计算数学 ›› 2022, Vol. 44 ›› Issue (3): 289-304.DOI: 10.12286/jssx.j2021-0887

• 青年评述 •    下一篇

非线性玻尔兹曼方程的傅里叶谱方法

胡婧玮   

  1. 华盛顿大学应用数学系, 美国华盛顿州西雅图市 98195
  • 收稿日期:2021-11-27 出版日期:2022-07-14 发布日期:2022-08-03
  • 作者简介:胡婧玮,现任美国华盛顿大学应用数学系副教授.2006年和2011年分别在北京大学和威斯康星大学麦迪逊分校获学士和博士学位;2011-2014年在德州大学奥斯汀分校从事博士后研究;2014-2021年任普渡大学助理教授和副教授.主要研究领域为空气动理学中的数值方法设计与分析,及它们在多尺度建模,科学工程计算中的应用.曾获美国国家科学基金授予的CAREER Award (2017);现任Kinetic\&Related Models和La Matematica杂志的编委.
  • 基金资助:
    美国国家科学基金DMS-1620250,DMS-2153208和CBET-1854829资助项目.
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胡婧玮. 非线性玻尔兹曼方程的傅里叶谱方法[J]. 计算数学, 2022, 44(3): 289-304.

Hu Jingwei. FOURIER SPECTRAL METHODS FOR NONLINEAR BOLTZMANN EQUATIONS[J]. Mathematica Numerica Sinica, 2022, 44(3): 289-304.

FOURIER SPECTRAL METHODS FOR NONLINEAR BOLTZMANN EQUATIONS

Hu Jingwei   

  1. Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA
  • Received:2021-11-27 Online:2022-07-14 Published:2022-08-03
玻尔兹曼方程作为空气动理学中最基本的方程之一,是连接微观牛顿力学和宏观连续介质力学的重要桥梁.该方程描述了一个由大量粒子组成的复杂系统的非平衡态时间演化:除了基本的输运项,其最重要的特性是粒子间的相互碰撞由一个高维,非局部且非线性的积分算子来描述,从而给玻尔兹曼方程的数值求解带来非常大的挑战.在过去的二十年间,基于傅里叶级数的谱方法成为了数值求解玻尔兹曼方程的一种很受欢迎且有效的确定性算法.这主要归功于谱方法的高精度及它可以被快速傅里叶变换加速的特质.本文将回顾玻尔兹曼方程的傅里叶谱方法,具体包括方法的导出,稳定性和收敛性分析,快速算法,以及在一大类基于碰撞的空气动理学方程中的推广.
关键词： 玻尔兹曼方程, 空气动理学, 谱方法, 稳定性, 快速傅里叶变换, 低秩逼近
The Boltzmann equation is one of the fundamental equations in kinetic theory, and serves as a basic building block connecting microscopic Newtonian mechanics and macroscopic continuum mechanics. Numerical approximation of the Boltzmann equation is a challenging problem mainly due to its high-dimensional, nonlocal, and nonlinear collision integral. Over the past 20 years, the spectral method based on Fourier series (or trigonometric polynomials) has become a popular and efficient deterministic method for solving the Boltzmann equation, manifested by its high accuracy and possibility of being accelerated by the fast Fourier transform. This paper aims to review the Fourier-Galerkin spectral method for the Boltzmann equation, stability and convergence of the method, fast algorithms, and generalizations to various Boltzmann-type collisional kinetic equations.
Key words: Boltzmann equation, kinetic theory, spectral method, stability, fast Fourier transform, low rank approximation

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