Convergence of the Fourier-Galerkin spectral method for the Boltzmann equation with uncertainties
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866929336591843328 |
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| author | Liu, Liu Qi, Kunlun |
| author_facet | Liu, Liu Qi, Kunlun |
| contents | It is well-known that the Fourier-Galerkin spectral method has been a popular approach for the numerical approximation of the deterministic Boltzmann equation with spectral accuracy rigorously proved. In this paper, we will show that such a spectral convergence of the Fourier-Galerkin spectral method also holds for the Boltzmann equation with uncertainties arising from both collision kernel and initial condition. Our proof is based on newly-established spaces and norms that are carefully designed and take the velocity variable and random variables with their high regularities into account altogether. For future studies, this theoretical result will provide a solid foundation for further showing the convergence of the full-discretized system where both the velocity and random variables are discretized simultaneously. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_04083 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Convergence of the Fourier-Galerkin spectral method for the Boltzmann equation with uncertainties Liu, Liu Qi, Kunlun Numerical Analysis Analysis of PDEs Primary 35Q20, 65M12, Secondary 65M70, 45G10 It is well-known that the Fourier-Galerkin spectral method has been a popular approach for the numerical approximation of the deterministic Boltzmann equation with spectral accuracy rigorously proved. In this paper, we will show that such a spectral convergence of the Fourier-Galerkin spectral method also holds for the Boltzmann equation with uncertainties arising from both collision kernel and initial condition. Our proof is based on newly-established spaces and norms that are carefully designed and take the velocity variable and random variables with their high regularities into account altogether. For future studies, this theoretical result will provide a solid foundation for further showing the convergence of the full-discretized system where both the velocity and random variables are discretized simultaneously. |
| title | Convergence of the Fourier-Galerkin spectral method for the Boltzmann equation with uncertainties |
| topic | Numerical Analysis Analysis of PDEs Primary 35Q20, 65M12, Secondary 65M70, 45G10 |
| url | https://arxiv.org/abs/2212.04083 |