{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:51:31Z","timestamp":1776721891012,"version":"3.51.2"},"reference-count":17,"publisher":"American Mathematical Society (AMS)","issue":"215","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    The construction of a Runge-Kutta pair of order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"5 left-parenthesis 4 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>5<\/mml:mn>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>4<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">5(4)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with the minimal number of stages requires the solution of a nonlinear system of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"25\">\n                        <mml:semantics>\n                          <mml:mn>25<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">25<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    order conditions in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"27\">\n                        <mml:semantics>\n                          <mml:mn>27<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">27<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    unknowns. We define a new family of pairs which includes pairs using\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"6\">\n                        <mml:semantics>\n                          <mml:mn>6<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">6<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    function evaluations per integration step as well as pairs which additionally use the first function evaluation from the next step. This is achieved by making use of Kutta\u2019s simplifying assumption on the original system of the order conditions, i.e., that all the internal nodes of a method contributing to the estimation of the endpoint solution provide, at these nodes, cost-free second-order approximations to the true solution of any differential equation. In both cases the solution of the resulting system of nonlinear equations is completely classified and described in terms of five free parameters. Optimal Runge-Kutta pairs with respect to minimized truncation error coefficients, maximal phase-lag order and various stability characteristics are presented. These pairs were selected under the assumption that they are used in Local Extrapolation Mode (the propagated solution of a problem is the one provided by the fifth-order formula of the pair). Numerical results obtained by testing the new pairs over a standard set of test problems suggest a significant improvement in efficiency when using a specific pair of the new family with minimized truncation error coefficients, instead of some other existing pairs.\n                  <\/p>","DOI":"10.1090\/s0025-5718-96-00718-1","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"1165-1181","source":"Crossref","is-referenced-by-count":15,"title":["A family of fifth-order Runge-Kutta pairs"],"prefix":"10.1090","volume":"65","author":[{"given":"S.","family":"Papakostas","sequence":"first","affiliation":[]},{"given":"G.","family":"Papageorgiou","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1017\/S1446788700023387","article-title":"On Runge-Kutta processes of high order","volume":"4","author":"Butcher, J. C.","year":"1964","journal-title":"J. Austral. Math. Soc."},{"key":"2","series-title":"A Wiley-Interscience Publication","isbn-type":"print","volume-title":"The numerical analysis of ordinary differential equations","author":"Butcher, J. C.","year":"1987","ISBN":"https:\/\/id.crossref.org\/isbn\/0471910465"},{"key":"3","doi-asserted-by":"publisher","first-page":"432","DOI":"10.1137\/0706038","article-title":"The complete solution of the fifth order Runge-Kutta equations","volume":"6","author":"Cassity, C. R.","year":"1969","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1","key":"4","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1093\/imamat\/16.1.35","article-title":"High-order explicit Runge-Kutta formulae, their uses, and limitations","volume":"16","author":"Curtis, A. R.","year":"1975","journal-title":"J. Inst. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0020-2932","issn-type":"print"},{"issue":"1","key":"5","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1016\/0771-050X(80)90013-3","article-title":"A family of embedded Runge-Kutta formulae","volume":"6","author":"Dormand, J. R.","year":"1980","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"key":"6","doi-asserted-by":"crossref","unstructured":"W. H. Enright and J. D. Pryce, Two FORTRAN packages for assessing initial value methods, ACM Trans. Math. Software 13 (1987), 1\u201327.","DOI":"10.1145\/23002.27645"},{"key":"7","unstructured":"E. Fehlberg, Classical fifth, sixth, seventh, and eighth order Runge-Kutta formulas with stepsize control, TR R\u2013287, NASA, 1968."},{"key":"8","unstructured":"\\bysame, Low order classical Runge-Kutta formulas with stepsize control and their application to some heat-transfer problems, TR R\u2013315, NASA, 1969."},{"key":"9","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-12607-3","volume-title":"Solving ordinary differential equations. I","volume":"8","author":"Hairer, E.","year":"1987","ISBN":"https:\/\/id.crossref.org\/isbn\/3540171452"},{"issue":"3","key":"10","doi-asserted-by":"publisher","first-page":"595","DOI":"10.1137\/0724041","article-title":"Explicit Runge-Kutta (-Nystr\u00f6m) methods with reduced phase errors for computing oscillating solutions","volume":"24","author":"van der Houwen, P. J.","year":"1987","journal-title":"SIAM J. Numer. 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Tsitouras, and S. N. Papakostas, Runge-Kutta pairs for periodic initial value problems, Rep. NA 93\u20131, Nat. Tech. Univ. Athens, Dept. Math., 1993.","DOI":"10.1007\/BF02243849"},{"key":"15","doi-asserted-by":"crossref","unstructured":"S. N. Papakostas, Ch. Tsitouras, and G. Papageorgiou, A general family of explicit Runge-Kutta pairs of orders 6(5), SIAM J. Numer. Anal. 33 (1996).","DOI":"10.1137\/0733046"},{"key":"16","doi-asserted-by":"publisher","first-page":"91","DOI":"10.2307\/2005249","article-title":"Local extrapolation in the solution of ordinary differential equations","volume":"27","author":"Shampine, L. F.","year":"1973","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"17","doi-asserted-by":"publisher","first-page":"21","DOI":"10.2307\/2004265","article-title":"Solutions of differential equations by evaluations of functions","volume":"20","author":"Shanks, E. Baylis","year":"1966","journal-title":"Math. 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