The Variable Time-stepping DLN-Ensemble Algorithms for incompressible Navier-Stokes Equations

The Variable Time-stepping DLN-Ensemble Algorithms for incompressible Navier-Stokes Equations | 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 The Variable Time-stepping DLN-Ensemble Algorithms for incompressible Navier-Stokes Equations Wenlong Pei This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6933885/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 06 Oct, 2025 Read the published version in Numerical Algorithms → Version 1 posted 9 You are reading this latest preprint version Abstract In the report, we propose a family of variable time-stepping ensemble algorithms for solving multiple incompressible Navier-Stokes equations (NSE) at one pass.The one-leg, two-step methods designed by Dahlquist, Liniger, and Nevanlinna (henceforth the DLN method) are non-linearly stable and second-order accurate under arbitrary time grids.We design the family of variable time-stepping DLN-Ensemble algorithms for multiple systems of NSE and prove that its numerical solutions are stable and second-order accurate in velocity under moderate time-step restrictions.Meanwhile, the family of algorithms can be equivalently implemented by a simple refactorization process: adding time filters on the backward Euler ensemble algorithm.In practice, we raise a time adaptive mechanism (based on the local truncation error criterion) for the family of DLN-Ensemble algorithms to balance accuracy and computational costs.Several numerical tests are to support the main conclusions of the report. The constant step test confirms the second-order convergence and time efficiency. The variable step test verifies the stability of the numerical solutions and the time efficiency of the adaptive mechanism. Ensemble variable time-stepping G-stability second-order time adaptivity Navier-Stokes Equations Refactorization Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 06 Oct, 2025 Read the published version in Numerical Algorithms → Version 1 posted Editorial decision: Revision requested 26 Aug, 2025 Reviews received at journal 25 Aug, 2025 Reviews received at journal 13 Jul, 2025 Reviewers agreed at journal 28 Jun, 2025 Reviewers agreed at journal 26 Jun, 2025 Reviewers invited by journal 26 Jun, 2025 Editor assigned by journal 22 Jun, 2025 Submission checks completed at journal 19 Jun, 2025 First submitted to journal 19 Jun, 2025 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. 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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-6933885","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":477785574,"identity":"e1f3f728-6602-4837-a723-9b0ae1173e7b","order_by":0,"name":"Wenlong Pei","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAwElEQVRIiWNgGAWjYFACxgaGxH82chAOG7FaHrClGZOiBajpAduhxAaitZizNzcwJPAcSO/nP2PA8KHsMGEtlj0HgVok7uTOnJFjwDjjHBFaDG4AnZRg8Cx3ww0eA2beNmK03H8I1JJwON3g/BkD5r9EabkBDLGEA4cTDA7kGDAzEqPFsiex4UBiQ5rhzBlpBQd7zqUT1mLOfvzhw58NNvL8/Ic3PvhRZk2Ew4D4AIxzAKcydC2jYBSMglEwCvADAFUwPlWVKh3mAAAAAElFTkSuQmCC","orcid":"","institution":"The Ohio State University","correspondingAuthor":true,"prefix":"","firstName":"Wenlong","middleName":"","lastName":"Pei","suffix":""}],"badges":[],"createdAt":"2025-06-19 22:38:12","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6933885/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6933885/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11075-025-02236-0","type":"published","date":"2025-10-06T15:58:03+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":93419773,"identity":"019936d8-919e-4ea9-8771-697f6779a745","added_by":"auto","created_at":"2025-10-13 16:07:23","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1531065,"visible":true,"origin":"","legend":"","description":"","filename":"DLNNSEEnsembelNumericalAlgorithmSubmissionSummer2025.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6933885/v1_covered_0d6ff3f8-5bf5-48b7-8034-6d3e90353493.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"The Variable Time-stepping DLN-Ensemble Algorithms for incompressible Navier-Stokes Equations","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":"numerical-algorithms","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"numa","sideBox":"Learn more about [Numerical Algorithms](http://link.springer.com/journal/11075)","snPcode":"11075","submissionUrl":"https://submission.nature.com/new-submission/11075/3","title":"Numerical Algorithms","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Ensemble, variable time-stepping, G-stability, second-order, time adaptivity, Navier-Stokes Equations, Refactorization","lastPublishedDoi":"10.21203/rs.3.rs-6933885/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6933885/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"In the report, we propose a family of variable time-stepping ensemble algorithms for solving multiple incompressible Navier-Stokes equations (NSE) at one pass.The one-leg, two-step methods designed by Dahlquist, Liniger, and Nevanlinna (henceforth the DLN method) are non-linearly stable and second-order accurate under arbitrary time grids.We design the family of variable time-stepping DLN-Ensemble algorithms for multiple systems of NSE and prove that its numerical solutions are stable and second-order accurate in velocity under moderate time-step restrictions.Meanwhile, the family of algorithms can be equivalently implemented by a simple refactorization process: adding time filters on the backward Euler ensemble algorithm.In practice, we raise a time adaptive mechanism (based on the local truncation error criterion) for the family of DLN-Ensemble algorithms to balance accuracy and computational costs.Several numerical tests are to support the main conclusions of the report. 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