Spectral convergence of a semi-discretized numerical system for the spatially homogeneous Boltzmann equation with uncertainties
<|reference_start|>Spectral convergence of a semi-discretized numerical system for the spatially homogeneous Boltzmann equation with uncertainties: 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].<|reference_end|>
@article{liu2024spectral,
    title={Spectral convergence of a semi-discretized numerical system for the
  spatially homogeneous Boltzmann equation with uncertainties},
    author={Liu Liu and Kunlun Qi},
    journal={arXiv preprint arXiv:2402.07060},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.07060},
    primaryClass={math.NA cs.NA math.AP}
}