[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81650-en":3,"doc-seo-81650-105":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},81650,8796095360427,"Lucas Martin","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",8,"Research & Report","A New Asymptotic-Preserving Dual Formulation Finite-Volume Method for the Compressible Euler Equations","The document presents a finite-volume numerical method for the compressible Euler equations targeting regimes with small Mach numbers. In such cases the system becomes stiff, causing explicit schemes to face severe time-step restrictions and inaccurate diffusion. The work develops an asymptotic-preserving (AP) approach that stays uniformly accurate and stable over all Mach numbers by using a primitive nonconservative formulation with nonconservative hyperbolic splitting, semi-implicit treatment of the stiff pressure through a Poisson-type elliptic equation, and a conservative central-upwind scheme for post-processed high-Mach solutions.","arXiv :2604 .26111v2 [math .NA] 10 Jul 2026  \nA New Asymptotic-Preserving Dual Formulation Finite-Volume Method for the Compressible Euler  \nEquations  \nAlina Chertock∗, Smadar Karni†, Alexander Kurganov‡, and Lorenzo Micalizzi§  \nAbstract  \nThe paper focuses on the development of numerical methods for the compressible Euler equations. It is well-known that if the Mach number is small, the system becomes stiff and hence explicit schemes suffer from severe time-step restrictions, making them inefficient or even impractical. Our objective is to develop an asymptotic preserving (AP) scheme that remains uniformly accurate and stable across all Mach numbers.  \nInstead of the conservative hyperbolic flux splitting approach, which is widely used to design AP schemes, we consider a primitive (nonconservative) formulation and introduce anonconservative hyperbolic splitting. The resulting system is discretized using a semi-implicit approach: the stiff part is handled semi-implicitly using second-order central differences, while the nonstiff part is treated explicitly using a second-order path-conservative centralupwind discretization. A key feature of our method is that the pressure at each time level is computed by solving a well-posed Poisson-type elliptic equation, thereby enforcing the AP property. Simultaneously, we evolve the conservative form of the system using a semidiscrete central-upwind (CU) scheme. At the end of each stage of the time discretization, we perform a special post-processing that selects the appropriate numerical solution depending on the Mach number. This guarantees that in low-Mach-number regimes, the solution is obtained by the AP nonconservative scheme, while in higher-Mach-number regimes, a sharp and physically relevant solution is computed by the conservative CU scheme.  \nNumerical experiments confirm that the proposed AP scheme achieves the expected second order of accuracy and that the time-step constraint is independent of the Mach number, making it a robust and efficient alternative to conventional explicit methods.  \nKey words: Compressible Euler equations; low Mach number; asymptotic preserving (AP) schemes; hyperbolic splitting; semi-implicit methods; deferred correction.  \nAMS subject classification: 65M08, 65M20, 76M12, 35L65, 76N15, 35B40 .  \n∗ Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA; cher[tock@math.ncsu.edu](tock@math.ncsu.edu)  \n†Department of Mathematics, University of Michigan, 48109, USA; [karni@umich.edu](karni@umich.edu)  \n‡Department of Mathematics and Shenzhen International Center for Mathematics, Southern University of Science and Technology, Shenzhen, 518055, China; [alexander@sustech.edu.cn](alexander@sustech.edu.cn)  \n§ Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA; [lmicali@ncsu.edu](lmicali@ncsu.edu)  \n2 A. Chertock, S. Karni, A. Kurganov & L. Micalizzi  \n1 Introduction  \nThe paper focuses on the compressible Euler equations, which, like any other hyperbolic system of PDEs, are characterized by a finite speed of propagation. This plays a crucial role in the development of explicit numerical methods, for which a major stability requirement is to keep the time steps inversely proportional to the maximum wave speed over the entire computational domain.  \nIt is well-known that low-Mach-number flows pose several major challenges for numerical simulations. A distinctive feature of such regimes is the appearance of both slow material waves, which transport quantities like entropy and vorticity, and fast acoustic waves, whose speeds scale inversely with the Mach number. As the Mach number decreases, the resulting stiffness imposes severe time-step restrictions on explicit methods and leads to excessive numerical diffusion, making such schemes inefficient or even impractical for real applications. Fully-implicit methods can address the stiffness, but have their own drawbacks: they tend to oversmear material wave","cbCairA0CVjoyTpQ","https://ap.wps.com/l/cbCairA0CVjoyTpQ","pdf",29258735,3,1,33,"English","en",105,"# Abstract\n# Key words\n# AMS subject classification\n# Introduction\n## Low-Mach-number challenges\n## Asymptotic-preserving (AP) schemes and motivation","[{\"question\":\"Why do explicit numerical schemes struggle for low-Mach-number compressible Euler flows?\",\"answer\":\"Low Mach numbers make the Euler system stiff because acoustic waves scale with the inverse of the Mach number. This leads to severe time-step restrictions for explicit methods and can cause excessive numerical diffusion, reducing efficiency and accuracy.\"},{\"question\":\"How does the proposed method maintain asymptotic-preserving (AP) behavior across Mach numbers?\",\"answer\":\"It combines a nonconservative formulation with nonconservative hyperbolic splitting and a semi-implicit discretization where the stiff pressure is obtained by solving a well-posed Poisson-type elliptic equation. A special post-processing selects between the AP nonconservative solution in low-Mach regimes and a conservative central-upwind solution in higher-Mach regimes.\"},{\"question\":\"What numerical strategy is used for the stiff and nonstiff parts of the discretization?\",\"answer\":\"The stiff part is handled semi-implicitly using second-order central differences, while the nonstiff part is treated explicitly using a second-order path-conservative central-upwind discretization. This design aims to keep the time-step constraint independent of the Mach number.\"}]",1784175157,83,{"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-new-asymptotic-preserving-dual-formulation-finite-volume-method-for-the-compressible-euler-equations","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/a-new-asymptotic-preserving-dual-formulation-finite-volume-method-for-the-compressible-euler-equations/81650/",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","2026-07-26","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},"Why do explicit numerical schemes struggle for low-Mach-number compressible Euler flows?","Question",{"text":75,"@type":76},"Low Mach numbers make the Euler system stiff because acoustic waves scale with the inverse of the Mach number. This leads to severe time-step restrictions for explicit methods and can cause excessive numerical diffusion, reducing efficiency and accuracy.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed method maintain asymptotic-preserving (AP) behavior across Mach numbers?",{"text":80,"@type":76},"It combines a nonconservative formulation with nonconservative hyperbolic splitting and a semi-implicit discretization where the stiff pressure is obtained by solving a well-posed Poisson-type elliptic equation. A special post-processing selects between the AP nonconservative solution in low-Mach regimes and a conservative central-upwind solution in higher-Mach regimes.",{"name":82,"@type":73,"acceptedAnswer":83},"What numerical strategy is used for the stiff and nonstiff parts of the discretization?",{"text":84,"@type":76},"The stiff part is handled semi-implicitly using second-order central differences, while the nonstiff part is treated explicitly using a second-order path-conservative central-upwind discretization. This design aims to keep the time-step constraint independent of the Mach number.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]