{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T00:32:49Z","timestamp":1767141169762,"version":"build-2238731810"},"reference-count":20,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,4,10]],"date-time":"2025-04-10T00:00:00Z","timestamp":1744243200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,4,10]],"date-time":"2025-04-10T00:00:00Z","timestamp":1744243200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100011033","name":"Agencia Estatal de Investigaci\u00f3n","doi-asserted-by":"publisher","award":["PID2022-136585NB-C22"],"award-info":[{"award-number":["PID2022-136585NB-C22"]}],"id":[{"id":"10.13039\/501100011033","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100011033","name":"Agencia Estatal de Investigaci\u00f3n","doi-asserted-by":"publisher","award":["PID2022-136585NB-C22"],"award-info":[{"award-number":["PID2022-136585NB-C22"]}],"id":[{"id":"10.13039\/501100011033","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100011033","name":"Agencia Estatal de Investigaci\u00f3n","doi-asserted-by":"publisher","award":["PID2022-136585NB-C22"],"award-info":[{"award-number":["PID2022-136585NB-C22"]}],"id":[{"id":"10.13039\/501100011033","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100014180","name":"Junta de Castilla y Le\u00f3n","doi-asserted-by":"publisher","award":["VA169P20"],"award-info":[{"award-number":["VA169P20"]}],"id":[{"id":"10.13039\/501100014180","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100015866","name":"Hezkuntza, Hizkuntza Politika Eta Kultura Saila, Eusko Jaurlaritza","doi-asserted-by":"crossref","award":["ITI456-22"],"award-info":[{"award-number":["ITI456-22"]}],"id":[{"id":"10.13039\/100015866","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/100015866","name":"Hezkuntza, Hizkuntza Politika Eta Kultura Saila, Eusko Jaurlaritza","doi-asserted-by":"crossref","award":["ITI456-22"],"award-info":[{"award-number":["ITI456-22"]}],"id":[{"id":"10.13039\/100015866","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2025,5]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    In this paper, we introduce an algorithm that provides approximate solutions to semi-linear ordinary differential equations with highly oscillatory solutions, which, after an appropriate change of variables, can be rewritten as non-autonomous systems with a\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$(2\\pi \/\\omega )$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>\u03c0<\/mml:mi>\n                            <mml:mo>\/<\/mml:mo>\n                            <mml:mi>\u03c9<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -periodic dependence on\n                    <jats:italic>t<\/jats:italic>\n                    . The proposed approximate solutions are given in closed form as functions\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$X(\\omega t,t)$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>X<\/mml:mi>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>\u03c9<\/mml:mi>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , where\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$X(\\theta ,t)$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>X<\/mml:mi>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>\u03b8<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    is (i) a truncated Fourier series in\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\theta $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>\u03b8<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    for fixed\n                    <jats:italic>t<\/jats:italic>\n                    and (ii) a truncated Taylor series in\n                    <jats:italic>t<\/jats:italic>\n                    for fixed\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\theta $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>\u03b8<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , which motivates the name of the method. These approximations are uniformly accurate in\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\omega $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>\u03c9<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , meaning that their accuracy does not degrade as\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\omega \\rightarrow \\infty $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>\u03c9<\/mml:mi>\n                            <mml:mo>\u2192<\/mml:mo>\n                            <mml:mi>\u221e<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . In addition, Taylor\u2013Fourier approximations enable the computation of high-order averaging equations for the original semi-linear system, as well as related maps that are particularly useful in the highly oscillatory regime (i.e., for sufficiently large\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\omega $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>\u03c9<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ). The main goal of this paper is to develop an efficient procedure for computing such approximations by combining truncated power series arithmetic with the Fast Fourier Transform (FFT). We present numerical experiments that illustrate the effectiveness of the proposed method, including applications to the nonlinear Schr\u00f6dinger equation with non-smooth initial data and a perturbed Kepler problem from satellite orbit dynamics.