[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82536-en":3,"doc-seo-82536-105":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},82536,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","Hermite Semi-Lagrangian schemes on triangular meshes for advection equations","High-order Hermite semi-Lagrangian schemes are developed for advection equations on unstructured triangular grids. The work combines Hermite finite elements based on Bell and Argyris elements with two semi-Lagrangian strategies: nodal transport of point values together with gradients and Hessians, and projection-based transport aligned with semi-Lagrangian discontinuous Galerkin ideas via shared degrees of freedom. Numerical stability and convergence studies compare the proposed variants.","arXiv :2607 .00619v1 [math .NA] 1 Jul 2026  \nHermite Semi-Lagrangian schemes on triangular meshes for  \nadvection equations  \nAli Elarif1,2,3 , Michel Mehrenberger4 , and Laurent Navoret 1,2  \n1 IRMA, UMR 7501 Universit´e de Strasbourg et CNRS, 7 rue Ren´e Descartes, 67000 Strasbourg, France  \n2 Inria, IRMA, Universit´e de Strasbourg, CNRS UMR 7501, 7 rue Ren´e Descartes, 67084 Strasbourg, France  \n3 Nodji Consulting, 236B rue du Faubourg des postes, 59000 Lille  \n4 Aix Marseille Universit´e, CNRS, I2M, Marseille, France  \nAbstract  \nHigh-order Hermite semi-Lagrangian schemes on unstructured triangular grids are proposed for advection equations, based on Bell and Argyris finite elements. Nodal semi-Lagrangian schemes transport point values together with gradients and Hessians, while projection semi-Lagrangian schemes are basically semi-Lagrangian Discontinuous Galerkin schemes (SLDG) with straight remaining backward triangles and shared degrees of freedoms. Stability and convergence studies of the different schemes are carried out numerically.  \n1 Introduction  \nSemi-Lagrangian (SL) schemes are a class of schemes widely used to solve advection equations. At each time iteration, the approximate solution is first advected exactly, i.e. expressed as the solution at the previous time step evaluated at the foot of the characteristics, and then pushed back to the approximation space by nodal evaluations (nodal SL) or by L2 projection (projection SL) . These schemes have the advantage of not being constrained by a CFL-type condition, which is of great advantage in the presence of unnecessarily small mesh cells. However, while high-order semi-Lagrangian schemes have been proposed on uniform Cartesian (e.g. odd degree Lagrange nodal SL [9, 14 , 22]) or non-uniform Cartesian grids (Hermite nodal SL, Semi-Lagrangian Discontinuous Galerkin), the extension to unstructured meshes is more challenging due to stability and computational issues.  \nRegarding nodal SL schemes on unstructured triangular grids, one popular solution is to use P2 interpolation [15], but P3 or higher order interpolation are unstable [11] . Another approach is to consider a larger stencil together with least square reconstruction [8, 3] . Finally, the third option consists in using Hermite interpolation by transporting nodal values of the function and its gradients [4] and based on the reduced Hsieh-Clough-Tucker (rHCT) approximation space. Note that in [4], only constant advection fields have been considered.  \nRegarding projection SL schemes, also called Semi-Lagrangian Discontinuous Galerkin schemes (SLDG) when combined with discontinuous polynomial approximation, stability issues appear to be ruled out, as the method is stable due to the L2 projection. However, the method requires the computation of integrals involving quantities defined on backward cells and their approximations can lead to instabilities [21, 20] . In order to prevent from such instabilities, further projection semiLagrangian schemes are based on mesh intersection algorithms and has been developed on Cartesian or polar grids [17, 5 , 10 , 6 , 12] . In the context of triangular meshes, a recent work proposes a SLDG scheme combined with a Runge-Kutta Discontinuous Galerkin method [7], where backward cells do not need to be curved to be still high order accurate. We also refer to [13, 1] for remapping schemeson triangular meshes, based on mesh intersections.  \nIn this work, we propose to explore the use of high-order Hermite elements on triangles, i.e. Bell and Argyris elements [2, 19] . In the context of nodal SL schemes, this requires to also advect the nodal  \nvalues of the Hessian. Moreover, the projection SL method can also be adapted to other discretizations than Discontinuous Galerkin: we will also explore the use of the Hermite elements in this context, which permit in particular to share some degrees of freedom. Note that, at the price of some loss of precision, the backward triangles will b","cbCaifhS5Sm8TURH","https://ap.wps.com/l/cbCaifhS5Sm8TURH","pdf",1699307,2,1,10,"English","en",105,"# Introduction\n# Hermite Semi-Lagrangian schemes","[{\"question\":\"What problem does the document address?\",\"answer\":\"It addresses high-order numerical solution of two-dimensional advection equations on unstructured triangular meshes using Hermite semi-Lagrangian methods.\"},{\"question\":\"How do nodal semi-Lagrangian schemes differ from projection semi-Lagrangian schemes here?\",\"answer\":\"Nodal semi-Lagrangian schemes transport point values along characteristics together with gradients and Hessians, while projection semi-Lagrangian schemes rely on L2 projection and connect to semi-Lagrangian discontinuous Galerkin ideas with shared degrees of freedom.\"},{\"question\":\"Which Hermite elements and approximation spaces are used?\",\"answer\":\"The schemes are built using Bell and Argyris Hermite finite elements, and the document also describes a reduced Hsieh-Clough-Tocher (rHCT) approximation space defined via nodal values and partial derivatives.\"}]",1784181361,25,{"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},"hermite-semi-lagrangian-schemes-on-triangular-meshes-for-advection-equations","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/hermite-semi-lagrangian-schemes-on-triangular-meshes-for-advection-equations/82536/",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","2026-07-21","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 problem does the document address?","Question",{"text":75,"@type":76},"It addresses high-order numerical solution of two-dimensional advection equations on unstructured triangular meshes using Hermite semi-Lagrangian methods.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do nodal semi-Lagrangian schemes differ from projection semi-Lagrangian schemes here?",{"text":80,"@type":76},"Nodal semi-Lagrangian schemes transport point values along characteristics together with gradients and Hessians, while projection semi-Lagrangian schemes rely on L2 projection and connect to semi-Lagrangian discontinuous Galerkin ideas with shared degrees of freedom.",{"name":82,"@type":73,"acceptedAnswer":83},"Which Hermite elements and approximation spaces are used?",{"text":84,"@type":76},"The schemes are built using Bell and Argyris Hermite finite elements, and the document also describes a reduced Hsieh-Clough-Tocher (rHCT) approximation space defined via nodal values and partial 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