[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82292-en":3,"doc-seo-82292-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},82292,13056703019662,"Evangeline","https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188",8,"Research & Report","Fifth-Order Well-Balanced Path-Conservative A-WENO Scheme for the Ripa Model","A fifth-order well-balanced (WB) path-conservative A-WENO scheme with central-upwind numerical fluxes (PCCU-5) is developed for the Ripa model. The method exactly preserves multiple steady states, including stillwater (lake-at-rest, isobaric, and constant water height) and moving-water equilibria. A flux globalization technique yields a quasi-conservative formulation by embedding source terms into fluxes, while equilibrium-based WENO interpolation and local characteristic equilibrium projection reduce oscillations near discontinuities. Numerical experiments confirm higher resolution than the second-order counterpart.","Journal of Computational Mathematics Vol.xx, No.x, 200x, 1–33 .  \n[http://www.global-sci.org/jcm](http://www.global-sci.org/jcm)  \n[doi:??](doi:??)  \narXiv :2607 .09293v1 [math .NA] 10 Jul 2026  \nFIFTH-ORDER WELL-BALANCED PATH-CONSERVATIVE  \nA-WENO SCHEME FOR THE RIPA MODEL*  \nYan-Ping Qiu  \nSchool of Mathematical Sciences and Laboratory of Marine Mathematics, Ocean University of China,  \nQingdao, China  \nEmail: [qyp@stu.ouc.edu.cn](qyp@stu.ouc.edu.cn)  \nZhen Gao  \nSchool of Mathematical Sciences and Laboratory of Marine Mathematics, Ocean University of China,  \nQingdao, China  \nEmail: [zhengao@ouc.edu.cn](zhengao@ouc.edu.cn)  \nAlexander Kurganov  \nDepartment of Mathematics and Shenzhen International Center for Mathematics, Southern University  \nof Science and Technology, Shenzhen, China  \nEmail: [alexander@sustech.edu.cn](alexander@sustech.edu.cn)  \nBao-Shan Wang  \nSchool of Mathematical Sciences and Laboratory of Marine Mathematics, Ocean University of China,  \nQingdao, China  \nEmail: [wbs@ouc.edu.cn](wbs@ouc.edu.cn)  \nXiao Wen  \nCollege of Mathematics and Systems Science, Shandong University of Science and Technology,  \nQingdao, China  \nEmail: [xiaowen@sdust.edu.cn](xiaowen@sdust.edu.cn)  \nAbstract  \nIn this work, we introduce a fifth-order well-balanced (WB) path-conservative A-WENO scheme with the central-upwind numerical fluxes (PCCU-5) for the Ripa model. The proposed scheme is capable of exactly preserving a variety of steady states, including stillwater, moving-water, isobaric, and constant water height ones. This goal is achieved with the help of a flux globalization technique: The source terms are incorporated into the fluxes, resulting in a quasi-conservative system, for which central-upwind numerical fluxes are computed using the path-conservative integration. The proposed A-WENO scheme utilizesa WENO interpolation of the equilibrium variables rather than the conservative ones to ensure the WB property. In addition, we perform the WENO interpolation of the local characteristic equilibrium variables to mitigate numerical oscillations near discontinuities. We perform a series of numerical experiments, which demonstrate that the proposed fifthorder WB PCCU-5 scheme achieves high resolution and clearly outperforms its secondorder counterpart. Our numerical results also demonstrate the importance of the local characteristic projection for significantly reducing (eliminating) numerical oscillations near discontinuities.  \nMathematics subject classification: 65M06, 65M20, 76M20, 86-08 .  \nKey words: Path-conservative methods, Central-upwind schemes, Well-balanced methods, A-WENO schemes, Local characteristic decomposition, Ripa model.  \n* Received xxx / Revised version received xxx / Accepted xxx /  \n2 Y.-P. QIU, Z. GAO, A. KURGANOV, B.-S. WANG, AND X. WEN  \n1. Introduction  \nWe consider the shallow water equations, in which the water temperature fluctuations are taken into account. The studied system was introduced in [45,46] for modeling ocean currentsand is often referred to as the Ripa model, which in the one-dimensional (1-D) case reads as  \n􀀾􀀸 ht + qx = 0 ,  \n􀀾  \n qt + 􀀐 hu2 + 12 h2 θ􀀑 x = −hθZx , (1.1)  \n􀀾  \n􀀺􀀾 (hθ)t + (huθ)x = 0 ,  \nwhere x is a spatial coordinate, t is time, h (x, t) is the water depth, u (x, t) is the velocity, q (x, t) = h (x, t)u (x, t) is the discharge, Z (x) is the bottom topography, and θ (x, t) is the potential temperature defined to be the reduced gravity g∆Θ/Θref , where ∆Θ is the difference in potential temperature from a certain reference value Θref and g is the acceleration due to gravity. Notice that when the potential temperature field θ ≡ g, the system (1.1) reduces to the classical Saint-Venant system of shallow water equations.  \nThe Ripa model (1.1) admits several steady states, including both still- and moving-water equilibria. There are three types of still-water equilibria:  \n• “Lake-at-rest” steady states satisfying  \nu ≡ 0, h + Z ≡ Const, θ ≡ Const; (1.2)  \n• Isobaric stea","cbCaieW6F025NAIT","https://ap.wps.com/l/cbCaieW6F025NAIT","pdf",6175342,3,1,33,"English","en",105,"# Introduction\n## Shallow water equations with temperature fluctuations (Ripa model)\n## Steady states of the Ripa model\n## Well-balanced numerical methods overview\n# Abstract and contributions (scheme design summary)\n# Numerical results (expected performance)","[{\"question\":\"What does the proposed PCCU-5 A-WENO scheme preserve for the Ripa model?\",\"answer\":\"It is designed to exactly preserve steady states including stillwater equilibria (lake-at-rest, isobaric, and constant water height) and moving-water steady states.\"},{\"question\":\"How is the well-balanced property achieved in the scheme?\",\"answer\":\"A flux globalization technique incorporates source terms into fluxes to form a quasi-conservative system, then central-upwind numerical fluxes are computed through path-conservative integration.\"},{\"question\":\"Why does using WENO interpolation of equilibrium variables and local characteristic projection help near discontinuities?\",\"answer\":\"Interpolating equilibrium variables supports the WB property, while interpolating local characteristic equilibrium variables mitigates numerical oscillations close to discontinuities.\"}]",1784179438,83,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"fifth-order-well-balanced-path-conservative-a-weno-scheme-for-the-ripa-model","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/fifth-order-well-balanced-path-conservative-a-weno-scheme-for-the-ripa-model/82292/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-22","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does the proposed PCCU-5 A-WENO scheme preserve for the Ripa model?","Question",{"text":75,"@type":76},"It is designed to exactly preserve steady states including stillwater equilibria (lake-at-rest, isobaric, and constant water height) and moving-water steady states.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the well-balanced property achieved in the scheme?",{"text":80,"@type":76},"A flux globalization technique incorporates source terms into fluxes to form a quasi-conservative system, then central-upwind numerical fluxes are computed through path-conservative integration.",{"name":82,"@type":73,"acceptedAnswer":83},"Why does using WENO interpolation of equilibrium variables and local characteristic projection help near discontinuities?",{"text":84,"@type":76},"Interpolating equilibrium variables supports the WB property, while interpolating local characteristic equilibrium variables mitigates numerical oscillations close to 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