A positivity-preserving hybrid DDG method for Poisson--Nernst--Planck systems
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866910975071879168 |
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| author | Liu, Hailiang Wang, Zhongming Yin, Peimeng |
| author_facet | Liu, Hailiang Wang, Zhongming Yin, Peimeng |
| contents | In earlier work [H. Liu and Z. Wang, J. Comput. Phys., 328(2017)], an arbitrary high-order conservative and energy-dissipative direct discontinuous Galerkin (DDG) scheme was developed. Although this scheme enforced solution positivity using cell averages as reference values, it lacked a theoretical guarantee for the positivity of those cell averages. In this study, we develop a novel arbitrary high-order DDG method with rigorously proven positivity-preserving properties. Specifically, the positivity of the cell averages is ensured through a modified numerical flux in combination with forward Euler time discretization. To achieve point-wise positivity of ion concentrations, we introduce a hybrid algorithm that integrates a positivity-preserving limiter. The proposed method is further extended to higher-dimensional problems with rectangular meshes. Numerical results confirm the scheme's high-order accuracy, guaranteed positivity preservation, and consistent discrete energy dissipation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_24058 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A positivity-preserving hybrid DDG method for Poisson--Nernst--Planck systems Liu, Hailiang Wang, Zhongming Yin, Peimeng Numerical Analysis In earlier work [H. Liu and Z. Wang, J. Comput. Phys., 328(2017)], an arbitrary high-order conservative and energy-dissipative direct discontinuous Galerkin (DDG) scheme was developed. Although this scheme enforced solution positivity using cell averages as reference values, it lacked a theoretical guarantee for the positivity of those cell averages. In this study, we develop a novel arbitrary high-order DDG method with rigorously proven positivity-preserving properties. Specifically, the positivity of the cell averages is ensured through a modified numerical flux in combination with forward Euler time discretization. To achieve point-wise positivity of ion concentrations, we introduce a hybrid algorithm that integrates a positivity-preserving limiter. The proposed method is further extended to higher-dimensional problems with rectangular meshes. Numerical results confirm the scheme's high-order accuracy, guaranteed positivity preservation, and consistent discrete energy dissipation. |
| title | A positivity-preserving hybrid DDG method for Poisson--Nernst--Planck systems |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2505.24058 |