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

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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].
Funding Information
  • Key Technologies Research and Development Program (2021YFA1001200)
  • Ministry of Science and Technology of the People's Republic of China (24301021)
  • Research Grants Council, University Grants Committee (14303022, 14301423)