{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,9,10]],"date-time":"2024-09-10T04:40:47Z","timestamp":1725943247836},"reference-count":6,"publisher":"Association for Computing Machinery (ACM)","issue":"2","content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["J. ACM"],"published-print":{"date-parts":[[1964,4]]},"abstract":"\n The order\n p<\/jats:italic>\n which is obtainable with a stable\n k<\/jats:italic>\n -step method in the numerical solution of\n y\u2032<\/jats:italic>\n =\n f<\/jats:italic>\n (\n x<\/jats:italic>\n ,\n y<\/jats:italic>\n ) is limited to\n p<\/jats:italic>\n =\n k<\/jats:italic>\n + 1 by the theorems of Dahlquist. In the present paper the customary schemes are modified by including the value of the derivative at one \u201cnonstep point;\u201d as usual, this value is gained from an explicit predictor. It is shown that the order of these generalized predictor-corrector methods is not subject to the above restrictions; stable\n k<\/jats:italic>\n -step schemes with\n p<\/jats:italic>\n = 2\n k<\/jats:italic>\n + 2 have been constructed for\n k<\/jats:italic>\n \u2264 4. Furthermore it is proved that methods of order\n p<\/jats:italic>\n actually converge like\n \n h\n p<\/jats:sup>\n <\/jats:italic>\n uniformly in a given interval of integration. Numerical examples give some first evidence of the power of the new methods.\n <\/jats:p>","DOI":"10.1145\/321217.321223","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T11:26:10Z","timestamp":1027769170000},"page":"188-209","update-policy":"http:\/\/dx.doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":130,"title":["Generalized Multistep Predictor-Corrector Methods"],"prefix":"10.1145","volume":"11","author":[{"given":"William B.","family":"Gragg","sequence":"first","affiliation":[{"name":"Bellcomm Inc., Washington, D. C."}]},{"given":"Hans J.","family":"Stetter","sequence":"additional","affiliation":[{"name":"Technische Hochschule M\u00fcnchen, Germany"}]}],"member":"320","published-online":{"date-parts":[[1964,4]]},"reference":[{"key":"e_1_2_1_1_2","doi-asserted-by":"crossref","first-page":"33","DOI":"10.7146\/math.scand.a-10454","article-title":"Convergmlee and stability in the numerical integration of ordinary differential equations","volume":"4","year":"1956","journal-title":"Math. Scand"},{"key":"e_1_2_1_2_2","volume-title":"Wiley","author":"HENRIVCI P.","year":"1962"},{"key":"e_1_2_1_3_2","unstructured":"UTRABE: M. YANAWANA H. AND SmNOmaRA Y. Periodic solutions of van der Pol's eqmtion with damping eoeflieient X = 2 10. J. Sci. Hiroshima Univ. {A} 23 (1960) 325-366. See also UaABE M. Theory of errors in numerical integration of ordinary differential equations Teeh. Rep. 183 U. Wisconsin Math. Res. Center 1960 88p. UTRABE: M. YANAWANA H. AND SmNOmaRA Y. Periodic solutions of van der Pol's eqmtion with damping eoeflieient X = 2 10. J. Sci. Hiroshima Univ. {A} 23 (1960) 325-366. See also UaABE M. Theory of errors in numerical integration of ordinary differential equations Teeh. Rep. 183 U. Wisconsin Math. Res. Center 1960 88p."},{"key":"e_1_2_1_4_2","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1007\/BF01386033","article-title":"On the convergence of characteristic finite-difference methods of high accuracy for quasi-linear hyperbolic equations","volume":"3","author":"SERTTER H .","year":"1961","journal-title":"Numer. Math."},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1145\/321172.321176"},{"key":"e_1_2_1_6_2","volume-title":"McGraw-Hill","author":"HILDEBRAND F .","year":"1956"}],"container-title":["Journal of the ACM"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/321217.321223","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,31]],"date-time":"2022-12-31T18:38:17Z","timestamp":1672511897000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/321217.321223"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1964,4]]},"references-count":6,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1964,4]]}},"alternative-id":["10.1145\/321217.321223"],"URL":"https:\/\/doi.org\/10.1145\/321217.321223","relation":{},"ISSN":["0004-5411","1557-735X"],"issn-type":[{"value":"0004-5411","type":"print"},{"value":"1557-735X","type":"electronic"}],"subject":[],"published":{"date-parts":[[1964,4]]},"assertion":[{"value":"1964-04-01","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}