{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,7]],"date-time":"2026-01-07T23:01:39Z","timestamp":1767826899835,"version":"3.49.0"},"reference-count":64,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2018,6,1]],"date-time":"2018-06-01T00:00:00Z","timestamp":1527811200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In recent years, the numerical solution of differential problems, possessing constants of motion, has been attacked by imposing the vanishing of a corresponding line integral. The resulting methods have been, therefore, collectively named (discrete) line integral methods, where it is taken into account that a suitable numerical quadrature is used. The methods, at first devised for the numerical solution of Hamiltonian problems, have been later generalized along several directions and, actually, the research is still very active. In this paper we collect the main facts about line integral methods, also sketching various research trends, and provide a comprehensive set of references.<\/jats:p>","DOI":"10.3390\/axioms7020036","type":"journal-article","created":{"date-parts":[[2018,6,1]],"date-time":"2018-06-01T04:37:46Z","timestamp":1527827866000},"page":"36","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":43,"title":["Line Integral Solution of Differential Problems"],"prefix":"10.3390","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6290-4107","authenticated-orcid":false,"given":"Luigi","family":"Brugnano","sequence":"first","affiliation":[{"name":"Dipartimento di Matematica e Informatica \u201cU. Dini\u201d, Universit\u00e0 di Firenze, Viale Morgagni 67\/A, 50134 Firenze, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Felice","family":"Iavernaro","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica, Universit\u00e0 di Bari, Via Orabona 4, 70125 Bari, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,6,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Brugnano, L., and Iavernaro, F. (2016). Line Integral Methods for Conservative Problems, Chapman et Hall\/CRC.","DOI":"10.1201\/b19319"},{"key":"ref_2","unstructured":"Hairer, E., Lubich, C., and Wanner, G. (2006). Geometric Numerical Integration, Springer. [2nd ed.]."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Leimkuhler, B., and Reich, S. (2004). 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