scholarly journals A stochastic Galerkin method for the Boltzmann equation with uncertainty

2016 ◽  
Vol 315 ◽  
pp. 150-168 ◽  
Author(s):  
Jingwei Hu ◽  
Shi Jin
Keyword(s):  
Boltzmann Equation ◽  
Galerkin Method ◽  
Stochastic Galerkin Method ◽  
The Boltzmann Equation ◽  
Stochastic Galerkin

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

2017 ◽  
Vol 10 (2) ◽  
pp. 465-488 ◽  
Author(s):  
Ruiwen Shu ◽  
Jingwei Hu ◽  
Shi Jin
Keyword(s):  
Boltzmann Equation ◽  
Galerkin Method ◽  
Collision Operator ◽  
Collision Kernel ◽  
Wavelet Bases ◽  
Random Inputs ◽  
Stochastic Galerkin Method ◽  
The Boltzmann Equation ◽  
Accuracy Result ◽  
Stochastic Galerkin

AbstractWe 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.

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