985","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"12","doi-asserted-by":"publisher","first-page":"497","DOI":"10.1137\/0722030","article-title":"Stability properties of implicit Runge-Kutta methods","volume":"22","author":"Frank, Reinhard","year":"1985","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"13","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-09947-6","volume-title":"Solving ordinary differential equations. II","volume":"14","author":"Hairer, E.","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/3540537759"},{"key":"14","unstructured":"Shoufu Li, \ud835\udc35-convergence of general linear methods, Proc. BAIL-V International conference, Shanghai, 1988, 203\u2014208, Boole Press Conf. Ser. 12, 1988."},{"key":"15","doi-asserted-by":"crossref","unstructured":"Shoufu Li, Stability and \ud835\udc35-convergence of general linear methods, Proc. 3rd International Congress on Comput. Appl. Math., Belgium, 1988. J. Comput. Appl. Math. 28(1989), 281\u2014296.","DOI":"10.1016\/0377-0427(89)90340-3"},{"key":"16","unstructured":"Shoufu Li, \ud835\udc35-convergence of General Multivalue Methods Applied to Stiff Problems in Banach Spaces, Proc. National Conf. on Comput. Math., Tianjin, 1990. SCIENCE IN CHINA (Series A), Chinese Series: 1992, 5:476-485, English Series: 36(1993), 1:1-13."},{"key":"17","unstructured":"Shoufu Li, Theory of Computational Methods for Stiff Differential Equations, Science and Technology Press, Hunan, China, 1997."},{"key":"18","unstructured":"Shoufu Li, Multistep Runge-Kutta methods with real eigenvalues and its parallel implementation, Proc. 5-th CSIAM Conference, Qinghua University Publishing House, 1998."},{"key":"19","unstructured":"Shoufu Li, High order algebraically stable and \ud835\udc35-convergent multistep Runge-Kutta methods with real eigenvalues, To appear."},{"key":"20","doi-asserted-by":"crossref","unstructured":"Shoufu Li, \ud835\udc35-convergence properties of multistep Runge-Kutta methods, Research Report, International Conference on Sci. Comput., Hangzhou, 1991. Math. 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