Structure-Preserving Numerical Methods for Fokker-Planck Equations
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866912121620529152 |
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| author | Bartel, Hanna Lampert, Joshua Ranocha, Hendrik |
| author_facet | Bartel, Hanna Lampert, Joshua Ranocha, Hendrik |
| contents | A common way to numerically solve Fokker-Planck equations is the Chang-Cooper method in space combined with one of the Euler methods in time. However, the explicit Euler method is only conditionally positive, leading to severe restrictions on the time step to ensure positivity. On the other hand, the implicit Euler method is robust but nonlinearly implicit. Instead, we propose to combine the Chang-Cooper method with unconditionally positive Patankar-type time integration methods, since they are unconditionally positive, robust for stiff problems, only linearly implicit, and also higher-order accurate. We describe the combined approach, analyse it, and present a relevant numerical example demonstrating advantages compared to schemes proposed in the literature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_07641 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Structure-Preserving Numerical Methods for Fokker-Planck Equations Bartel, Hanna Lampert, Joshua Ranocha, Hendrik Numerical Analysis A common way to numerically solve Fokker-Planck equations is the Chang-Cooper method in space combined with one of the Euler methods in time. However, the explicit Euler method is only conditionally positive, leading to severe restrictions on the time step to ensure positivity. On the other hand, the implicit Euler method is robust but nonlinearly implicit. Instead, we propose to combine the Chang-Cooper method with unconditionally positive Patankar-type time integration methods, since they are unconditionally positive, robust for stiff problems, only linearly implicit, and also higher-order accurate. We describe the combined approach, analyse it, and present a relevant numerical example demonstrating advantages compared to schemes proposed in the literature. |
| title | Structure-Preserving Numerical Methods for Fokker-Planck Equations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2404.07641 |