A stochastic Galerkin method for the Boltzmann equation with uncertainty

作者：Hu, Jingwei^*; Jin, Shi

来源：Journal of Computational Physics, 2016, 315: 150-168.

DOI：10.1016/j.jcp.2016.03.047

• Uncertainty quantification
• The Boltzmann equation
• Random input
• Generalized polynomial chaos
• Stochastic Galerkin method
• Singular value decomposition
• Fast spectral method

We develop a stochastic Galerkin method for the Boltzmann equation with uncertainty. The method is based on the generalized polynomial chaos (gPC) approximation in the stochastic Galerkin framework, and can handle random inputs from collision kernel, initial data or boundary data. We show that a simple singular value decomposition of gPC related coefficients combined with the fast Fourier-spectral method (in velocity space) allows one to compute the high-dimensional collision operator very efficiently. In the spatially homogeneous case, we first prove that the analytical solution preserves the regularity of the initial data in the random space, and then use it to establish the spectral accuracy of the proposed stochastic Galerkin method. Several numerical examples are presented to illustrate the validity of the proposed scheme.

• 出版日期2016-6-15
• 单位上海交通大学