[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83704-en":3,"doc-seo-83704-105":30,"detail-sidebar-cat-0-en-105":92},{"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},83704,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","Isotropy and Galilean Invariance of Lattice Boltzmann Method","This work investigates the two-dimensional, nine-velocity (D2Q9) lattice Boltzmann model and its symmetry properties. It shows that second-order accuracy fails unless cubic velocity terms are removed and explains ways to eliminate parasitic contributions. The equilibrium distribution choice does not affect the second-order equivalent PDE, but it strongly influences third-order effects. A new Oblique Dipole Benchmark quantifies cubic terms and higher-order equilibrium choices via obliquely propagating vortex dipoles with periodic boundaries.","arXiv :2607 .03222v1 [math .AP] 3 Jul 2026  \nIsotropy and Galilean invariance of Lattice Boltzmann Method: Theoretical and numerical analysis using  \noblique dipole benchmark ∗  \nFran¸cois Dubois†1,2, Souleyman Kadri-Harouna‡3, Pierre Lallemand§4, and  \nMohamed Mahdi Tekitek¶3  \n1 Conservatoire National des Arts et M´etiers, Laboratoire de M´ecanique des Structures et des Syst`emes Coupl´es, F-75003 Paris, France  \n2 Laboratoire de Math´ematiques d’Orsay, CNRS, Universit´e Paris-Saclay, Bˆat. 307,  \nF-91405 Orsay, France  \n3 MIA Laboratory, La Rochelle Universit´e, Ave. Albert Einstein, BP 33 060, 17 031 La Rochelle, France  \n4 Beijing Computational Science Research Center, Zhangguancun Software Park II, Haidian District, Beijing 100094, China  \nJuly 7, 2026  \nAbstract  \nThis work focuses on the two-dimensional, nine-velocity (D2Q9) lattice Boltzmann model. First, we show that the D2Q9 scheme cannot achieve secondorder accuracy unless the cubic velocity terms are neglected, and we explain how some of these parasitic terms can be eliminated. Second, we demonstrate that the standard choice of the equilibrium distribution has no effect on the equivalent PDE at second order. Finally, we numerically investigate the effect of these cubic terms and study different choices of equilibrium distributions using a new benchmark called the Oblique Dipole Benchmark, which describes obliquely propagating 2D vortex dipoles with periodic boundary conditions.  \nKeywords: LB scheme with projection, Cubic Terms, MRT, Numerical dissipation, Taylor expansion, ABCD method, Isotropy, Galilean invariance, double vortex  \nIntroduction  \nDubois and Lallemand [9] show that the choice of the equilibrium distribution function is central to recovering the Navier-Stokes equations at second order in the lattice Boltzmann framework. However, residual second-order defects persist in D2Q9 and D3Q19 schemes, primarily due to cubic terms. This issue was highlighted by Dellar [5] and others [26, 15, 14] . It raises questions regarding equilibrium construction  \n∗ Contribution submitted for publication in Computers & Fluids and presented at the ICMMES Conference, 2025 .  \n†[fdubois@lecnam.fr](fdubois@lecnam.fr)  \n‡[souleyman.kadri-harouna@univ-lr.fr](souleyman.kadri-harouna@univ-lr.fr)  \n§ plalleman1@free.fr  \n¶ Corresponding [author: mohamed.tekitek@univ-lr.fr](author: mohamed.tekitek@univ-lr.fr)  \nIsotropy and Galilean invariance of LBM  \nin BGK, multiple-relaxation-time (MRT), and elaborate schemes like the projected D2Q9 [9] . These residual defects impact Galilean invariance, which must be assessed when the effective shear viscosity is independent of the advection velocity.  \nThe choice of equilibrium deserves particular attention. This choice has no effect on the second-order equivalent PDE, so it can be selected freely at that level. However, it does impact the third order, as previously emphasized by Qian et al.  \n[25] and recently by Lallemand et al. [16] . This impact is crucial because it can fundamentally affect stability and physical modeling. To assess these effects, we designed a numerical test case. This test reveals dispersive errors associated with odd higher-order terms. It also shows the effective numerical viscosity induced by even higher-order truncation errors, particularly regarding their impact on isotropy and Galilean invariance.  \nThe proposed benchmark, inspired by Bruneau and Clercx [2], consists of a double-vortex (dipole) flow with periodic boundary conditions instead of no-slip (Dirichlet) conditions. Unlike standard tests such as the Taylor-Green vortex orlid-driven cavity, this case involves vortical structures advected by a spatially nonuniform velocity field. Consequently, errors stemming from cubic terms and higherorder numerical viscosity (often negligible in classical configurations) become pronounced enough for quantitative analysis. In this benchmark, the double-vortex is self-advected by a non-constant velocity field. The iso","cbCaiheKDv2c76L3","https://ap.wps.com/l/cbCaiheKDv2c76L3","pdf",4998933,5,1,20,"English","en",105,"# Abstract\n# Introduction\n# Isotropy and Galilean invariance of LBM\n# 1 Lattice Boltzmann scheme and equivalent PDE","[{\"question\":\"Why does the D2Q9 lattice Boltzmann scheme not reach second-order accuracy?\",\"answer\":\"Second-order accuracy requires neglecting cubic velocity terms; otherwise parasitic contributions remain. The work explains how some of these terms can be eliminated to restore the desired accuracy level.\"},{\"question\":\"Does the choice of equilibrium distribution affect the second-order equivalent PDE?\",\"answer\":\"No. The paper states that equilibrium choice has no effect on the second-order equivalent PDE, so it can be selected freely at that order.\"},{\"question\":\"How is the Oblique Dipole Benchmark used to study isotropy and Galilean invariance?\",\"answer\":\"The benchmark propagates obliquely moving 2D vortex dipoles under periodic boundary conditions. It measures isotropy via symmetry loss or deviation from a pseudo-spectral reference and makes cubic-term and numerical dissipation effects prominent for quantitative comparison.\"}]",1784189844,50,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"isotropy-and-galilean-invariance-of-lattice-boltzmann-method","",{"@graph":36,"@context":86},[37,54,69],{"@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":53},"https://docshare.wps.com/document/isotropy-and-galilean-invariance-of-lattice-boltzmann-method/83704/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"Why does the D2Q9 lattice Boltzmann scheme not reach second-order accuracy?","Question",{"text":76,"@type":77},"Second-order accuracy requires neglecting cubic velocity terms; otherwise parasitic contributions remain. The work explains how some of these terms can be eliminated to restore the desired accuracy level.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"Does the choice of equilibrium distribution affect the second-order equivalent PDE?",{"text":81,"@type":77},"No. The paper states that equilibrium choice has no effect on the second-order equivalent PDE, so it can be selected freely at that order.",{"name":83,"@type":74,"acceptedAnswer":84},"How is the Oblique Dipole Benchmark used to study isotropy and Galilean invariance?",{"text":85,"@type":77},"The benchmark propagates obliquely moving 2D vortex dipoles under periodic boundary conditions. It measures isotropy via symmetry loss or deviation from a pseudo-spectral reference and makes cubic-term and numerical dissipation effects prominent for quantitative comparison.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,110,114,119,122,126,129,133],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":29,"slug":113},6,"Technology","technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":22,"slug":125},9,"Religion & Spirituality","religion-spirituality",{"id":22,"doc_module":4,"doc_module_name":46,"category_name":127,"show_sort_weight":22,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":46,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":46,"category_name":135,"show_sort_weight":20,"slug":136},19,"General","general"]