\n                  <\/jats:p>","DOI":"10.1007\/s10915-025-02880-7","type":"journal-article","created":{"date-parts":[[2025,4,10]],"date-time":"2025-04-10T04:49:29Z","timestamp":1744260569000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Taylor\u2013Fourier Approximation"],"prefix":"10.1007","volume":"103","author":[{"given":"M. P.","family":"Calvo","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7453-4844","authenticated-orcid":false,"given":"J.","family":"Makazaga","sequence":"additional","affiliation":[]},{"given":"A.","family":"Murua","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,4,10]]},"reference":[{"key":"2880_CR1","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1137\/141000671","volume":"59","author":"J Bezanson","year":"2017","unstructured":"Bezanson, J., Edelman, A., Karpimski, S., Shah, V.B.: Julia: a fresh approach to numerical computing. SIAM Rev. 59, 65\u201398 (2017)","journal-title":"SIAM Rev."},{"key":"2880_CR2","doi-asserted-by":"publisher","first-page":"020003","DOI":"10.1063\/1.4992149","volume":"1863","author":"L Brugnano","year":"2017","unstructured":"Brugnano, L., Calvo, M., Montijano, J.I., R\u00e1ndez, L.: Fourier methods for oscillatory differential problems with a constant high frequency. AIP Confer. Proc. 1863, 020003 (2017)","journal-title":"AIP Confer. Proc."},{"key":"2880_CR3","doi-asserted-by":"publisher","first-page":"26","DOI":"10.1016\/j.apnum.2015.01.001","volume":"91","author":"B Cano","year":"2015","unstructured":"Cano, B., Gonz\u00e1lez-Pach\u00f3n, A.: Exponential time integration of solitary waves of cubic Schr\u00f6dinger equation. Appl. Numer. Math. 91, 26\u201345 (2015)","journal-title":"Appl. Numer. Math."},{"key":"2880_CR4","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1007\/s10208-012-9118-8","volume":"12","author":"Ph Chartier","year":"2012","unstructured":"Chartier, Ph., Murua, A., Sanz-Serna, J.M.: Higher-order averaging, formal series and numerical integration II: the quasi-periodic case. Found. Comput. Math. 12, 471\u2013508 (2012)","journal-title":"Found. Comput. Math."},{"key":"2880_CR5","doi-asserted-by":"publisher","first-page":"3009","DOI":"10.3934\/dcds.2012.32.3009","volume":"32","author":"P Chartier","year":"2012","unstructured":"Chartier, P., Murua, A., Sanz-Serna, J.M.: A formal series approach to averaging: exponentially small error estimates. Discret. Contin. Dyn. Syst. 32, 3009\u20133027 (2012)","journal-title":"Discret. Contin. Dyn. Syst."},{"key":"2880_CR6","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1007\/s00211-014-0638-9","volume":"129","author":"P Chartier","year":"2015","unstructured":"Chartier, P., Crouseilles, N., Lemou, M., M\u00e9hats, F.: Uniformly accurate numerical schemes for highly oscillatory Klein\u2013Gordon and nonlinear Schr\u00f6dinger equations. Numer. Math. 129, 211\u2013250 (2015)","journal-title":"Numer. Math."},{"key":"2880_CR7","doi-asserted-by":"publisher","first-page":"991","DOI":"10.3934\/dcds.2014.34.991","volume":"34","author":"G Chen","year":"2014","unstructured":"Chen, G., Olver, P.J.: Numerical simulation of nonlinear dispersive quantization. Discret. Contin. Dyn. Syst. 34, 991\u20131008 (2014)","journal-title":"Discret. Contin. Dyn. Syst."},{"key":"2880_CR8","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1007\/s10208-002-0062-x","volume":"3","author":"D Cohen","year":"2003","unstructured":"Cohen, D., Hairer, E., Lubich, Ch.: Modulated Fourier expansions of highly oscillatory differential equations. Found. Comput. Math. 3, 327\u2013345 (2003)","journal-title":"Found. Comput. Math."},{"key":"2880_CR9","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1016\/0771-050X(80)90013-3","volume":"6","author":"JR Dormand","year":"1980","unstructured":"Dormand, J.R., Prince, P.J.: A family of embedded Runge\u2013Kutta formulae. J. Comput. Appl. Math. 6, 19\u201326 (1980)","journal-title":"J. Comput. Appl. Math."},{"key":"2880_CR10","unstructured":"Hairer, E., Lubich, Ch., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer (2006)"},{"key":"2880_CR11","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1017\/S0962492910000048","volume":"19","author":"M Hochbruck","year":"2010","unstructured":"Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209\u2013286 (2010)","journal-title":"Acta Numer."