{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,22]],"date-time":"2025-11-22T11:34:47Z","timestamp":1763811287950,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,12,6]],"date-time":"2023-12-06T00:00:00Z","timestamp":1701820800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Two-Derivative Runge\u2013Kutta methods have been proposed by Chan and Tsai in 2010 and order conditions up to the fifth order are given. In this work, for the first time, we derive order conditions for order six. Simplifying assumptions that reduce the number of order conditions are also given. The procedure for constructing sixth-order methods is presented. A specific method is derived in order to illustrate the procedure; this method is of the sixth algebraic order with a reduced phase-lag and amplification error. For numerical comparison, five well-known test problems have been solved using a seventh-order Two-Derivative Runge\u2013Kutta method developed by Chan and Tsai and several Runge\u2013Kutta methods of orders 6 and 8. Diagrams of the maximum absolute error vs. computation time show the efficiency of the new method.<\/jats:p>","DOI":"10.3390\/a16120558","type":"journal-article","created":{"date-parts":[[2023,12,6]],"date-time":"2023-12-06T08:48:32Z","timestamp":1701852512000},"page":"558","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Construction of Two-Derivative Runge\u2013Kutta Methods of Order Six"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4840-1810","authenticated-orcid":false,"given":"Zacharoula","family":"Kalogiratou","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Western Macedonia, 50100 Kozani, Greece"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1679-1962","authenticated-orcid":false,"given":"Theodoros","family":"Monovasilis","sequence":"additional","affiliation":[{"name":"Department of Economics, University of Western Macedonia, 50100 Kozani, Greece"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1016\/S0377-0427(02)00475-2","article-title":"Optimized Runge-Kutta pairs for problems with oscillating solutions","volume":"147","author":"Tsitouras","year":"2002","journal-title":"J. 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