A stable multiplicative dynamical low-rank discretization for the linear Boltzmann-BGK equation
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916065588543488 |
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| author | Baumann, Lena Einkemmer, Lukas Klingenberg, Christian Kusch, Jonas |
| author_facet | Baumann, Lena Einkemmer, Lukas Klingenberg, Christian Kusch, Jonas |
| contents | The numerical method of dynamical low-rank approximation (DLRA) has recently been applied to various kinetic equations showing a significant reduction of the computational effort. In this paper, we apply this concept to the linear Boltzmann-Bhatnagar-Gross-Krook (Boltzmann-BGK) equation which due its high dimensionality is challenging to solve. Inspired by the special structure of the non-linear Boltzmann-BGK problem, we consider a multiplicative splitting of the distribution function. We propose a rank-adaptive DLRA scheme making use of the basis update & Galerkin integrator and combine it with an additional basis augmentation to ensure numerical stability, for which an analytical proof is given and a classical hyperbolic Courant-Friedrichs-Lewy (CFL) condition is derived. This allows for a further acceleration of computational times and a better understanding of the underlying problem in finding a suitable discretization of the system. Numerical results of a series of different test examples confirm the accuracy and efficiency of the proposed method compared to the numerical solution of the full system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06844 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A stable multiplicative dynamical low-rank discretization for the linear Boltzmann-BGK equation Baumann, Lena Einkemmer, Lukas Klingenberg, Christian Kusch, Jonas Numerical Analysis 35Q20, 65L04, 65M12, 65M22 The numerical method of dynamical low-rank approximation (DLRA) has recently been applied to various kinetic equations showing a significant reduction of the computational effort. In this paper, we apply this concept to the linear Boltzmann-Bhatnagar-Gross-Krook (Boltzmann-BGK) equation which due its high dimensionality is challenging to solve. Inspired by the special structure of the non-linear Boltzmann-BGK problem, we consider a multiplicative splitting of the distribution function. We propose a rank-adaptive DLRA scheme making use of the basis update & Galerkin integrator and combine it with an additional basis augmentation to ensure numerical stability, for which an analytical proof is given and a classical hyperbolic Courant-Friedrichs-Lewy (CFL) condition is derived. This allows for a further acceleration of computational times and a better understanding of the underlying problem in finding a suitable discretization of the system. Numerical results of a series of different test examples confirm the accuracy and efficiency of the proposed method compared to the numerical solution of the full system. |
| title | A stable multiplicative dynamical low-rank discretization for the linear Boltzmann-BGK equation |
| topic | Numerical Analysis 35Q20, 65L04, 65M12, 65M22 |
| url | https://arxiv.org/abs/2411.06844 |