[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85809-en":3,"doc-seo-85809-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"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},85809,8796095461564,"Liam","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","An Asymptotic-Preserving Reduced-Order Method for Parametrised Rarefied Gas Flow by Proper Generalised Decomposition","Modelling rarefied gas flow using the Boltzmann equation is essential yet difficult because high dimensionality and multiple characteristic scales make conventional solvers too expensive for engineering design simulations and inadequate for fast parametric response. Based on proper generalised decomposition (PGD), an a priori reduced-order approach is developed for the high-dimensional, parametrised Shakhov kinetic model. Separated low-rank representations reduce the problem to low-dimensional subproblems. Synthetic-equation solutions are integrated into PGD to preserve hydrodynamic asymptotics, enabling automatic reduction to Navier–Stokes solvers and efficient evaluation across rarefaction parameters. Numerical tests confirm accuracy with major computational cost reductions.","arXiv :2607 . 10085v1 [math .NA] 11 Jul 2026  \nHighlights  \nAn asymptotic-preserving reduced-order method for parametrised rarefied gas flow by proper generalised decomposition  \nLuowei Yin, Wei Su  \n• Proposed PGD reduced-order model for high-dimensional/parametrised Boltzmann equation  \n• Captured hydrodynamic limit by combining synthetic iterative method  \n• Proposed separated representation for dealing with arbitrary geometries  \n• Significantly reduced computational cost and maintained certain accuracy  \nAn asymptotic-preserving reduced-order method for parametrised rarefied gas flow by proper generalised  \ndecomposition Luowei Yina , Wei Sua,b,∗  \na Department of Mathematics, The Hong Kong University of Science and  \nTechnology, Clear Water Bay, Kowloon, Hong Kong, China  \nb Division of Emerging Interdisciplinary Areas, The Hong Kong University of Science and  \nTechnology, Clear Water Bay, Kowloon, Hong Kong, China  \nAbstract  \nModelling rarefied gas flow using the Boltzmann equation is vital in many areas. Due to the high dimensionality and coexistence of multiple characteristic scales, conventional solution strategies to this equation incur prohibitively high computational costs and are inadequate for rapid response in engineering design simulations. Based on proper generalised decomposition (PGD), we propose an a priori, asymptotic-preserving reduced-order method to solve the high-dimensional, parametrised Shakhov kinetic model equation. The method reduces the original problem to a few low-dimensional problems by formulating separated representations for the low-rank solution, thereby mitigating the curse of dimensionality. To capture the hydrodynamic asymptotics, we incorporated solutions of some synthetic equations into the PGD algorithm. This treatment allows the PGD solver to automatically reduce toa macroscopic solver for the Navier-Stokes equations, whose solution naturally exhibits low-rank structure. By treating the rarefaction parameter asan additional coordinate, a parametrised solution can be computed once and for all over the entire range of rarefaction, enabling fast multiple queries to any points in the parameter space. Numerical examples are presented to demonstrate the capability of the method to simulate rarefied gas flow with certain accuracy and a significant reduction in computational costs. Keywords: reduced-order modelling, proper generalised decomposition,  \n∗[weisu@ust.hk](weisu@ust.hk)  \nasymptotic preserving, parametrised Boltzmann equation  \n1. Introduction  \nKinetic theory has demonstrated its practical significance in describing the dynamics of rarefied gas flows encountered in various engineering applications, such as microelectromechanical systems, high-altitude flights, unconventional natural gas production, extreme ultraviolet lithography, 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 gas reads as follows:  \n∂f   ∂f   ∂f  \n+ v′ · + F · = C (f, f)  \n∂t ∂x′ ∂v′ ,  \nwhere f (t, x′, v′) is the one-particle velocity distribution function, which is a function of time t, spatial position x′ and molecular velocity v ′ ; F is an 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 stochastic [2] or deterministic [3] . Both methods are expensive in computational time and memory requirements, due to the high dimensionality and multiscale nature of the equation. To solve the equation deterministically, one first discretises the velocity space by Nv discrete points, resulting in Nv partial differential equations (PDEs) continuous in time and spatial space that can be solved using conventional techniques of computational fluid","cbCaiiEVOllO8Vlk","https://ap.wps.com/l/cbCaiiEVOllO8Vlk","pdf",1925167,1,35,"English","en",105,"# Highlights\n# Abstract\n# Keywords\n# Introduction","[{\"question\":\"What problem does the proposed method address?\",\"answer\":\"It targets the high computational cost of solving the high-dimensional, parametrised Boltzmann/Shakhov kinetic models for rarefied gas flows, especially across ranges of Knudsen numbers required for engineering design.\"},{\"question\":\"How does the method reduce computational complexity?\",\"answer\":\"It uses proper generalised decomposition (PGD) with separated representations to express the low-rank solution and reduce the original problem to a small set of low-dimensional subproblems.\"},{\"question\":\"How does the method ensure correct hydrodynamic behavior in limiting regimes?\",\"answer\":\"It incorporates solutions of synthetic equations into the PGD algorithm so that the solver preserves hydrodynamic asymptotics and automatically reduces to a macroscopic Navier–Stokes solver structure.\"}]",1784206390,88,{"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":85,"head_meta":87,"extra_data":89,"updated_unix":27},"an-asymptotic-preserving-reduced-order-method-for-parametrised-rarefied-gas-flow-by-proper-generalised-decomposition","",{"@graph":35,"@context":84},[36,53,67],{"@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/an-asymptotic-preserving-reduced-order-method-for-parametrised-rarefied-gas-flow-by-proper-generalised-decomposition/85809/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What problem does the proposed method address?","Question",{"text":74,"@type":75},"It targets the high computational cost of solving the high-dimensional, parametrised Boltzmann/Shakhov kinetic models for rarefied gas flows, especially across ranges of Knudsen numbers required for engineering design.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does the method reduce computational complexity?",{"text":79,"@type":75},"It uses proper generalised decomposition (PGD) with separated representations to express the low-rank solution and reduce the original problem to a small set of low-dimensional subproblems.",{"name":81,"@type":72,"acceptedAnswer":82},"How does the method ensure correct hydrodynamic behavior in limiting regimes?",{"text":83,"@type":75},"It incorporates solutions of synthetic equations into the PGD algorithm so that the solver preserves hydrodynamic asymptotics and automatically reduces to a macroscopic Navier–Stokes solver 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