{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,27]],"date-time":"2025-10-27T21:07:16Z","timestamp":1761599236175,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,4]],"date-time":"2022-08-04T00:00:00Z","timestamp":1659571200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100006769","name":"Russian Science Foundation (RSF)","doi-asserted-by":"publisher","award":["22-19-00573"],"award-info":[{"award-number":["22-19-00573"]}],"id":[{"id":"10.13039\/501100006769","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Composition is a powerful and simple approach for obtaining numerical integration methods of high accuracy order while preserving the geometric properties of a basic integrator. Adaptive step size control allows one to significantly increase the performance of numerical integration methods. However, there is a lack of efficient step size control algorithms for composition solvers due to some known difficulties in constructing a low-cost embedded local error estimator. In this paper, we propose a novel local error estimator based on a difference between the semi-implicit CD method and semi-explicit midpoint methods within a common composition scheme. We evaluate the performance of adaptive composition schemes with the proposed local error estimator, comparing it with the other state-of-the-art approaches. We show that composition ODE solvers with the proposed step size control algorithm possess higher numerical efficiency than known methods, by using a comprehensive set of nonlinear test problems.<\/jats:p>","DOI":"10.3390\/a15080275","type":"journal-article","created":{"date-parts":[[2022,8,4]],"date-time":"2022-08-04T21:52:48Z","timestamp":1659649968000},"page":"275","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["New Step Size Control Algorithm for Semi-Implicit Composition ODE Solvers"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9878-4568","authenticated-orcid":false,"given":"Petr","family":"Fedoseev","sequence":"first","affiliation":[{"name":"Youth Research Institute, Saint Petersburg Electrotechnical University \u201cLETI\u201d, 197376 Saint Petersburg, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5119-2274","authenticated-orcid":false,"given":"Dmitriy","family":"Pesterev","sequence":"additional","affiliation":[{"name":"Youth Research Institute, Saint Petersburg Electrotechnical University \u201cLETI\u201d, 197376 Saint Petersburg, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2591-0962","authenticated-orcid":false,"given":"Artur","family":"Karimov","sequence":"additional","affiliation":[{"name":"Department of Computer-Aided Design, Saint Petersburg Electrotechnical University \u201cLETI\u201d, 197376 Saint Petersburg, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8941-4220","authenticated-orcid":false,"given":"Denis","family":"Butusov","sequence":"additional","affiliation":[{"name":"Youth Research Institute, Saint Petersburg Electrotechnical University \u201cLETI\u201d, 197376 Saint Petersburg, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"597","DOI":"10.1080\/10556780500140664","article-title":"Derivation of symmetric composition constants for symmetric integrators","volume":"20","author":"Sofroniou","year":"2005","journal-title":"Optim. Methods Softw."},{"key":"ref_2","unstructured":"Haier, E., Lubich, C., and Wanner, G. (2006). Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Leimkuhler, B., and Reich, S. (2004). Simulating Hamiltonian Dynamics (No. 14), Cambridge University Press.","DOI":"10.1017\/CBO9780511614118"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"204113","DOI":"10.1063\/1.5094046","article-title":"Efficient geometric integrators for nonadiabatic quantum dynamics. II. The diabatic representation","volume":"150","author":"Roulet","year":"2019","journal-title":"J. Chem. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"262","DOI":"10.1016\/0375-9601(90)90092-3","article-title":"Construction of higher order symplectic integrators","volume":"150","author":"Yoshida","year":"1990","journal-title":"Phys. Lett. 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Nonstiff Problems, Springer.","DOI":"10.1007\/978-3-662-12607-3"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Butusov, D. (2021). Adaptive Stepsize Control for Extrapolation Semi-Implicit Multistep ODE Solvers. Mathematics, 9.","DOI":"10.3390\/math9090950"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Butusov, D., Tutueva, A., Fedoseev, P., Terentev, A., and Karimov, A. (2020). Semi-implicit multistep extrapolation ODE solvers. Mathematics, 8.","DOI":"10.3390\/math8060943"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"S\u00fcli, E., and David, F.M. (2003). An Introduction to Numerical Analysis, Cambridge University Press.","DOI":"10.1017\/CBO9780511801181"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1007\/BF01462235","article-title":"Extrapolation at stiff differential equations","volume":"52","author":"Hairer","year":"1988","journal-title":"Numer. 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