[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82904-en":3,"doc-seo-82904-105":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},82904,8796095461564,"Liam","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Convergence Analysis of a MAC Scheme for the Barotropic Euler System","Convergence analysis is developed for a Marker-and-Cell (MAC) numerical scheme applied to the barotropic Euler system. A recently established Lax-type convergence theorem is used to prove unconditional convergence to a dissipative weak solution, and convergence to a strong solution on its existence interval. Relative energy error estimates are derived up to the lifespan of a strong solution without requiring uniform boundedness. Under boundedness of the numerical solutions, an optimal relative energy rate of 1 is shown, yielding a convergence rate of 1/2 for the numerical solutions, supported by numerical experiments.","arXiv :2607 .05105v1 [math .NA] 6 Jul 2026  \nConvergence analysis of a MAC scheme for the barotropic Euler system  \nBangwei She∗ Congying Wang† Jin Zhao‡  \nAbstract  \nWe study a Marker-and-Cell (MAC) scheme for the barotropic Euler system. First, we apply the recently developed Lax-type convergence theorem to show the convergence of the MAC scheme to i) a dissipative weak solution unconditionally and ii) a strong solution as long as it exists. Second, We derive relative energy error estimates up to the lifespan of a strong solution, without assuming uniform boundedness of the numerical sequence. Additionally, assuming the boundedness of the numerical solutions, we obtain the optimal relative energy rate of 1, corresponding to a convergence rate of 1/2 for the numerical solutions. Finally, we corroborate our theoretical results by numerical experiments.  \nKeywords: Barotropic Euler system; Marker and Cell scheme; Convergence analysis.  \nAMS Subject Classification: 65M12, 76M20  \nContents  \n1 Introduction 2  \n2 Solution concepts and consistent approximations 3  \n2.1 Solution concepts ............................... 3  \n2.2 Consistent approximations .......................... 4  \n3 The MAC scheme and convergence 5  \n3.1 The MAC scheme ............................... 5  \n3.2 Stability, consistency, and convergence ................... 7  \n4 Error estimates 10  \n5 Numerical experiments 13  \n∗ Academy for Multidisciplinary Studies, Capital Normal University, West 3rd Ring North Road 105, Beijing 100048, P. R. China. [bangweishe@cnu.edu.cn](bangweishe@cnu.edu.cn)  \n†School of Mathematical Sciences, Capital Normal University, West 3rd Ring North Road 105, Beijing 100048, P. R. China. [congyingw@163.com](congyingw@163.com)  \n‡Academy for Multidisciplinary Studies, Capital Normal University, West 3rd Ring North Road 105, Beijing 100048, P. R. China. [zjin@cnu.edu.cn](zjin@cnu.edu.cn)  \n6 Conclusion 15  \nA Proof of Lemma 3.5 18  \nB Relative energy norm 20  \n1 Introduction  \nWe consider the barotropic Euler system  \n􀀸 ∂t ϱ + ∇ · m = 0 ,  \n􀀼􀀺 ∂tm + ∇ · m~~ ~~~~ ~~m + ∇p(ϱ) = 0 (1)  \nin the time-space cylinder [0, T] × Ω . Here ϱ , m, and p(ϱ) = ϱγ are the density, momentum, and pressure of the fluid, respectively, and γ > 1 is the adiabatic exponent. The system is complemented by the initial data  \nϱ(0 , ·) = ϱ0 > 0 , m(0 , ·) = m 0 , (2)  \nand periodic boundary conditions, where the domain Ω is identified with the flat torus  \nΩ = Td ≡ 􀀀[0 , 1]| {0 , 1}􀀁 d .  \nOver the past few decades, many numerical methods have been successfully applied to the approximation of the Euler equations; see, e.g., Toro [24], Feistauer et al. [6], BenArtzi et al. [3], Godunov [11], Shu and Osher [23], and LeVeque [18] . However, a rigorous convergence analysis of numerical methods remains difficult in general. Concerning the full Euler system, Feireisl et al. [7] analyzed the convergence of the fundamental Lax– Friedrichs and Rusanov schemes. They established a convergence framework based on the dissipative measure-valued weak–strong uniqueness principle, valid on the lifespan of the strong solution. Moreover, using this convergence theory, Luk´aˇcov´a-Medvid’ov´a and Yuan demonstrated the convergence of the classical Godunov method [22], which has also been extended to other methods, see e.g. [19, 1, 16, 5] . Concerning the barotropic Euler system, Arun and Krishnamurthy [2] have shown the convergence of a semi-implicit, entropy-stable finite volume scheme, under the boundedness of numerical solutions, to a dissipative measure-valued solution. We also mention the vanishing-viscosity approach by Feireisl et al. [10] . In the isentropic case, they introduced viscosity solutions and a viscosity finite volume method, and proved convergence of the numerical solutions to a dissipative measure-valued solution of the barotropic Euler system.  \nIn this paper, we study a Marker-and-Cell (MAC) scheme for the barotropic Euler system. The MAC scheme was originally introdu","cbCaiiGYk0yNKvle","https://ap.wps.com/l/cbCaiiGYk0yNKvle","pdf",600254,2,1,21,"English","en",105,"# Introduction\n# Solution Concepts and Consistent Approximations\n## Solution concepts\n## Consistent approximations\n# The MAC Scheme and Convergence\n## The MAC scheme\n## Stability, consistency, and convergence\n# Error Estimates\n# Numerical Experiments\n# Conclusion\n# Appendix A: Proof of Lemma 3.5\n# Appendix B: Relative energy norm","[{\"question\":\"What problem does the paper address?\",\"answer\":\"The paper studies convergence of a Marker-and-Cell (MAC) scheme for the barotropic Euler system and seeks results without imposing assumptions on the numerical sequences.\"},{\"question\":\"How is unconditional convergence established?\",\"answer\":\"The approach introduces Navier–Stokes-type artificial diffusion to the MAC scheme, yielding additional weak BV/negative L2 control on the velocity gradient, then applies a generalized Lax equivalence theory.\"},{\"question\":\"What convergence rate is obtained and under what condition?\",\"answer\":\"Assuming boundedness of the numerical solutions, the paper proves an optimal relative energy rate of 1, corresponding to a convergence rate of 1/2 for the numerical solutions.\"}]",1784183834,53,{"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},"convergence-analysis-of-a-mac-scheme-for-the-barotropic-euler-system","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/convergence-analysis-of-a-mac-scheme-for-the-barotropic-euler-system/82904/",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","2026-07-24","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 problem does the paper address?","Question",{"text":75,"@type":76},"The paper studies convergence of a Marker-and-Cell (MAC) scheme for the barotropic Euler system and seeks results without imposing assumptions on the numerical sequences.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is unconditional convergence established?",{"text":80,"@type":76},"The approach introduces Navier–Stokes-type artificial diffusion to the MAC scheme, yielding additional weak BV/negative L2 control on the velocity gradient, then applies a generalized Lax equivalence theory.",{"name":82,"@type":73,"acceptedAnswer":83},"What convergence rate is obtained and under what condition?",{"text":84,"@type":76},"Assuming boundedness of the numerical solutions, the paper proves an optimal relative energy rate of 1, corresponding to a convergence rate of 1/2 for the numerical 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