Spectral Convergence of a Semi-discretized Numerical System for the Spatially Homogeneous Boltzmann Equation with Uncertainties

SIAM/ASA Journal on Uncertainty Quantification, Volume 12, Issue 3, Page 812-841, September 2024.
Abstract.In this paper, we study the Boltzmann equation with uncertainties and prove that the spectral convergence of the semi-discretized numerical system holds in a combined velocity and random space, where the Fourier spectral method is applied for approximation in the velocity space, whereas the generalized polynomial chaos (gPC)-based stochastic Galerkin (SG) method is employed to discretize the random variable. Our proof is based on a delicate energy estimate for showing the well-posedness of the numerical solution as well as a rigorous control of its negative part in our well-designed functional space that involves high-order derivatives of both the velocity and random variables. This paper rigorously justifies the statement proposed in Remark 4.4 of [J. Hu and S. Jin, J. Comput. Phys., 315 (2016), pp. 150–168].