[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82074-en":3,"doc-seo-82074-105":29,"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82074,13056703019404,"Miles","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","A Splitting Scheme for Dispersive Shallow Moment Equations","The well-known Shallow Water Equations (SWE) model incompressible free-surface flows when vertical effects are negligible after vertical averaging, but averaging removes vertical-axis information. Dispersive Shallow Moment Models (DSM) retain vertical velocity and pressure profiles through a non-hydrostatic pressure formulation, yet non-stationary solution methods were missing due to difficulty expressing the pressure equation as a divergence-free constraint. The work rewrites DSM pressure equations into a Poisson-like problem to enable a projection-type splitting scheme, presenting linear and nonlinear cases and reporting non-stationary numerical results via a hybrid Finite-Volume Finite-Difference method.","arXiv :2607 .08870v1 [physics .flu-dyn] 9 Jul 2026  \nA Splitting Scheme for Dispersive Shallow Moment Equations  \nUllika Scholz 1 , Robin Paar 1,2 , and Manuel Torrilhon 1  \n1 Chair of Applied and Computational Mathematics  \n2 Research Training Group EDDy  \nJuly 13, 2026  \nAbstract  \nThe well-known Shallow Water Equations (SWE) are used for modeling incompressible freesurface flows whenever the shallowness allows for a vertical-averaging; i.e., vertical effects are negligible in comparison to horizontal ones. But vertical averaging comes with the price of losing information along the vertical axis. Moment models for shallow flow contain information on the vertical velocity and pressure profile despite being dimensionally reduced. A class of these models incorporating a non-hydrostatic pressure have been introduced before as Dispersive Shallow Moment Models (DSM) . However, no method for solving the non-stationary equations has been presented yet, mainly because it was unclear how to compute the pressure equation in the form of the divergence-free constraint. We rewrite the pressure equations of the DSM models in the form of a Poisson-like problem to enable their solution with a projection-type splitting scheme. For the linear equations, we present the calculations for the generalized model and discuss the non-linear case. We state the first two linear models and the corresponding nonlinear counterparts. Finally, we introduce a hybrid Finite-Volume Finite-Difference method and discuss the non-stationary numerical results for an experiment with periodic boundary and uneven bottom topography.  \n1 Introduction  \nFree-surface flows occur in many different contexts in nature and industry. Therefore, their efficient computation is an important problem. Numerous models for incompressible free-surface flows are available, many of which are tailored to specific situations of applications.  \nA particularly simple and well-known model is the classical Shallow Water Equations (SWE) . From a mathematical perspective, it is a hyperbolic system of PDEs derived from the Euler equations via vertical integration; see e.g. [20] for a rigorous mathematical derivation of the SWE and other reduced models. A prerequisite for successfully modeling the flow using the vertically integrated SWE model is that the velocity profile of the horizontal flow is nearly constant.  \nThe SWE model is accurate for shallow flows where the ratio between the fluid height and the vertical length scale is small. If this situation does not apply, two things happen: First, the averaging of the horizontal velocity is no longer justified. One possible solution to this problem is to switch to the – still dimensionally reduced but slightly more complex – so-called Shallow Water Moment Equations (SWME) [19] . They rely on a polynomial expansion in vertical direction, encoding information on the vertical profile in additional flow variables, also called moments. Since the introduction of the SWME in 2019, these have been further developed in many ways. Some noticeable examples are hyperbolic regularization [18, 2], bedload transport [6], the combination with multilayer models [12], the change of the basis functions in the expansion to Chebyshev polynomials or splines [30, 27] as well as the integration into larger computational frameworks [32, 29] . On the other hand, as shallowness decreases, the influence of dispersive effects increases, but frequency dispersion is not modeled by the SWE. This means in practice that not only the shape of waves is poorly captured by the SWE, but the speed of wave packages is generally overestimated. This problem can be remedied in mathematical modeling by incorporating a non-hydrostatic pressure in the PDE system. One well-known system in  \nthis context are the Green-Naghdi equations [13, 11] . See [8] for more examples of dispersive models and their first-order hyperbolic formulations.  \nThe Dispersive Shallow Moment Equations (DSM) form a hie","cbCaihYoo1rVr0FN","https://ap.wps.com/l/cbCaihYoo1rVr0FN","pdf",997732,1,34,"English","en",105,"# Abstract\n# Introduction\n## Shallow Water and Moment Models\n## Dispersive Models and Non-Hydrostatic Pressure\n## DSM Motivation and Projection-Type Splitting Approach\n## Numerical Method Outline","[{\"question\":\"Why do Shallow Water Equations become insufficient for less shallow flows?\",\"answer\":\"When the flow is less shallow, vertical averaging is no longer justified and the influence of dispersive effects increases. As a result, SWE poorly captures wave shape and overestimates wave package speeds.\"},{\"question\":\"What key idea enables solving non-stationary DSM equations in this work?\",\"answer\":\"The paper rewrites the DSM pressure equations into a Poisson-like problem, making it possible to solve them with a projection-type splitting scheme despite the constraint form of the pressure equation.\"},{\"question\":\"What numerical approach is used for the non-stationary experiments?\",\"answer\":\"A hybrid Finite-Volume Finite-Difference method is introduced, and the paper discusses non-stationary numerical results for an experiment with periodic boundary conditions and uneven bottom topography.\"}]",1784178064,86,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"a-splitting-scheme-for-dispersive-shallow-moment-equations","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/a-splitting-scheme-for-dispersive-shallow-moment-equations/82074/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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},"Why do Shallow Water Equations become insufficient for less shallow flows?","Question",{"text":75,"@type":76},"When the flow is less shallow, vertical averaging is no longer justified and the influence of dispersive effects increases. As a result, SWE poorly captures wave shape and overestimates wave package speeds.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What key idea enables solving non-stationary DSM equations in this work?",{"text":80,"@type":76},"The paper rewrites the DSM pressure equations into a Poisson-like problem, making it possible to solve them with a projection-type splitting scheme despite the constraint form of the pressure equation.",{"name":82,"@type":73,"acceptedAnswer":83},"What numerical approach is used for the non-stationary experiments?",{"text":84,"@type":76},"A hybrid Finite-Volume Finite-Difference method is introduced, and the paper discusses non-stationary numerical results for an experiment with periodic boundary conditions and uneven bottom topography.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]