Young Researchers Workshop: Stochastic and deterministic methods in kinetic theory

A stochastic Galerkin method for the Boltzmann equation with high dimensional random inputs using sparse grids

Jingwei Hu

Purdue University


We propose a stochastic Galerkin method using sparse grids for the Boltzmann equation with high dimensional random inputs. The method uses locally supported piecewise polynomials as an orthonormal basis of the random space. By a sparse grid technique, only a moderate number of basis functions are required to achieve good accuracy in high 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 multidimensional case. The efficiency of the method is illustrated by numerical examples with uncertainties from the initial data, boundary data and collision kernel.