[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86042-en":3,"doc-seo-86042-105":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},86042,1099514067438,"River Wang","https://ap-avatar.wpscdn.com/avatar/100002539ee87300030?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780474512215547542",8,"Research & Report","A compact simple HWENO scheme with ADER time discretization for hyperbolic conservation laws II: triangular meshes","A compact high-order HWENO method is developed for hyperbolic conservation laws on triangular meshes, using ADER time discretization. The Lax–Wendroff procedure converts time derivatives into spatial derivatives so that cell averages of the derivatives are obtained from a time-accurate solution evaluated as Gaussian points on cell interfaces via the Green–Gauss theorem, rather than through direct evolution as in conventional HWENO schemes. Compared with RK-HWENO on unstructured meshes, the new approach avoids extra reconstruction and time-advancing equations, improving efficiency while retaining non-oscillatory high-order accuracy near discontinuities.","arXiv :2607 . 10809v1 [math .NA] 12 Jul 2026  \nA compact simple HWENO scheme with ADER time discretization for hyperbolic conservation laws II: triangular  \nmeshes  \nDongmi Luo 1 , Zhuang Zhao*2 , Jianxian Qiu3 , Yibing Chen4  \nAbstract  \nA compact and high order HWENO scheme using ADER (Arbitrary high order using DERivatives) time discretization is developed for hyperbolic conservation lawson the triangular mesh, which is the extension of the work on the structured mesh ([Luo et. al](Luo et. al). (2024) [27]). The Lax-Wendroff procedure is employed to convert time derivatives to spatial derivatives. Thanks to this, the cell averages of the derivatives of the solution can be obtained by the time accurate solution as Gaussian points along the cell interfaces through the Green-Gauss theorem instead of by the evolution solution directly in the conventional HWENO methods. Comparing with the existing RungeKutta HWENO (RK-HWENO) method on the unstructured mesh ([Zhao et. al](Zhao et. al). (2025)  \n[42]), the new method has the following advantages. Firstly, the RK-HWENO method must solve the additional equations for reconstructions and time advancing, which is avoided for the new method. Secondly, the HWENO reconstruction in the new method is performed once per time step and is different from the RK-HWENO method, in which the reconstruction is performed several times every time step. Because of these advantages the new method is more efficient than the RK-HWENO method with smaller numerical errors and less computational costs. Besides, comparing with the existing ADER-WENO methods [6, 7] under the same order of accuracy, the stencil of the new method is more compact since the both the function and its first derivative values are used in the reconstruction of the HWENO schemes. Numerical examples demonstrate that the new method can achieve the high order for smooth solutions both in space and time, keep non-oscillatory near discontinuities.  \nKeywords: hyperbolic conservation laws, HWENO, Lax-Wendroff procedure, triangular mesh, high order  \n1 Institute of Applied Physics and Computational Mathematics, Beijing 100088, China. E-mail: dong[miluo@stu.xmu.edu.cn](miluo@stu.xmu.edu.cn).  \n2 School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, Xiamen University, Xiamen, Fujian 361005, China. E-mail: [zzhao@xmu.edu.cn](zzhao@xmu.edu.cn).  \n3 School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, Xiamen University, Xiamen, Fujian 361005, China. E-mail: [jxqiu@xmu.edu.cn](jxqiu@xmu.edu.cn).  \n4 Institute of Applied Physics and Computational Mathematics, Beijing 100088, China, E-mail: chen [yibing@iapcm.ac.cn](yibing@iapcm.ac.cn).  \n1 Introduction  \nIn [27], a compact simple Hermite weighted essentially non-oscillatory (HWENO) scheme with ADER (Arbitrary high order using DERivatives) time discretization for hyperbolic conservation laws was proposed on the structured meshes. In this paper, we go on with the work to develop the compact simple HWENO scheme for the numerical solutions of hyperbolic conservation laws in the form  \n􀀚 WWt(+x,  ·=W00(x, ) , (1.1)  \non the triangular meshes, where x = (x, y) ∈ Ω ⊂ R2 , Ω is a bounded domain, F =(f(W), g (W)), and W, f (W), g (W) are either scalars or vectors.  \nThe HWENO schemes are first proposed for solving one-dimensional nonlinear hyperbolic conservation law systems [30], whose idea of the reconstruction comes from the original WENO schemes [14, 23] . However both the function and its derivative values are evolved in time and employed in the reconstructions while the function values are adopted in the original WENO schemes [14, 23] . Then the HWENO schemes are extended to the twodimensional structured meshes [31] and unstructured meshes [45] . But these methods have some drawbacks. Firstly, the linear weights of the point values in two","cbCaitSZuDxn96FH","https://ap.wps.com/l/cbCaitSZuDxn96FH","pdf",1807640,4,1,30,"English","en",105,"# Introduction\n## Background and motivation for HWENO schemes\n## Extending HWENO from structured to unstructured/triangular meshes\n## Time discretization strategies: RK, ADER, GRP and Lax–Wendroff based methods","[{\"question\":\"What main numerical idea does the paper introduce for time discretization?\",\"answer\":\"It uses ADER time discretization together with the Lax–Wendroff procedure to convert time derivatives into spatial derivatives.\"},{\"question\":\"How does the method compute derivative information on triangular meshes?\",\"answer\":\"It obtains cell averages of solution derivatives using a Green–Gauss approach by evaluating the time-accurate solution as Gaussian points along cell interfaces.\"},{\"question\":\"What efficiency advantages are claimed over RK-HWENO and other existing methods?\",\"answer\":\"The approach avoids solving additional equations for reconstruction and time advancing, performs HWENO reconstruction once per time step, and uses a more compact stencil than comparable ADER-WENO methods while achieving smaller numerical errors and lower computational cost.\"}]",1784208040,76,{"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},"a-compact-simple-hweno-scheme-with-ader-time-discretization-for-hyperbolic-conservation-laws-ii-triangular-meshes","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"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":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/a-compact-simple-hweno-scheme-with-ader-time-discretization-for-hyperbolic-conservation-laws-ii-triangular-meshes/86042/",{"url":52,"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","2026-07-27","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 main numerical idea does the paper introduce for time discretization?","Question",{"text":75,"@type":76},"It uses ADER time discretization together with the Lax–Wendroff procedure to convert time derivatives into spatial derivatives.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the method compute derivative information on triangular meshes?",{"text":80,"@type":76},"It obtains cell averages of solution derivatives using a Green–Gauss approach by evaluating the time-accurate solution as Gaussian points along cell interfaces.",{"name":82,"@type":73,"acceptedAnswer":83},"What efficiency advantages are claimed over RK-HWENO and other existing methods?",{"text":84,"@type":76},"The approach avoids solving additional equations for reconstruction and time advancing, performs HWENO reconstruction once per time step, and uses a more compact stencil than comparable ADER-WENO methods while achieving smaller numerical errors and lower computational 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