{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T07:50:49Z","timestamp":1740124249368,"version":"3.37.3"},"reference-count":8,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,1,5]],"date-time":"2023-01-05T00:00:00Z","timestamp":1672876800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,1,5]],"date-time":"2023-01-05T00:00:00Z","timestamp":1672876800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100009934","name":"E\u00f6tv\u00f6s Lor\u00e1nd University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100009934","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Period Math Hung"],"published-print":{"date-parts":[[2023,9]]},"abstract":"Abstract<\/jats:title>In the numerical integration of nonlinear autonomous initial value problems, the computational process depends on the step size scaled vector field hf<\/jats:italic> as a distinct entity. This paper considers a parameterized transformation $$\\begin{aligned} hf \\mapsto hf \\circ (I-\\gamma hf)^{-1}, \\end{aligned}$$<\/jats:tex-math>\n \n \n \n \n \n h<\/mml:mi>\n f<\/mml:mi>\n \u21a6<\/mml:mo>\n h<\/mml:mi>\n f<\/mml:mi>\n \u2218<\/mml:mo>\n \n \n (<\/mml:mo>\n I<\/mml:mi>\n -<\/mml:mo>\n \u03b3<\/mml:mi>\n h<\/mml:mi>\n f<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n ,<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mtd>\n <\/mml:mtr>\n <\/mml:mtable>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:disp-formula>and its role in the finite step size stability of singly diagonally implicit Runge\u2014Kutta (SDIRK) methods. For a suitably chosen $$\\gamma > 0$$<\/jats:tex-math>\n \n \u03b3<\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, the transformed map is Lipschitz continuous with a reasonably small constant, whenever hf<\/jats:italic> is negative monotone. With this transformation, an SDIRK method is equivalent to an explicit Runge\u2013Kutta (ERK) method applied to the transformed vector field. Through this mapping, the SDIRK methods\u2019 A-stability, and linear order conditions are investigated. The latter are closely related to approximations of the exponential function $$\\textrm{e}^z$$<\/jats:tex-math>\n \n e<\/mml:mtext>\n z<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> that are polynomial in z<\/jats:italic>, when considering ERK methods, and polynomial in terms of the transformed variable $$z(1-\\gamma z)^{-1}$$<\/jats:tex-math>\n \n z<\/mml:mi>\n \n \n (<\/mml:mo>\n 1<\/mml:mn>\n -<\/mml:mo>\n \u03b3<\/mml:mi>\n z<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, in case of SDIRK methods. Considering the second family of methods, and expanding the exponential function in terms of this transformed variable, the coefficients can be expressed in terms of Laguerre polynomials. Lastly, a family of methods is constructed using the transformed vector field, and its order conditions, A-stability, and B-stability are investigated.<\/jats:p>","DOI":"10.1007\/s10998-022-00510-5","type":"journal-article","created":{"date-parts":[[2023,1,5]],"date-time":"2023-01-05T18:02:42Z","timestamp":1672941762000},"page":"167-181","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Runge\u2013Kutta\u2013M\u00f6bius methods"],"prefix":"10.1007","volume":"87","author":[{"given":"Andr\u00e1s","family":"Moln\u00e1r","sequence":"first","affiliation":[]},{"given":"Imre","family":"Fekete","sequence":"additional","affiliation":[]},{"given":"Gustaf","family":"S\u00f6derlind","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,1,5]]},"reference":[{"key":"510_CR1","doi-asserted-by":"publisher","first-page":"358","DOI":"10.1007\/BF01931672","volume":"15","author":"JC Butcher","year":"1975","unstructured":"J.C. Butcher, A stability property of implicit Runge-Kutta methods. BIT 15, 358\u2013361 (1975)","journal-title":"BIT"},{"key":"510_CR2","doi-asserted-by":"crossref","unstructured":"J. C. Butcher, Numerical methods for ordinary differential equations, 3rd ed., Wiley (2016)","DOI":"10.1002\/9781119121534"},{"key":"510_CR3","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1007\/BF01963532","volume":"3","author":"G Dahlquist","year":"1963","unstructured":"G. Dahlquist, A special stability problem for linear multistep methods. BIT 3, 27\u201343 (1963)","journal-title":"BIT"},{"key":"510_CR4","volume-title":"Solving Ordinary Differential Equations I: Nonstiff Problems, Springer Series in Computational Mathematics","author":"E Hairer","year":"1993","unstructured":"E. Hairer, S. Norsett, G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer Series in Computational Mathematics, 2nd edn. (Springer-Verlag, Berlin Heidelberg, 1993)","edition":"2"},{"key":"510_CR5","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-05221-7","volume-title":"Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics","author":"E Hairer","year":"1996","unstructured":"E. Hairer, G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics, 2nd edn. (Springer-Verlag, Berlin Heidelberg, 1996)","edition":"2"},{"key":"510_CR6","unstructured":"C. A. Kennedy and M. H. Carpenter, Diagonally implicit Runge\u2013Kutta methods for ordinary differential equations, A Review. NASA Report. Langley research center. Hampton VA 23681 (2016)"},{"key":"510_CR7","doi-asserted-by":"publisher","first-page":"3388","DOI":"10.22436\/jnsa.009.05.124","volume":"9","author":"WK Shao","year":"2016","unstructured":"W.K. Shao, Y. He, J. Pan, Some identities for the generalized Laguerre polynomials. J Nonlinear Sci Appl 9, 3388\u20133396 (2016)","journal-title":"J Nonlinear Sci Appl"},{"key":"510_CR8","first-page":"631","volume":"46","author":"G S\u00f6derlind","year":"2006","unstructured":"G. S\u00f6derlind, The logarithmic norm. History and modern theory, BIT 46, 631\u2013652 (2006)","journal-title":"History and modern theory, BIT"}],"container-title":["Periodica Mathematica Hungarica"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10998-022-00510-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10998-022-00510-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10998-022-00510-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,8,25]],"date-time":"2023-08-25T16:09:07Z","timestamp":1692979747000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10998-022-00510-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,5]]},"references-count":8,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,9]]}},"alternative-id":["510"],"URL":"https:\/\/doi.org\/10.1007\/s10998-022-00510-5","relation":{},"ISSN":["0031-5303","1588-2829"],"issn-type":[{"type":"print","value":"0031-5303"},{"type":"electronic","value":"1588-2829"}],"subject":[],"published":{"date-parts":[[2023,1,5]]},"assertion":[{"value":"1 July 2022","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 January 2023","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}