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A Stochastic Galerkin Method for the Boltzmann Equation with Multi-Dimensional Random Inputs Using Sparse Wavelet Bases

Part of: Equations of mathematical physics and other areas of application Numerical analysis: Ordinary differential equations

Published online by Cambridge University Press:  09 May 2017

Ruiwen Shu ,
Jingwei Hu  and
Shi Jin
Ruiwen Shu*
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA
Jingwei Hu*
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
Shi Jin*
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA Institute of Natural Sciences, School of Mathematical Science, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240, China
*
*Corresponding author. Email addresses:rshu2@math.wisc.edu (R. Shu), jingweihu@purdue.edu (J. Hu), sjin@wisc.edu (S. Jin)
*Corresponding author. Email addresses:rshu2@math.wisc.edu (R. Shu), jingweihu@purdue.edu (J. Hu), sjin@wisc.edu (S. Jin)
*Corresponding author. Email addresses:rshu2@math.wisc.edu (R. Shu), jingweihu@purdue.edu (J. Hu), sjin@wisc.edu (S. Jin)
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Abstract

We propose a stochastic Galerkin method using sparse wavelet bases for the Boltzmann equation with multi-dimensional random inputs. Themethod uses locally supported piecewise polynomials as an orthonormal basis of the random space. By a sparse approach, only a moderate number of basis functions is required to achieve good accuracy in multi-dimensional random spaces. We discover a sparse structure of a set of basis-related coefficients, which allows us to accelerate the computation of the collision operator. Regularity of the solution of the Boltzmann equation in the random space and an accuracy result of the stochastic Galerkin method are proved in multi-dimensional cases. The efficiency of the method is illustrated by numerical examples with uncertainties from the initial data, boundary data and collision kernel.

Keywords

Uncertainty quantificationBoltzmann equationstochastic Galerkin methodssparse wavelets

MSC classification

Secondary: 35Q20: Boltzmann equations 65L60: Finite elements, Rayleigh-Ritz, Galerkin and collocation methods
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 2 , May 2017 , pp. 465 - 488
Copyright
Copyright © Global-Science Press 2017 

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