[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84698-en":3,"doc-seo-84698-105":29,"detail-sidebar-cat-0-en-105":95},{"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},84698,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","Accelerated iterative method for solving the steady-state Boltzmann equation","Efficient simulation of steady-state rarefied gas flows is hindered by the high-dimensional collision integral and severe numerical stiffness in the near-continuum regime. A modified Newton method combined with a macroscopic synthetic system (Newton-MS) is introduced to solve the steady-state Boltzmann equation with a quadratic collision operator. Outer nonlinear iterations use modified Newton updates, while each Newton correction is handled by an inner source iteration accelerated through Chapman–Enskog closure and discretized via discontinuous Galerkin for reduced cost. Numerical tests confirm high efficiency for multiple flow configurations.","arXiv :2607 .03910v1 [math .NA] 4 Jul 2026  \nAccelerated iterative method for solving the steady-state  \nBoltzmann equation  \nPei Zhang∗, Zhenning Cai†, Yanli Wang‡  \nAbstract  \nThe efficient simulation of steady-state rarefied gas flows remains a significant computational challenge due to the high dimensionality of the collision integral and the severe numerical stiffness in the near-continuum regime. In this work, we propose a modified Newton method equipped with a macroscopic synthetic system (Newton-MS) for the steady-state Boltzmann equation with the quadratic collision operator. In Newton-MS, the modified Newton iteration is utilized as the outer nonlinear solver, while each Newton correction equation is solved by an inner source iteration, where the linearized collision operator is utilized to approximate the quadratic collision model, and it is reduced into a linear iteration. Moreover, a macroscopic synthetic system based on Chapman-Enskog closure is derived to accelerate the convergence of the linear inner iteration in the continuum limit. Besides, the fully discrete macroscopic synthetic system is deduced under the framework of the discontinuous Galerkin method to reduce computational cost compared to directly discretizing the continuous macroscopic synthetic system. Several numerical examples, including the 1D Fourier, Couette flow problem, and the 2D cavity flow and thermal-driven cavity flow, are studied to validate the high efficiency of Newton-MS.  \nKeywords: steady-state Boltzmann equation; Newton-MS; macroscopic synthetic acceleration; fast Fourier spectral method  \n1 Introduction  \nRarefied gas dynamics constitutes a pivotal branch of fluid mechanics, essential for understanding multiscale flow phenomena in high-altitude aerothermodynamics, micro-electromechanical systems (MEMS), and specialized gas transport processes. The flow regimes are characterized by the Knudsen number (Kn), defined as the ratio of the molecular mean free path to a characteristic macroscopic length scale. Although the Euler or Navier-Stokes equations suffice for the continuum regime (Kn ≪ 1), they become physically inadequate as the rarefaction effects increase. Consequently, the Boltzmann equation, which governs the evolution of the single-particle probability distribution function in phase space, serves asthe fundamental framework that is valid across all flow regimes. However, the high dimensionality of phase space and the complex quadratic collision term pose substantial challenges for efficient numerical simulations.  \nTraditional numerical methods for solving the Boltzmann equation are generally categorized into stochastic methods and deterministic methods. Direct Monte Carlo simulation (DSMC) [1 , 9 , 22] is one of the most popular stochastic methods that is effective in simulating high-speed rarefied gas flows. However, it suffers from computational inefficiency and inherent statistical noise in low-speed regimes. The Unified Gas-Kinetic Wave-Particle (UGKWP) method couples deterministic waves for continuum flows with stochastic particles for kinetic non-equilibrium, covering the full Knudsen number regime. Unlike DSMC, which is costly and time-step limited in near-continuum flows, UGKWP adaptively weights waves and particles by local Knudsen number, recovering a fluid solver in continuum and a particle method in rarefied regimes. This hybrid reduces noise and cost while preserving kinetic accuracy for multiscale gas dynamics [20] .  \n∗ Beijing Computational Science Research Center, Beijing, China, email: [zhangpei@csrc.ac.cn](zhangpei@csrc.ac.cn).  \n†Department of Mathematics, National University of Singapore, Singapore, 119076, email: [matcz@nus.edu.sg](matcz@nus.edu.sg).‡Beijing Computational Science Research Center, Beijing, China, email: [ylwang@csrc.ac.cn](ylwang@csrc.ac.cn).  \nConversely, deterministic methods are often preferred for continuous or low-speed flows where the statistical fluctuations must be minimized. The discrete velocity method (DVM) [2 , ","cbCaiqzCs6tjCmnr","https://ap.wps.com/l/cbCaiqzCs6tjCmnr","pdf",837797,1,26,"English","en",105,"# Introduction\n## Rarefied gas dynamics and governing equations\n## Existing numerical approaches for the Boltzmann equation\n## Convergence issues near the continuum limit\n# Method and formulation\n## Modified Newton method and Newton-MS framework\n## Inner source iteration with linearized collision operator\n## Macroscopic synthetic system and continuum acceleration\n## Discontinuous Galerkin discretization for cost reduction\n# Numerical validation\n## 1D Fourier and Couette flow cases\n## 2D cavity flow and thermal-driven cavity flow cases","[{\"question\":\"What problem does Newton-MS address in steady-state Boltzmann simulations?\",\"answer\":\"It targets slow convergence and stiffness in the near-continuum regime, where source iteration becomes prohibitively inefficient and numerical errors can contaminate the converged solution.\"},{\"question\":\"How is the modified Newton method integrated into Newton-MS?\",\"answer\":\"Modified Newton iterations act as the outer nonlinear solver, while each Newton correction equation is solved using an inner source iteration.\"},{\"question\":\"How does the macroscopic synthetic system accelerate convergence?\",\"answer\":\"A macroscopic synthetic system derived via Chapman–Enskog closure accelerates the convergence of the linear inner iteration in the continuum limit.\"},{\"question\":\"Which discretization strategy is used to reduce computational cost?\",\"answer\":\"The fully discrete macroscopic synthetic system is derived within the discontinuous Galerkin framework, avoiding direct discretization of the continuous synthetic 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problem does Newton-MS address in steady-state Boltzmann simulations?","Question",{"text":75,"@type":76},"It targets slow convergence and stiffness in the near-continuum regime, where source iteration becomes prohibitively inefficient and numerical errors can contaminate the converged solution.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the modified Newton method integrated into Newton-MS?",{"text":80,"@type":76},"Modified Newton iterations act as the outer nonlinear solver, while each Newton correction equation is solved using an inner source iteration.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the macroscopic synthetic system accelerate convergence?",{"text":84,"@type":76},"A macroscopic synthetic system derived via Chapman–Enskog closure accelerates the convergence of the linear inner iteration in the continuum limit.",{"name":86,"@type":73,"acceptedAnswer":87},"Which discretization strategy is used to reduce computational cost?",{"text":88,"@type":76},"The fully discrete macroscopic synthetic system is derived within the discontinuous Galerkin framework, avoiding direct discretization of the continuous synthetic system.","https://schema.org",{"og:url":51,"og:type":91,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":93,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":96},[97,101,105,109,114,119,124,127,132,135,139],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Story & 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