A high-order accurate well-balanced entropy stable finite difference scheme for shallow water equations over bottom topography

A high-order accurate well-balanced entropy stable finite difference scheme for shallow water equations over bottom topography | 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 A high-order accurate well-balanced entropy stable finite difference scheme for shallow water equations over bottom topography Changxi Xu, Rendi Liu, Xiangyu Zhou, Zhizhuang Zhang, Shouguo Qian, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8016540/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 article establishes a finite difference scheme for shallow water equations; the proposed scheme enjoys high-order accuracy and is entropy stable. Firstly, we construct a second-order entropy conservative scheme, which meets the entropy identity for a given entropy pair and maintains the lake at rest steady state exactly. The main idea is to make the discretizations to the source term match the one to the flux gradient. Secondly, high-order entropy conservative scheme is realized by taking the second-order entropy conservative schemes with affordable entropy conservative fluxes as a building block. Thirdly, adding an appropriate dissipation term to the existing entropy conservative scheme leads to a semi-discrete entropy stable scheme, which meets the discrete entropy inequality. Especially, the entropy stable scheme can effectively suppress oscillations from the entropy conservative scheme. Herein, the dissipation term is built using the weighted essentially non-oscillatory reconstruction according to the scaled entropy variables. Finally, the temporal discretization is by the Runge-Kutta approach. Extensive examples are implemented to show the accuracy, the well-balanced property, and the ability to well resolve small perturbations. shallow water equations high-order entropy stable finite difference well-balanced 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-8016540","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":552099982,"identity":"2f65bd19-96d0-44c5-9f2d-1969a7eb871b","order_by":0,"name":"Changxi Xu","email":"","orcid":"","institution":"Qingdao University","correspondingAuthor":false,"prefix":"","firstName":"Changxi","middleName":"","lastName":"Xu","suffix":""},{"id":552099983,"identity":"0067d9b5-2e50-4b82-b3d7-dd84af71d6ae","order_by":1,"name":"Rendi Liu","email":"","orcid":"","institution":"Qingdao University","correspondingAuthor":false,"prefix":"","firstName":"Rendi","middleName":"","lastName":"Liu","suffix":""},{"id":552099984,"identity":"dbab3aee-b7ed-498c-ae33-0eaa154217d6","order_by":2,"name":"Xiangyu Zhou","email":"","orcid":"","institution":"Qingdao University","correspondingAuthor":false,"prefix":"","firstName":"Xiangyu","middleName":"","lastName":"Zhou","suffix":""},{"id":552099985,"identity":"3a467ba7-1dac-4dd8-abe5-65e4bef9a0cd","order_by":3,"name":"Zhizhuang Zhang","email":"","orcid":"","institution":"Qingdao University","correspondingAuthor":false,"prefix":"","firstName":"Zhizhuang","middleName":"","lastName":"Zhang","suffix":""},{"id":552099986,"identity":"ee5aa22b-5b45-416e-b643-8b637fc22cb0","order_by":4,"name":"Shouguo Qian","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAyElEQVRIiWNgGAWjYJCCAwwMNmCSgYcELWkkagGCwyRoke9fvPHAzx3nZftuJDA+eNvGIG9OSIvBjWcFB3vP3DaeeSOB2XBuG4PhzgZCWiTOGBzgbbuduOFGAps0bxtDgsEBQg6bccbg4N+2cyAt7L+J0sJwvsfgMG/bAbAtzERpMbjBVnBYti3ZeOaZh82Sc85JGG4g6LD+w5s/vm2zk+07nnzww5syG3nCDpNIMABRjA0gBOQSUg8E/AdgWkbBKBgFo2AU4AAAvKtKc+WCqEUAAAAASUVORK5CYII=","orcid":"","institution":"Qingdao University","correspondingAuthor":true,"prefix":"","firstName":"Shouguo","middleName":"","lastName":"Qian","suffix":""},{"id":552099987,"identity":"0a03b905-0c3b-4758-9610-40da997139e2","order_by":5,"name":"Gang Li","email":"","orcid":"","institution":"Qingdao University","correspondingAuthor":false,"prefix":"","firstName":"Gang","middleName":"","lastName":"Li","suffix":""}],"badges":[],"createdAt":"2025-11-03 08:23:16","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8016540/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8016540/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":97105628,"identity":"9bb14a4e-4959-42bf-8b49-fbd309230215","added_by":"auto","created_at":"2025-12-01 04:47:19","extension":"json","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":7336,"visible":true,"origin":"","legend":"","description":"","filename":"682df6fab56f428c8a9f7789d8c3b010.json","url":"https://assets-eu.researchsquare.com/files/rs-8016540/v1/604215ef9de53b482102703e.json"},{"id":99789639,"identity":"9f2892f4-f8a5-42cb-96ce-2c9f90138aee","added_by":"auto","created_at":"2026-01-08 12:50:13","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1088488,"visible":true,"origin":"","legend":"","description":"","filename":"Paper.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8016540/v1_covered_57519f33-57cb-4d6c-a91e-19b877c5da86.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":" A high-order accurate well-balanced entropy stable finite difference scheme for shallow water equations over bottom topography ","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":"shallow water equations, high-order, entropy stable, finite difference, well-balanced","lastPublishedDoi":"10.21203/rs.3.rs-8016540/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8016540/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis article establishes a finite difference scheme for shallow water equations; the proposed scheme enjoys high-order accuracy and is entropy stable. Firstly, we construct a second-order entropy conservative scheme, which meets the entropy identity for a given entropy pair and maintains the lake at rest steady state exactly. The main idea is to make the discretizations to the source term match the one to the flux gradient. Secondly, high-order entropy conservative scheme is realized by taking the second-order entropy conservative schemes with affordable entropy conservative fluxes as a building block. Thirdly, adding an appropriate dissipation term to the existing entropy conservative scheme leads to a semi-discrete entropy stable scheme, which meets the discrete entropy inequality. Especially, the entropy stable scheme can effectively suppress oscillations from the entropy conservative scheme. Herein, the dissipation term is built using the weighted essentially non-oscillatory reconstruction according to the scaled entropy variables. Finally, the temporal discretization is by the Runge-Kutta approach. Extensive examples are implemented to show the accuracy, the well-balanced property, and the ability to well resolve small perturbations.\u003c/p\u003e","manuscriptTitle":"A high-order accurate well-balanced entropy stable finite difference scheme for shallow water equations over bottom topography","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-12-01 04:47:14","doi":"10.21203/rs.3.rs-8016540/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":"da9430d3-fdc1-4d85-9e94-43ba647a4a26","owner":[],"postedDate":"December 1st, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-01-03T17:23:45+00:00","versionOfRecord":[],"versionCreatedAt":"2025-12-01 04:47:14","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8016540","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8016540","identity":"rs-8016540","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}