An efficient two-step explicit/Crank-Nicolson with finite element approach for three-dimensional coupled Burgers' equations with source terms

An efficient two-step explicit/Crank-Nicolson with finite element approach for three-dimensional coupled Burgers' equations with source terms | 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 An efficient two-step explicit/Crank-Nicolson with finite element approach for three-dimensional coupled Burgers' equations with source terms Eric Ngondiep This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8482681/v2 This work is licensed under a CC BY 4.0 License Status: Posted Version 2 posted You are reading this latest preprint version Show more versions Abstract This paper constructs an efficient two-step explicit/Crank-Nicolson technique combined with finite element method to simulate a three-dimensional coupled Burgers equations with source terms, subject to appropriate initial and boundary conditions. The space derivatives are approximated using the finite element formulation whereas a combination of an efficient explicit scheme and crank-Nicolson approach is employed to interpolate the time derivative. The developed computational technique efficiently treats the time derivative term, ensuring its stability across small time steps. Both stability and convergence order of the new algorithm are numerically analyzed using the L^{\infty}(0,T;L^{2})-norm. The computational results suggest that the proposed two-step explicit/Crank-Nicolson approach is temporal second-order accurate and spatial third-order convergent. Four numerical examples are carried out to show the applicability and the efficiency of the new strategy. AMS Subject Classification (MSC): 65M12, 65M06. Numerical Analysis Computational Mathematics Computational Physics three-dimensional coupled Burgers' equations with source terms explicit scheme Crank- Nicolson method nite element method stability and convergence order Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 2 posted You are reading this latest preprint version Show more versions 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-8482681","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":612882408,"identity":"3401db89-539d-4d43-a3ed-190408bdfdf7","order_by":0,"name":"Eric Ngondiep","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA60lEQVRIiWNgGAWjYFACxgYIfRiIPwAxGzspWhhngLQwE23bAQYGZh4Qg5AWfunDjZ8Lftnk8R1nPvbY5tc2eT5mBsYPH3Nwa5HsS2yWntmXVix5mC3dOLfvtmEbMwOz5MxtuLUYnGFsY+btOZy44TCPmXRuz20gF+gdXjxa7BFa+L9JW/bctieoxYAHqIXnB9gWNmmGH7cTCWqROMPYLM3bkJY48zCbuWFvw+3kNmbGZrx+4e9hf/iZ549NYt/5w88e/Phz23Z+e/PBDx/xaAEDxjYwxQZlwCIXL/gD1QJljIJRMApGwShAAQBYiU/mE68vWAAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0002-7487-3301","institution":"Imam Mohammad Ibn saud Islamic University","correspondingAuthor":true,"prefix":"","firstName":"Eric","middleName":"","lastName":"Ngondiep","suffix":""}],"badges":[],"createdAt":"2025-12-30 15:23:52","currentVersionCode":2,"declarations":{"humanSubjects":false,"vertebrateSubjects":true,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":true},"doi":"10.21203/rs.3.rs-8482681/v2","doiUrl":"https://doi.org/10.21203/rs.3.rs-8482681/v2","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":106092932,"identity":"1b2bba6c-6e8b-4bee-87ec-f2f2119fc8f2","added_by":"auto","created_at":"2026-04-03 11:30:54","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2681618,"visible":true,"origin":"","legend":"","description":"","filename":"Arevisedexplcranknicol3dburgers.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8482681/v2_covered_60ad7213-434c-4c22-9a8a-1973f35de49b.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"An efficient two-step explicit/Crank-Nicolson with finite element approach for three-dimensional coupled Burgers' equations with source terms","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Imam Mohammad Ibn saud Islamic University","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":"three-dimensional coupled Burgers' equations with source terms, explicit scheme, Crank- Nicolson method, \fnite element method, stability and convergence order","lastPublishedDoi":"10.21203/rs.3.rs-8482681/v2","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8482681/v2","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis paper constructs an efficient two-step explicit/Crank-Nicolson technique combined with finite element method to simulate a three-dimensional coupled Burgers equations with source terms, subject to appropriate initial and boundary conditions. The space derivatives are approximated using the finite element formulation whereas a combination of an efficient explicit scheme and crank-Nicolson approach is employed to interpolate the time derivative. The developed computational technique efficiently treats the time derivative term, ensuring its stability across small time steps. Both stability and convergence order of the new algorithm are numerically analyzed using the L^{\\infty}(0,T;L^{2})-norm. The computational results suggest that the proposed two-step explicit/Crank-Nicolson approach is temporal second-order accurate and spatial third-order convergent. Four numerical examples are carried out to show the applicability and the efficiency of the new strategy.\u003c/p\u003e\n\u003cp\u003eAMS Subject Classification (MSC): 65M12, 65M06.\u003c/p\u003e","manuscriptTitle":"An efficient two-step explicit/Crank-Nicolson with finite element approach for three-dimensional coupled Burgers' equations with source terms","msid":"","msnumber":"","nonDraftVersions":[{"code":2,"date":"2026-03-27 19:28:09","doi":"10.21203/rs.3.rs-8482681/v2","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}},{"code":1,"date":"2026-01-06 16:25:00","doi":"10.21203/rs.3.rs-8482681/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":"ba7b4c4e-3b64-443c-9298-baf93fd3c9bc","owner":[],"postedDate":"March 27th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":65217191,"name":"Numerical Analysis"},{"id":65217192,"name":"Computational Mathematics"},{"id":65217193,"name":"Computational Physics"}],"tags":[],"updatedAt":"2026-03-23T17:07:03+00:00","versionOfRecord":[],"versionCreatedAt":"2026-03-27 19:28:09","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v2","identity":"rs-8482681","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8482681","identity":"rs-8482681","version":["v2"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}