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arXiv · 2402.07060

Spectral convergence of a semi-discretized numerical system for the spatially homogeneous Boltzmann equation with uncertainties

Liu Liu·Kunlun Qi
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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, J. Hu and S. Jin, J. Comput. Phys., 315 (2016), pp. 150-168].

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Liu Liu, Kunlun Qi. 2024-02-10. Spectral convergence of a semi-discretized numerical system for the spatially homogeneous Boltzmann equation with uncertainties. https://arxiv.org/abs/2402.07060

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