A stochastic Galerkin method for the Boltzmann equation with uncertainty

JOURNAL OF COMPUTATIONAL PHYSICS

卷:315

Article

Hu, Jingwei¹  Jin, Shi^2,3,4 

[1] Purdue Univ, Dept Math, W Lafayette, IN 47907 USA

[2] Univ Wisconsin, Dept Math, Madison, WI 53706 USA

[3] Shanghai Jiao Tong Univ, MOE LSEC, Inst Nat Sci, Dept Math, Shanghai 200240, Peoples R China

[4] Shanghai Jiao Tong Univ, SHL MAC, Shanghai 200240, Peoples R China

关键词: Uncertainty quantification;    The Boltzmann equation;    Random input;    Generalized polynomial chaos;    Stochastic Galerkin method;    Singular value decomposition;    Fast spectral method;   

DOI  :  10.1016/j.jcp.2016.03.047

来源: Elsevier

【 摘 要 】

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. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free