Spectral convergence of a semi-discretized numerical system for the spatially homogeneous Boltzmann equation with uncertainties
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909192010334208 |
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| author | Liu, Liu Qi, Kunlun |
| author_facet | Liu, Liu Qi, Kunlun |
| contents | 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]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_07060 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral convergence of a semi-discretized numerical system for the spatially homogeneous Boltzmann equation with uncertainties Liu, Liu Qi, Kunlun Numerical Analysis Analysis of PDEs Primary 35Q20, 65M12, Secondary 65M70, 45G10 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]. |
| title | Spectral convergence of a semi-discretized numerical system for the spatially homogeneous Boltzmann equation with uncertainties |
| topic | Numerical Analysis Analysis of PDEs Primary 35Q20, 65M12, Secondary 65M70, 45G10 |
| url | https://arxiv.org/abs/2402.07060 |