Asymptotic-preserving and positivity-preserving discontinuous Galerkin method for the semiconductor Boltzmann equation in the diffusive scaling
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916662046883840 |
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| author | Ding, Huan Liu, Liu Zhong, Xinghui |
| author_facet | Ding, Huan Liu, Liu Zhong, Xinghui |
| contents | In this paper, we develop an asymptotic-preserving and positivity-preserving discontinuous Galerkin (DG) method for solving the semiconductor Boltzmann equation in the diffusive scaling. We first formulate the diffusive relaxation system based on the even-odd decomposition method, which allows us to split into one relaxation step and one transport step. We adopt a robust implicit scheme that can be explicitly implemented for the relaxation step that involves the stiffness of the collision term, while the third-order strong-stability-preserving Runge-Kutta method is employed for the transport step. We couple this temporal scheme with the DG method for spatial discretization, which provides additional advantages including high-order accuracy, $h$-$p$ adaptivity, and the ability to handle arbitrary unstructured meshes. A positivity-preserving limiter is further applied to preserve physical properties of numerical solutions. The stability analysis using the even-odd decomposition is conducted for the first time. We demonstrate the accuracy and performance of our proposed scheme through several numerical examples. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_19487 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic-preserving and positivity-preserving discontinuous Galerkin method for the semiconductor Boltzmann equation in the diffusive scaling Ding, Huan Liu, Liu Zhong, Xinghui Numerical Analysis 65M60, 65M70, 35Q20 In this paper, we develop an asymptotic-preserving and positivity-preserving discontinuous Galerkin (DG) method for solving the semiconductor Boltzmann equation in the diffusive scaling. We first formulate the diffusive relaxation system based on the even-odd decomposition method, which allows us to split into one relaxation step and one transport step. We adopt a robust implicit scheme that can be explicitly implemented for the relaxation step that involves the stiffness of the collision term, while the third-order strong-stability-preserving Runge-Kutta method is employed for the transport step. We couple this temporal scheme with the DG method for spatial discretization, which provides additional advantages including high-order accuracy, $h$-$p$ adaptivity, and the ability to handle arbitrary unstructured meshes. A positivity-preserving limiter is further applied to preserve physical properties of numerical solutions. The stability analysis using the even-odd decomposition is conducted for the first time. We demonstrate the accuracy and performance of our proposed scheme through several numerical examples. |
| title | Asymptotic-preserving and positivity-preserving discontinuous Galerkin method for the semiconductor Boltzmann equation in the diffusive scaling |
| topic | Numerical Analysis 65M60, 65M70, 35Q20 |
| url | https://arxiv.org/abs/2503.19487 |