},{"key":"2880_CR12","doi-asserted-by":"publisher","first-page":"204","DOI":"10.1515\/crll.1965.218.204","volume":"218","author":"P Kustaanheimo","year":"1965","unstructured":"Kustaanheimo, P., Stiefel, E.: Perturbation theory of Kepler motion based on spinor regularization. J. Reine Angew. Math. 218, 204\u2013219 (1965)","journal-title":"J. Reine Angew. Math."},{"key":"2880_CR13","doi-asserted-by":"publisher","first-page":"372","DOI":"10.1137\/0704033","volume":"4","author":"JD Lawson","year":"1967","unstructured":"Lawson, J.D.: Generalized Runge\u2013Kutta processes for stable systems with large Lipschitz constants. SIAM J. Numer. Anal. 4, 372\u2013380 (1967)","journal-title":"SIAM J. Numer. Anal."},{"key":"2880_CR14","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-58351-3","volume-title":"Satellite Orbits . Models Methods and Applications","author":"O Montenbruck","year":"2000","unstructured":"Montenbruck, O., Gill, E.: Satellite Orbits . Models Methods and Applications. Springer, Berlin (2000)"},{"key":"2880_CR15","doi-asserted-by":"publisher","first-page":"731","DOI":"10.1007\/s10208-017-9352-1","volume":"18","author":"A Ostermann","year":"2018","unstructured":"Ostermann, A., Schratz, K.: Low regularity exponential-type integrators for semilinear Schr\u00f6dinger equations. Found. Comput. Math. 18, 731\u2013755 (2018)","journal-title":"Found. Comput. Math."},{"key":"2880_CR16","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1007\/s10915-022-01786-y","volume":"91","author":"A Ostermann","year":"2022","unstructured":"Ostermann, A., Yao, F.: A fully discrete low-regularity integrator for the nonlinear Schr\u00f6dinger equation. J. Sci. Comput. 91, 9 (2022)","journal-title":"J. Sci. Comput."},{"issue":"1","key":"2880_CR17","doi-asserted-by":"publisher","first-page":"15","DOI":"10.5334\/jors.151","volume":"5","author":"C Rackauckas","year":"2017","unstructured":"Rackauckas, C., Nie, Q.: Differentialequations.jl-a performant and feature-rich ecosystem for solving differential equations in julia. J. Open Res. Softw. 5(1), 15 (2017)","journal-title":"J. Open Res. Softw."},{"key":"2880_CR18","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-65027-7","volume-title":"Linear and Regular Celestial Mechanics","author":"E Stiefel","year":"1971","unstructured":"Stiefel, E., Scheifele, G.: Linear and Regular Celestial Mechanics. Springer Verlag, Berlin (1971)"},{"key":"2880_CR19","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719598","volume-title":"Spectral Methods in Matlab","author":"LlN Trefethen","year":"2000","unstructured":"Trefethen, Ll.N.: Spectral Methods in Matlab. SIAM, Philadelphia (2000)"},{"key":"2880_CR20","unstructured":"Verner, J.: https:\/\/www.sfu.ca\/~jverner\/"}],"updated-by":[{"DOI":"10.1007\/s10915-025-02955-5","type":"correction","label":"Correction","source":"publisher","updated":{"date-parts":[[2025,6,11]],"date-time":"2025-06-11T00:00:00Z","timestamp":1749600000000}}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-025-02880-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-025-02880-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-025-02880-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,10]],"date-time":"2025-09-10T06:43:57Z","timestamp":1757486637000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-025-02880-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,10]]},"references-count":20,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,5]]}},"alternative-id":["2880"],"URL":"https:\/\/doi.org\/10.1007\/s10915-025-02880-7","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,10]]},"assertion":[{"value":"8 June 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 February 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 March 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 April 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 September 2025","order":5,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Update","order":6,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Retrospective Open Access Order has been processed.","order":7,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 June 2025","order":8,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Correction","order":9,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"A Correction to this paper has been published:","order":10,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"https:\/\/doi.org\/10.1007\/s10915-025-02955-5","URL":"https:\/\/doi.org\/10.1007\/s10915-025-02955-5","order":11,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no Conflict of interest","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"69"}}