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

A Stochastic Galerkin Method for the Boltzmann Equation with Multi-Dimensional Random Inputs Using Sparse Wavelet Bases
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Year:    2017

Numerical Mathematics: Theory, Methods and Applications, Vol. 10 (2017), Iss. 2 : pp. 465–488

Abstract

We propose a stochastic Galerkin method using sparse wavelet bases for the Boltzmann equation with multi-dimensional random inputs. The method 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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/nmtma.2017.s12

Numerical Mathematics: Theory, Methods and Applications, Vol. 10 (2017), Iss. 2 : pp. 465–488

Published online:    2017-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    24

Keywords:   

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