非线性玻尔兹曼方程的傅里叶谱方法

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.

FOURIER SPECTRAL METHODS FOR NONLINEAR BOLTZMANN EQUATIONS

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.