Cost-reduction implicit exponential Runge-Kutta methods for highly oscillatory systems

Cost-reduction implicit exponential Runge-Kutta methods for highly oscillatory systems | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Cost-reduction implicit exponential Runge-Kutta methods for highly oscillatory systems Xianfa Hu, Wansheng Wang, Bin Wang, Yonglei Fang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3299925/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 07 Jul, 2024 Read the published version in Journal of Mathematical Chemistry → Version 1 posted You are reading this latest preprint version Abstract In this paper, two novel classes of implicit exponential Runge--Kutta (ERK) methods are studied for solving highly oscillatory systems. Firstly, we analyze the symplectic conditions for two kinds of exponential integrators and obtain the symplectic method. In order to effectively solve highly oscillatory problems, we try to design the highly accurate implicit ERK integrators. By comparing the Taylor series expansion of numerical solution with exact solution, it can be verified that the order conditions of two new kinds of exponential methods are identical to classical Runge--Kutta (RK) methods, which implies that using the coefficients of RK methods, some highly accurate numerical methods are directly formulated. Furthermore, we also investigate the linear stability properties for these exponential methods. Finally, numerical results not only display the long time energy preservation of the symplectic method, but also present the accuracy and efficiency of these formulated methods in comparison with standard ERK methods. Implicit exponential Runge--Kutta methods Symplectic conditions Order conditions Linear stability analysis Highly oscillatory systems Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 07 Jul, 2024 Read the published version in Journal of Mathematical Chemistry → Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3299925","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":229608525,"identity":"14f8b4c4-3375-4b60-b5ca-40df00649d28","order_by":0,"name":"Xianfa Hu","email":"","orcid":"","institution":"Shanghai Normal University","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Xianfa","middleName":"","lastName":"Hu","suffix":""},{"id":229608526,"identity":"cc073bb0-7e63-480c-9334-e7d3e46ac16d","order_by":1,"name":"Wansheng Wang","email":"","orcid":"","institution":"Shanghai Normal University","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Wansheng","middleName":"","lastName":"Wang","suffix":""},{"id":229608527,"identity":"c38e558a-7a80-49db-9af2-c0403abfcfcc","order_by":2,"name":"Bin Wang","email":"","orcid":"","institution":"Xi'an Jiaotong University","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Bin","middleName":"","lastName":"Wang","suffix":""},{"id":229608528,"identity":"6547b6c7-da03-48ee-aed3-19919b1f8673","order_by":3,"name":"Yonglei Fang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1klEQVRIiWNgGAWjYDACdsYHEIYE84EDH34Qo4WZ2QBM80iwJR6c2UOaFh7jwxxsROgwOMzM+Lng1zZ5e+meD4cZeBjk+cUOENTCLD2z77Zhj8zZDYcLLBgMZ85OIKSF/4A0b89txh6J3A2HZ/AwJBjcJqiFmfk3UIt9j0TOg8M8bMRpYZPm+XE7EaiFgTgtkkAt1rwNt5N7bqQZAANZgrBf+I43M9/m+XPbtn1G8uMPH37YyPNLE9CicABIMLbB+RL4lYOAfAOI/ENY4SgYBaNgFIxgAAATREaomtXStwAAAABJRU5ErkJggg==","orcid":"","institution":"Zaozhuang University","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Yonglei","middleName":"","lastName":"Fang","suffix":""}],"badges":[],"createdAt":"2023-08-27 08:14:20","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3299925/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3299925/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s10910-024-01646-0","type":"published","date":"2024-07-08T00:32:13+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":59885525,"identity":"ebd83d11-b78b-42d0-84cc-f730e9751764","added_by":"auto","created_at":"2024-07-09 00:32:19","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1335880,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3299925/v1_covered_19d732b1-5a20-4e6f-ac3b-54fa10312076.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Cost-reduction implicit exponential Runge-Kutta methods for highly oscillatory systems","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Implicit exponential Runge--Kutta methods, Symplectic conditions, Order conditions, Linear stability analysis, Highly oscillatory systems","lastPublishedDoi":"10.21203/rs.3.rs-3299925/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3299925/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003e In this paper, two novel classes of implicit exponential Runge--Kutta (ERK) methods are studied for solving highly oscillatory systems. Firstly, we analyze the symplectic conditions for two kinds of exponential integrators and obtain the symplectic method. In order to effectively solve highly oscillatory problems, we try to design the highly accurate implicit ERK integrators. By comparing the Taylor series expansion of numerical solution with exact solution, it can be verified that the order conditions of two new kinds of exponential methods are identical to classical Runge--Kutta (RK) methods, which implies that using the coefficients of RK methods, some highly accurate numerical methods are directly formulated. Furthermore, we also investigate the linear stability properties for these exponential methods. Finally, numerical results not only display the long time energy preservation of the symplectic method, but also present the accuracy and efficiency of these formulated methods in comparison with standard ERK methods.\u003c/p\u003e","manuscriptTitle":"Cost-reduction implicit exponential Runge-Kutta methods for highly oscillatory systems","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2023-09-01 08:25:02","doi":"10.21203/rs.3.rs-3299925/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"f5f4c87c-a9fa-43d5-8def-27e174017993","owner":[],"postedDate":"September 1st, 2023","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2024-07-09T00:32:13+00:00","versionOfRecord":{"articleIdentity":"rs-3299925","link":"https://doi.org/10.1007/s10910-024-01646-0","journal":{"identity":"journal-of-mathematical-chemistry","isVorOnly":false,"title":"Journal of Mathematical Chemistry"},"publishedOn":"2024-07-08 00:32:13","publishedOnDateReadable":"July 8th, 2024"},"versionCreatedAt":"2023-09-01 08:25:02","video":"","vorDoi":"10.1007/s10910-024-01646-0","vorDoiUrl":"https://doi.org/10.1007/s10910-024-01646-0","workflowStages":[]},"version":"v1","identity":"rs-3299925","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3299925","identity":"rs-3299925","version":["v1"]},"buildId":"7rjqhiLT3MXkJMwkYKINL","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}