[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85415-en":3,"doc-seo-85415-105":29,"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":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},85415,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1782109480056885918",8,"Research & Report","Reduced-Order Solution for Rarefied Gas Flow by Proper Generalised Decomposition","Modeling rarefied gas flow through the Boltzmann equation demands substantial computing resources because the kinetic formulation is high-dimensional and involves multiple characteristic scales, making conventional methods too slow for parametric studies and engineering optimization loops. This paper develops an a priori reduced-order approach using proper generalised decomposition to solve the parametrised Shakhov kinetic model equation. Separated representations of low-rank solutions, data, and operators transform the problem into a few low-dimensional subproblems, avoiding the curse of dimensionality. A global solution across the rarefaction parameter enables fast, repeated queries. Numerical results show high accuracy with major reductions in CPU time and memory usage.","arXiv :2505 . 19555v1 [math .NA] 26 May 2025  \nReduced-Order Solution for Rarefied Gas Flow by Proper Generalised Decomposition  \nWei Sua,b,∗, Xi Zouc  \na Division of Emerging Interdisciplinary Areas, The Hong Kong University of Science  \nand Technology, Clear Water Bay, Kowloon, Hong Kong, China b Department of Mathematics, The Hong Kong University of Science and  \nTechnology, Clear Water Bay, Kowloon, Hong Kong, China cZienkiewicz Institute for Modelling, Data and AI, Swansea University, Swansea, SA1  \n8EN, United Kingdom  \nAbstract  \nModelling rarefied gas flow via the Boltzmann equation plays a vital role in many areas. Due to the high dimensionality of this kinetic equation and the coexistence of multiple characteristic scales in the transport processes, conventional solution strategies incur prohibitively high computational costs and are inadequate for rapid response for parametric analysis and optimisation loops in engineering design simulations. This paper proposes an a priori reduced-order method based on the proper generalised decomposition to solve the high-dimensional, parametrised Shakhov kinetic model equation. This method reduces the original problem into a few low-dimensional problem by formulating separated representations for the low-rank solution, as well as data and operators in the equation, thereby overcoming the curse of dimensionality. Furthermore, a general solution can be calculated once and for all in the whole range of the rarefaction parameter, enabling fast and multiple queries to a specific solution at any point in the parameter space. Numerical examples are presented to demonstrate the capability of the method to simulate rarefied gas flow with high accuracy and significant reduction in CPU time and memory requirements.  \nKeywords: reduced-order modelling, proper generalised decomposition, Boltzmann equation, parametrised rarefied gas flow  \n∗[weisu@ust.hk](weisu@ust.hk)  \nPreprint submitted to Journal of Computational Physics May 27, 2025  \n1. Introduction  \nKinetic theory has demonstrated its practical significance since the 1950sin describing the dynamics of rarefied gas flows encountered in various engineering applications, such as nano-/microelectromechanical systems, highaltitude flights, unconventional natural gas production, vacuum science, etc. The centre of the theory is the Boltzmann equation, which determines the thermofluid properties of a gaseous system by providing evolution information on the probability distribution of gas molecules [1] . The Boltzmann equation for monatomic gases reads  \n∂f   ∂f   ∂f  \n∂t + v′ · ∂x′ + F · ∂v′ = C (f, f) ,  \nwhere f (t, x′, v′) is the one-particle velocity distribution function, which is a function of the time t, spatial position x′ and molecular velocity v′ ; Fis external driven force; and C is the Boltzmann collision integral operator, representing the variation rate of the velocity distribution function due to molecular collisions. Methods for solving the Boltzmann equation are generally categorised as stochastic and deterministic. Stochastic methods use simulation particles to trace the movements and collisions of molecules statistically [2], whereas deterministic methods discretise the velocity distribution function in all independent variables and solve the resultant system algebraically [3] . Both methods are expensive in computational time and memory requirements, mainly because of the high dimensionality of the equation and the multiscale nature of rarefied gas transport.  \nThe Boltzmann equation is seven-dimensional. To solve it deterministically, one first discretises the velocity space by Nv discrete points, resulting in Nv partial differential equations (PDE) continuous in time and spatial space that can be solved by finite difference, finite volume, or finite element techniques combing with a time-stepping scheme. Typically, Nv is around 104 ∼ 106 for a satisfactory solution, leading to a large number of degrees of freedom (DoF) . ","cbCaioCdxIiw37lR","https://ap.wps.com/l/cbCaioCdxIiw37lR","pdf",1559232,1,24,"English","en",105,"# Introduction\n## Kinetic theory and the Boltzmann equation\n## Stochastic vs deterministic solution strategies\n## Computational challenges from dimensionality and multiscale effects\n## Need for reduced-order modeling","[{\"question\":\"Why are conventional Boltzmann-equation solution strategies computationally expensive for rarefied gas flow?\",\"answer\":\"They face prohibitively high costs due to the equation’s high dimensionality and the coexistence of multiple characteristic scales, making them inadequate for rapid parametric analysis and optimization loops.\"},{\"question\":\"What reduced-order method does the paper propose, and what is its main idea?\",\"answer\":\"It proposes an a priori reduced-order method based on proper generalised decomposition. By using separated representations for low-rank solutions, data, and operators, the high-dimensional parametrised Shakhov kinetic model is reduced to a few low-dimensional subproblems.\"},{\"question\":\"How does the proposed approach support fast evaluation over the rarefaction parameter?\",\"answer\":\"A general solution can be computed once for the entire range of the rarefaction parameter, allowing fast and repeated queries to obtain the solution at any point in the parameter space.\"}]",1784203236,60,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":27},"reduced-order-solution-for-rarefied-gas-flow-by-proper-generalised-decomposition","",{"@graph":35,"@context":85},[36,53,68],{"@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/reduced-order-solution-for-rarefied-gas-flow-by-proper-generalised-decomposition/85415/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 are conventional Boltzmann-equation solution strategies computationally expensive for rarefied gas flow?","Question",{"text":75,"@type":76},"They face prohibitively high costs due to the equation’s high dimensionality and the coexistence of multiple characteristic scales, making them inadequate for rapid parametric analysis and optimization loops.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What reduced-order method does the paper propose, and what is its main idea?",{"text":80,"@type":76},"It proposes an a priori reduced-order method based on proper generalised decomposition. By using separated representations for low-rank solutions, data, and operators, the high-dimensional parametrised Shakhov kinetic model is reduced to a few low-dimensional subproblems.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the proposed approach support fast evaluation over the rarefaction parameter?",{"text":84,"@type":76},"A general solution can be computed once for the entire range of the rarefaction parameter, allowing fast and repeated queries to obtain the solution at any point in the parameter space.","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":24},{"code":4,"msg":5,"data":92},[93,97,101,105,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":28,"slug":108},5,"Comic","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":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]