[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85340-en":3,"doc-seo-85340-105":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},85340,13056703020460,"Valentina","https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923",8,"Research & Report","Algebraic Invariant Quadratization Schemes for Cahn–Hilliard Equations","This paper introduces the Algebraic Invariant Quadratization (AIQ) framework for rational-like energy functions by adding auxiliary variables that act as Casimir functions in an extended dynamical system. AIQ is combined with symplectic Runge–Kutta time discretization and Fourier pseudo-spectral spatial discretization to produce fully discrete schemes. The method is applied to Cahn–Hilliard equations in isotropic and anisotropic settings, analyzing dispersion, spinodal instability, coarsening, and missing-orientation effects while preserving the original energy evolution and capturing physical phenomena.","arXiv :2607 . 11569v1 [math .NA] 13 Jul 2026  \nAlgebraic Invariant Quadratization Schemes for Cahn–Hilliard Equations  \nFei Xie 1 Nan Lu2 Yajuan Sun 1 ∗  \n1 State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; University of Chinese Academy of Sciences, Beijing 100049, China  \n2 Department of Mathematics, Shanghai Normal University, Shanghai 200234, China  \n[xiefei2021@lsec.cc.ac.cn](xiefei2021@lsec.cc.ac.cn) [lunan@shnu.edu.cn](lunan@shnu.edu.cn) [sunyj@lsec.cc.ac.cn](sunyj@lsec.cc.ac.cn)  \nAbstract  \nIn this paper, we propose the Algebraic Invariant Quadratization (AIQ) framework for rational-like energy functions by introducing auxiliary variables, which are interpreted as Casimir functions of the extended system. Combining AIQ with symplectic Runge–Kutta (SRK) methods in time and Fourier pseudo-spectral discretization in space, we obtain fully discrete schemes. The resulting schemes are applied to Cahn–Hilliard equations in both the isotropic and anisotropic cases. We analyze the discrete dispersion relation, spinodal instability, coarsening behavior, and missing-orientation phenomena. Numerical comparisons demonstrate the improved performance superiority of the proposed method over the stabilized invariant energy quadratization (S-IEQ) and scalar auxiliary variable (SAV) methods in preserving the original energy evolution and capturing the underlying physical phenomena.  \nKeywords: Cahn–Hilliard equation; anisotropic phase-field model; energy-stable scheme; dispersion relation; coarsening rate.  \nMSC 2020: 65P10; 65L05; 65M12 .  \n1 Introduction  \nFor many physical systems, preserving the intrinsic geometric structures is essential for accurately capturing their qualitative behavior. These structures may include symplecticity, mass conservation, momentum conservation, Casimir invariants, and energy conservation or dissipation. Typical examples arise from Hamiltonian systems in celestial mechanics, plasma and fluid models with geometric invariants, and biological or phase-field models governed by dissipative dynamics. In systems where the dynamics are constrained by energy surfaces or driven by energy dissipation, the energy law plays an important role in stability and longtime evolution. A numerical method that fails to respect this structure may introduce artificial energy drift and lead to unreliable long-time predictions. Therefore, constructing energy-preserving or energy-dissipative schemes is a central issue in structure-preserving numerical methods.  \nVarious numerical methods have been developed to preserve energy structures, including the discrete variational derivative method [7], the average vector field (AVF) method [4], and discrete-gradient-based integral-preserving frameworks for PDEs [5] . These methods usually require more complicated constructionsand may involve numerical quadrature. Recently, auxiliary-variable methods such as IEQ [22] and SAV [17] have provided more systematic and efficient reformulation strategies and have been widely used for gradient flows. These methods usually require less detailed structural analysis and are easier to construct and implement. They are effective and flexible. However, they usually preserve a modified energy rather than the original energy itself, which may lead to energy errors or drift and affect the accuracy of long-time physical simulations.  \n∗ The work of the third author is supported by the National Natural Science Foundation of China (Grant No. 12271513) .  \nTo maintain consistency with the original energy law and improve the reliability of long-time simulations, recent studies have focused on auxiliary-variable formulations that preserve the original energy structure. For Hamiltonian systems with polynomial first integrals, the multiple quadratic auxiliary variable (MQAV) method was introduced in [18] to preserve the original invariants. More recently, for systems refor","cbCaiaLpXvsGMDWe","https://ap.wps.com/l/cbCaiaLpXvsGMDWe","pdf",5974099,4,1,27,"English","en",105,"# Introduction\n## Structure-preserving numerical methods\n## Auxiliary-variable frameworks (IEQ, SAV)\n## Algebraic Invariant Quadratization (AIQ) framework\n# Cahn–Hilliard equations as model problems\n## Structural properties of the CH model","[{\"question\":\"What is the main idea behind the Algebraic Invariant Quadratization (AIQ) framework?\",\"answer\":\"AIQ introduces auxiliary variables interpreted as Casimir functions of an extended system, enabling rational-like energy functions to be quadratized while supporting preservation of key invariants.\"},{\"question\":\"How are the fully discrete numerical schemes constructed in this work?\",\"answer\":\"The paper combines AIQ with symplectic Runge–Kutta methods in time and Fourier pseudo-spectral discretization in space, yielding fully discrete schemes.\"},{\"question\":\"What aspects of the Cahn–Hilliard dynamics are analyzed to assess the proposed method?\",\"answer\":\"The discrete dispersion relation, spinodal instability, coarsening behavior, and missing-orientation phenomena are analyzed for both isotropic and anisotropic cases.\"}]",1784202627,68,{"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},"algebraic-invariant-quadratization-schemes-for-cahnhilliard-equations","",{"@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/algebraic-invariant-quadratization-schemes-for-cahnhilliard-equations/85340/",{"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","2026-07-24","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 is the main idea behind the Algebraic Invariant Quadratization (AIQ) framework?","Question",{"text":75,"@type":76},"AIQ introduces auxiliary variables interpreted as Casimir functions of an extended system, enabling rational-like energy functions to be quadratized while supporting preservation of key invariants.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are the fully discrete numerical schemes constructed in this work?",{"text":80,"@type":76},"The paper combines AIQ with symplectic Runge–Kutta methods in time and Fourier pseudo-spectral discretization in space, yielding fully discrete schemes.",{"name":82,"@type":73,"acceptedAnswer":83},"What aspects of the Cahn–Hilliard dynamics are analyzed to assess the proposed method?",{"text":84,"@type":76},"The discrete dispersion relation, spinodal instability, coarsening behavior, and missing-orientation phenomena are analyzed for both isotropic and anisotropic 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