Tendler-like Formulas for Stiff ODEs

Tendler-like Formulas for Stiff ODEs | 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Share Download PDF Research Article Tendler-like Formulas for Stiff ODEs Elmar Klausmeier This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8688554/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This paper proves a convergence result for a general class of methods for the solution of ordinary differential equations (initial value problems). The proof uses standard results from the theory of matrix polynomials. We present new cyclic linear multistep formulas of orders 3 to 9 for stiff equations, which, order by order, outperform the cyclic composite multistep methods of Tendler with respect to the Widlund-wedge angle and Widlund-distance. The Tendler formulas had already outperformed the BDF order by order. We present numerical accuracy comparisons on Dahlquist’s and Runge’s test equations with the BDF, the original Tendler, the new methods, and the Tischer methods. Mathematics Subject Classification: 65L04 65L06 CR: G.1.7 cyclic linear multistep methods matrix polynomials stiff differential equations convergence analysis Widlund-wedge Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted 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-8688554","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":628889557,"identity":"39cf08d4-f847-4ab8-92e7-2ce85627e536","order_by":0,"name":"Elmar Klausmeier","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABFElEQVRIiWNgGAWjYBAC+3YwdVgGwjVgYOAH0QkFuLUYMIOpYzxwLZINIC0GBLWc4UGIHIDqxa2FO/FzBcMNHv4Z6U83fCiwyTc+vzrxwwMDBnl+sQPY/cLMu1nyDMMHHokbOWY3ZxikWW678XazBNBhhjNnJ+CwhXcD0PUfeBhu5LDd5jE4bGB24+wGkJYEg9s4tWz+2QB0mPyN9Ge3/xj8NzCecXbzDwJatgFtOcNjcCPB7DbQ7wYG/L3bCNmyzbLB4BiP4Zk3Zjd7DJINJG7wbrNIMJDA6Rf79t7NNxsqDsvJHU9/duPHHzsD/v6zm2/+qLCR55fGrgVqFzJHAqxSAo9yDMB/gBTVo2AUjIJRMAIAAAdUX78CLt2BAAAAAElFTkSuQmCC","orcid":"","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Elmar","middleName":"","lastName":"Klausmeier","suffix":""}],"badges":[],"createdAt":"2026-01-24 18:08:30","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8688554/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8688554/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":107920133,"identity":"755697f6-5e2d-4c37-8550-a636e92a6470","added_by":"auto","created_at":"2026-04-27 14:42:04","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":640319,"visible":true,"origin":"","legend":"","description":"","filename":"tendlerlikeformulasforstiffodes.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8688554/v1_covered_07b2ebc2-489d-4cf5-bb99-84ba182b6c5e.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Tendler-like Formulas for Stiff ODEs","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"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":"cyclic linear multistep methods, matrix polynomials, stiff differential equations, convergence analysis, Widlund-wedge","lastPublishedDoi":"10.21203/rs.3.rs-8688554/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8688554/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis paper proves a convergence result for a general class of methods for the solution of ordinary differential equations (initial value problems). 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