On Reciprocity and Splitting Errors in Direct-Forcing Immersed Boundary Methods

On Reciprocity and Splitting Errors in Direct-Forcing Immersed Boundary Methods | 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 On Reciprocity and Splitting Errors in Direct-Forcing Immersed Boundary Methods Francesco Capuano This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9107598/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 9 You are reading this latest preprint version Abstract Direct-forcing immersed boundary methods (IBMs) are widely used to enforce no-slip boundary conditions in incompressible flow solvers owing to their flexibility and ease of implementation. In this work we introduce a semi-discrete framework that encompasses both Lagrangian and Eulerian direct-forcing IBMs. Within this formulation, classical algorithms can be interpreted as approximate block-LU factorizations of the fully coupled monolithic system. This perspective reveals two distinct sources of error in the enforcement of the no-slip condition: a reciprocity error, associated with the non-exact adjoint relationship between interpolation and spreading operators, and a splitting error, arising from the pressure predictor used in projection-based time integration. The framework clarifies how the relative importance of these mechanisms depends on the interpolation kernel, the time step, and the physical configuration. Representative numerical experiments illustrate the behavior of the two error mechanisms and confirm the predictions of the analysis. In particular, a simple oscillating-balloon configuration is proposed as a diagnostic benchmark that directly exposes inconsistencies between boundary motion and the Eulerian mass balance. The analysis provides a unified interpretation of previously observed inaccuracies in direct-forcing IBMs and suggests possible directions for improvement, including the development of stable higher-order pressure predictors and improved constructions of interpolation-spreading operators. immersed boundary methods direct forcing scheme time integration projection method incompressible flow Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviews received at journal 14 May, 2026 Reviews received at journal 29 Mar, 2026 Reviewers agreed at journal 19 Mar, 2026 Reviewers agreed at journal 18 Mar, 2026 Reviewers agreed at journal 18 Mar, 2026 Reviewers invited by journal 17 Mar, 2026 Editor assigned by journal 17 Mar, 2026 Submission checks completed at journal 17 Mar, 2026 First submitted to journal 12 Mar, 2026 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-9107598","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":608717826,"identity":"a4710e8c-21ae-4e56-a3c7-ac280f1e5dd6","order_by":0,"name":"Francesco Capuano","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+UlEQVRIiWNgGAWjYBAC9gZcQoyYMhDAcwCXEClaJBIIaJE+/OzBj4o6eX4G5oefC9vs8vgl35g9+MFgI7sB0zSIFr40c8OeM4cNZzawGUvPbEsulpydAxRhSDPGpcWeh8FMmrHtQILBAR4Gad425sQNt3PMpBkYDifitIWH/Zs04786kBbm37xt9Ykbbp4BafmPRwsP0JYGZpAWNqAtQMNv8IC0HMCnpUyy5xjQL81sZtYzzh1PnNmTVm7YY5BsPBO3w7ZJ/KgBhhh78+PbBWXVif3sh7cBw9BOtg+HFgRgBiMwYGNgMCCkHKELpmUUjIJRMApGAQIAAJGOVG7DY5hvAAAAAElFTkSuQmCC","orcid":"","institution":"Universitat Politècnica de Catalunya","correspondingAuthor":true,"prefix":"","firstName":"Francesco","middleName":"","lastName":"Capuano","suffix":""}],"badges":[],"createdAt":"2026-03-12 18:23:29","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9107598/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9107598/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":105035913,"identity":"b0acc32e-f993-427e-8c39-049c9abde4a0","added_by":"auto","created_at":"2026-03-20 07:26:55","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":920871,"visible":true,"origin":"","legend":"","description":"","filename":"submit.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9107598/v1_covered_5e1f427f-7ed1-4284-9b69-a04d3ac502bd.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"On Reciprocity and Splitting Errors in Direct-Forcing Immersed Boundary Methods","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"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":"flow-turbulence-and-combustion","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"appl","sideBox":"Learn more about [Flow, Turbulence and Combustion](http://link.springer.com/journal/10494)","snPcode":"10494","submissionUrl":"https://submission.nature.com/new-submission/10494/3","title":"Flow, Turbulence and Combustion","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"immersed boundary methods, direct forcing scheme, time integration, projection method, incompressible flow","lastPublishedDoi":"10.21203/rs.3.rs-9107598/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9107598/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Direct-forcing immersed boundary methods (IBMs) are widely used to enforce no-slip boundary conditions in incompressible flow solvers owing to their flexibility and ease of implementation. 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