[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82427-en":3,"doc-seo-82427-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"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},82427,7971461741311,"Ophelia","https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826",8,"Research & Report","Generalized skew-gradient embedding for thermodynamically consistent systems","The skew-gradient embedding (SGE) framework rewrites a thermodynamically consistent system as a generalized gradient flow by embedding its zero-energy contribution into a skew-symmetric operator. In a time-discrete method, the operator profiles can be evaluated at earlier time levels, preserving skew-symmetry so the discrete energy balance contribution vanishes, often enabling explicit decoupling of multiphysics. Non-uniqueness yields generalized skew-gradient embeddings (GSGE) via admissible gauge freedom; least-squares in any positive definite metric selects a unique minimum Hilbert–Schmidt gauge. The theory supports regularized approximations and invariant-preserving gauges, with rank-two cases characterized by a Jacobi criterion. Applied to compatible MAC discretizations of incompressible Navier–Stokes, it produces finite-dimensional rank-two Poisson–GENERIC form and preserves exact discrete energy under implicit midpoint. For Cahn–Hilliard–Navier–Stokes, GSGE–BDF2 preserves mass, dissipates discrete energy unconditionally, and permits decoupled implementation.","arXiv :2607 .09617v1 [math .NA] 10 Jul 2026  \nGeneralized skew-gradient embedding for thermodynamically  \nconsistent systems  \nXuelong Gua,1 , Qi Wanga,1,∗  \na Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA  \nAbstract  \nThe skew-gradient embedding (SGE) framework [16] reformulates a thermodynamically consistent system as a generalized gradient flow by embedding its zero-energy contribution in a skew-symmetric operator. In a time-discrete scheme, the profiles defining this operator may be evaluated at previous time levels. The resulting operator remains skew-symmetric, so its contribution to the discrete energy balance vanishes; this explicit treatment often decouples multiphysics systems. We show that this operator is not unique: the admissible gauges form an affine space, and we call the resulting family generalized skew-gradient embeddings (GSGE) . For any positive definite metric, least squares selects a unique minimum-Hilbert–Schmidt gauge, and the native metric recovers SGE. This construction also gives regularized approximations, corrections of non-neutral residuals, and gauges that preserve prescribed invariants. For rank-two gauges, we use a necessary and sufficient Jacobi criterion. Applying this criterion to a compatible MAC discretization of the incompressible Navier–Stokes equations gives a finite-dimensional rank-two Poisson–GENERIC formulation at the semi-discrete level; the implicit midpoint rule preserves this rank-two GENERIC structure at the fully discrete level and satisfies the exact discrete energy law. For the Cahn–Hilliard– Navier–Stokes system, the regularized GSGE–BDF2 scheme preserves mass, dissipates the discrete energy unconditionally, and admits a decoupled implementation.  \nKeywords: generalized Onsager principle, skew-gradient embedding, gauge freedom, Poisson structure, GENERIC, structure-preserving discretization  \n1. Introduction  \nThermodynamically consistent models in classical electrodynamics, fluid and solid mechanics, quantum mechanics, complex fluids, phase-field hydrodynamics, and statistical physics often arise from conservation laws coupled with constitutive relations.  \n∗ Corresponding author.  \nEmail address: [QWANG@math.sc.edu](QWANG@math.sc.edu) (Qi Wang)  \n[1](1 X.G. is)[ X.G. is](1 X.G. is) supported by NSF [award OIA-2242812. Q.W. is](award OIA-2242812. Q.W. is) partially supported by NSF awards DMS-2038080 and OIA-2242812, and DOE award DE-SC0025229 .  \nTheir coupled reversible and irreversible parts admit several structural descriptions, such as the port-Hamiltonian, metriplectic, and GENERIC (General Equation for the Non-Equilibrium Reversible–Irreversible Coupling) formulations [31 , 43 , 2 , 37] . After a compatible spatial discretization, or at the formal PDE level used to identify the structural identities, such systems can be written in the following generalized form  \n∂t Φ = M (Φ)G(Φ) + J(Φ), G(Φ) := ∇F (Φ) . (1)  \nHere Φ denotes the state variable in a real inner-product space H , F (Φ) is the free energy, G(Φ) = ∇F (Φ) is the thermodynamic force, and M (Φ) is a symmetric negative semidefinite mobility operator [33, 34 , 44] . The generalized Onsager principle provides a systematic route to such models and has been applied to complex fluids, liquid crystals, active matter, and interacting particle systems [51 , 52 , 26 , 23 , 18] . The vector field J(Φ) satisfying ⟨G(Φ), J(Φ)⟩ = 0 is the zero-energy contribution (ZEC)[47 , 48]; it collects reversible mechanisms such as convection, transport, rotation, and electromagnetic coupling. This orthogonality condition gives the energy dissipation law  \nddt F (Φ(t)) = ⟨G(Φ), ∂t Φ⟩ = ⟨G(Φ), M (Φ)G(Φ)⟩ ≤ 0. (2)  \nThe skew-gradient embedding (SGE) framework [16] exploits the zero-energy property of J by embedding it into a skew-symmetric operator. Specifically, (1) is rewritten as  \n∂t Φ = 􀀐 M (Φ) + S∗ (Φ) 􀀑 G(Φ), S∗ (Φ) = G∥(ΦG)(J)~~ ~~∥(Φ2) . (3)  \nSince S∗ (Φ) is skew-symmetric, the reversible ","cbCaiuIsXQb84UA9","https://ap.wps.com/l/cbCaiuIsXQb84UA9","pdf",430157,1,19,"English","en",105,"# Abstract\n# Introduction\n## Generalized form and energy dissipation\n## Skew-gradient embedding (SGE) and zero-energy contribution\n## Generalized skew-gradient embeddings (GSGE) and gauge space\n## Least-squares gauge selection and regularization\n## Invariant preservation and rank-two Poisson geometry","[{\"question\":\"What does the skew-gradient embedding (SGE) framework achieve for thermodynamically consistent systems?\",\"answer\":\"SGE embeds the zero-energy contribution into a skew-symmetric operator so that its discrete contribution to the energy balance vanishes, preserving the discrete energy law and enabling efficient explicit or higher-order energy-stable schemes.\"},{\"question\":\"Why are generalized skew-gradient embeddings (GSGE) introduced?\",\"answer\":\"The skew operator used in SGE is not unique; admissible gauges form an affine space. GSGE describes the resulting family of embeddings that all preserve the energy balance while allowing different gauge choices.\"},{\"question\":\"How are rank-two gauges characterized in the paper?\",\"answer\":\"For rank-two gauges, the paper uses a necessary and sufficient Jacobi criterion. This criterion enables a rank-two Poisson–GENERIC formulation at the semi-discrete level and is preserved by the implicit midpoint rule at the fully discrete level with an exact discrete energy law.\"}]",1784180329,48,{"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":85,"head_meta":87,"extra_data":89,"updated_unix":27},"generalized-skew-gradient-embedding-for-thermodynamically-consistent-systems","",{"@graph":35,"@context":84},[36,53,67],{"@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/generalized-skew-gradient-embedding-for-thermodynamically-consistent-systems/82427/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What does the skew-gradient embedding (SGE) framework achieve for thermodynamically consistent systems?","Question",{"text":74,"@type":75},"SGE embeds the zero-energy contribution into a skew-symmetric operator so that its discrete contribution to the energy balance vanishes, preserving the discrete energy law and enabling efficient explicit or higher-order energy-stable schemes.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"Why are generalized skew-gradient embeddings (GSGE) introduced?",{"text":79,"@type":75},"The skew operator used in SGE is not unique; admissible gauges form an affine space. GSGE describes the resulting family of embeddings that all preserve the energy balance while allowing different gauge choices.",{"name":81,"@type":72,"acceptedAnswer":82},"How are rank-two gauges characterized in the paper?",{"text":83,"@type":75},"For rank-two gauges, the paper uses a necessary and sufficient Jacobi criterion. This criterion enables a rank-two Poisson–GENERIC formulation at the semi-discrete level and is preserved by the implicit midpoint rule at the fully discrete level with an exact discrete energy law.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":21,"doc_module":4,"doc_module_name":45,"category_name":135,"show_sort_weight":105,"slug":136},"General","general"]