On the numerical signature of blow-up in hydrodynamic equations
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866915062042591232 |
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| author | Jansson, Erik Modin, Klas |
| author_facet | Jansson, Erik Modin, Klas |
| contents | The phenomenon of finite time blow-up in hydrodynamic partial differential equations is central in analysis and mathematical physics. While numerical studies have guided theoretical breakthroughs, it is challenging to determine if the observed computational results are genuine or mere numerical artifacts. Here we identify numerical signatures of blow-up. Our study is based on the complexified Euler equations in two dimensions, where instant blow-up is expected. Via a geometrically consistent spatiotemporal discretization, we perform several numerical experiments and verify their computational stability. We then identify a signature of blow-up based on the growth rates of the supremum norm of the vorticity with increasing spatial resolution. The study aims to be a guide for cross-checking the validity for future numerical experiments of suspected blow-up in equations where the analysis is not yet resolved. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_02328 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the numerical signature of blow-up in hydrodynamic equations Jansson, Erik Modin, Klas Numerical Analysis Differential Geometry 35Q31, 35B44, 81S10, 65G50, 65M12 The phenomenon of finite time blow-up in hydrodynamic partial differential equations is central in analysis and mathematical physics. While numerical studies have guided theoretical breakthroughs, it is challenging to determine if the observed computational results are genuine or mere numerical artifacts. Here we identify numerical signatures of blow-up. Our study is based on the complexified Euler equations in two dimensions, where instant blow-up is expected. Via a geometrically consistent spatiotemporal discretization, we perform several numerical experiments and verify their computational stability. We then identify a signature of blow-up based on the growth rates of the supremum norm of the vorticity with increasing spatial resolution. The study aims to be a guide for cross-checking the validity for future numerical experiments of suspected blow-up in equations where the analysis is not yet resolved. |
| title | On the numerical signature of blow-up in hydrodynamic equations |
| topic | Numerical Analysis Differential Geometry 35Q31, 35B44, 81S10, 65G50, 65M12 |
| url | https://arxiv.org/abs/2210.02328 |