[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86506-en":3,"doc-seo-86506-105":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},86506,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","Symplectic Hamiltonian Direct Discontinuous Galerkin Method for Wave Propagation","This paper develops a symplectic Hamiltonian direct discontinuous Galerkin (DDG) approach for wave propagation, covering both linear and semilinear wave equations. In an auxiliary-variable-free DG setting, the symmetry of the numerical flux bilinear form is proved equivalent to the existence of a discrete Hamiltonian structure. As a consequence, symmetric interior penalty and symmetric DDG (SDDG) schemes inherit this structure, while Baumann–Oden, DDG, and BR2 do not. Fully discrete symplectic methods are built by pairing SDDG with symplectic time integrators, including error bounds and numerical validation.","arXiv :2607 . 10652v1 [math .NA] 12 Jul 2026  \nSymplectic Hamiltonian Direct Discontinuous Galerkin Method  \nfor Wave Propagation  \nHaomiao Li 1,2 Yumiao Li3 Jiaxin Wang2 Tiegang Liu2,4 Kun Wang2,4,*  \n1 Sino-French Carbon Neutrality Research Center, Centrale Pekin, Beihang University, Beijing 100191, China  \n2 LMIB and School of Mathematical Sciences, Beihang University, Beijing 100191, China  \n3 Hunan Key Laboratory for Computation and Simulation in Science and Engineering, National Center for Applied Mathematics in Hunan, Xiangtan University, Xiangtan 411105, Hunan, PR China  \n4 International Research Center for Mathematics and Interdisciplinary Sciences, Hangzhou International Innovation Institute of Beihang University, Hangzhou 311115, China  \n* Corresponding author. Email: [wangkun@buaa.edu](wangkun@buaa.edu)  \nAbstract  \nThis paper presents a symplectic Hamiltonian direct discontinuous Galerkin (DDG) method for approximating wave propagation problems, including the linear and semilinear wave equations. Within an auxiliary-variable-free DG framework, we prove that the symmetry of the numerical flux bilinear form is equivalent to the existence of a discrete Hamiltonian structure. It follows that methods such as the symmetric interior penalty method and the symmetric DDG (SDDG) method admit a discrete Hamiltonian structure, whereas schemes including the Baumann–Oden, DDG, and BR2 methods do not possess this property. Exploiting this structure, we construct fully discrete symplectic schemes by combining the SDDG spatial discretization with symplectic time integrators. We further derive error estimates for the SDDG method applied to semilinear wave equations, showing the optimal convergence rate for the displacement and the suboptimal convergence rate for the velocity. Numerical experiments validate the theoretical convergence rates and demonstrate that the symplectic Hamiltonian DDG method achieves superior long-time energy conservation and accuracy.  \nKeywords. Wave equations, Discontinuous Galerkin methods, Hamiltonian systems, Symplectic methods  \n1 Introduction  \nIn this paper, we consider the following semi-linear wave equation  \n∂ttu(x, t) + g(u(x, t)) = ∇ · (κ∇u(x, t)) + f(t), in Ω × J. (1.1)  \nwhere J is a finite time interval and Ω is an open and bounded domain. It arise in a wide range of applications, including plasmas, hydrodynamics, and magnetohydrodynamics [12] . The governing equation (1.1) belongs to the class of Hamiltonian partial differential equations (PDEs) . Hamiltonian systems preserve a symplectic structure on phase space. This property has motivated the development of symplectic integrators, which generate symplectic maps when applied to Hamiltonian ordinary differential equations (ODEs) [14] . Owing to their ability to respect the underlying geometric structure, such methods exhibit favourable long-time stability and nearenergy conservation, and have therefore become a standard tool for the numerical simulation of conservative dynamical systems [28] . Motivated by these successes, the concept of symplecticity has been extended from finite-dimensional ODEs to infinite-dimensional Hamiltonian PDEs.  \nTwo main approaches have been developed for this purpose, namely the multisymplectic method and the Hamiltonian method-of-lines method.  \nIn this work, we focus on the Hamiltonian method-of-lines framework, in contrast to the multisymplectic formulation which treats space and time on an equal footing [4, 21 , 22] but can be computationally demanding. In the method-of-lines approach, the Hamiltonian PDEis first discretized in space to obtain a finite-dimensional Hamiltonian ODE system, which is then integrated in time using symplectic time integrators. This separation of space and time simplifies the numerical analysis and allows one to directly exploit mature high-ordersymplectic integrators with well-understood stability and accuracy properties. Within this framework, the main challenge lies in the cons","cbCaimqZ0MuER1g9","https://ap.wps.com/l/cbCaimqZ0MuER1g9","pdf",2558412,4,1,41,"English","en",105,"# Introduction\n## Semi-linear wave equation and Hamiltonian PDE background\n## Method-of-lines vs multisymplectic approaches\n## DG discretizations for ∇·(κ∇u) and auxiliary-variable vs direct formulations\n## Interface treatment, stability/consistency, and Hamiltonian structure requirement","[{\"question\":\"What problem does the paper address?\",\"answer\":\"The paper targets wave propagation problems modeled by linear and semilinear wave equations, formulated as Hamiltonian PDEs.\"},{\"question\":\"How is the discrete Hamiltonian structure characterized in the method?\",\"answer\":\"It is shown that symmetry of the numerical flux bilinear form is equivalent to the existence of a discrete Hamiltonian structure within an auxiliary-variable-free DG framework.\"},{\"question\":\"Which DG schemes preserve the discrete Hamiltonian structure and which do not?\",\"answer\":\"Symmetric interior penalty and symmetric DDG (SDDG) admit a discrete Hamiltonian structure, whereas Baumann–Oden, DDG, and BR2 schemes do not.\"}]",1784212263,103,{"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},"symplectic-hamiltonian-direct-discontinuous-galerkin-method-for-wave-propagation","",{"@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/symplectic-hamiltonian-direct-discontinuous-galerkin-method-for-wave-propagation/86506/",{"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 problem does the paper address?","Question",{"text":75,"@type":76},"The paper targets wave propagation problems modeled by linear and semilinear wave equations, formulated as Hamiltonian PDEs.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the discrete Hamiltonian structure characterized in the method?",{"text":80,"@type":76},"It is shown that symmetry of the numerical flux bilinear form is equivalent to the existence of a discrete Hamiltonian structure within an auxiliary-variable-free DG framework.",{"name":82,"@type":73,"acceptedAnswer":83},"Which DG schemes preserve the discrete Hamiltonian structure and which do not?",{"text":84,"@type":76},"Symmetric interior penalty and symmetric DDG (SDDG) admit a discrete Hamiltonian structure, whereas Baumann–Oden, DDG, and BR2 schemes do 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