论文标题
对于不确定性的Boltzmann方程的傅立叶 - 加利尔金光谱法的收敛
Convergence of the Fourier-Galerkin spectral method for the Boltzmann equation with uncertainties
论文作者
论文摘要
众所周知，傅立叶 - 加盖尔光谱方法一直是确定性玻尔兹曼方程数的数值近似值的流行方法，并严格证明了光谱精度。在本文中，我们将表明，傅立叶 - 加盖金光谱法的这种光谱收敛也适用于玻尔兹曼方程，并且碰撞内核和初始条件都会引起不确定性。我们的证明是基于经过精心设计的新建立的空间和规范，并将速度变量和随机变量完全考虑到了速度变量和随机变量。对于将来的研究，这一理论结果将为进一步显示全局部化系统的收敛提供稳固的基础，在该系统同时将速度和随机变量同时离散化。
It is well-known that the Fourier-Galerkin spectral method has been a popular approach for the numerical approximation of the deterministic Boltzmann equation with spectral accuracy rigorously proved. In this paper, we will show that such a spectral convergence of the Fourier-Galerkin spectral method also holds for the Boltzmann equation with uncertainties arising from both collision kernel and initial condition. Our proof is based on newly-established spaces and norms that are carefully designed and take the velocity variable and random variables with their high regularities into account altogether. For future studies, this theoretical result will provide a solid foundation for further showing the convergence of the full-discretized system where both the velocity and random variables are discretized simultaneously.
