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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.05193 |
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| _version_ | 1866910876028633088 |
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| author | Terrab, Soraya Fung, Samy Wu Ryan, Jennifer K. |
| author_facet | Terrab, Soraya Fung, Samy Wu Ryan, Jennifer K. |
| contents | We present a hybrid filter that is only applied to the approximation at the final time and allows for reducing errors away from a shock as well as near a shock. It is designed for discontinuous Galerkin approximations to PDEs and combines a rigorous moment-based Smoothness-Increasing Accuracy-Conserving (SIAC) filter with a data-driven CNN filter. While SIAC improves accuracy in smooth regions, it fails to reduce the $\mathcal{O}(1)$ errors near discontinuities, particularly in inviscid compressible flows with shocks. Our hybrid SIAC-CNN filter, trained exclusively on top-hat functions, enforces consistency constraints globally and higher-order moment conditions in smooth regions, reducing both $\ell_2$ and $\ell_\infty$ errors near discontinuities and preserving theoretical accuracy in smooth regions. We demonstrate its effectiveness on the Euler equations for the Lax, Sod, and Shu-Osher shock-tube problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_05193 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A hybrid SIAC -- data-driven post-processing filter for discontinuities in solutions to numerical PDEs Terrab, Soraya Fung, Samy Wu Ryan, Jennifer K. Numerical Analysis 65M99 We present a hybrid filter that is only applied to the approximation at the final time and allows for reducing errors away from a shock as well as near a shock. It is designed for discontinuous Galerkin approximations to PDEs and combines a rigorous moment-based Smoothness-Increasing Accuracy-Conserving (SIAC) filter with a data-driven CNN filter. While SIAC improves accuracy in smooth regions, it fails to reduce the $\mathcal{O}(1)$ errors near discontinuities, particularly in inviscid compressible flows with shocks. Our hybrid SIAC-CNN filter, trained exclusively on top-hat functions, enforces consistency constraints globally and higher-order moment conditions in smooth regions, reducing both $\ell_2$ and $\ell_\infty$ errors near discontinuities and preserving theoretical accuracy in smooth regions. We demonstrate its effectiveness on the Euler equations for the Lax, Sod, and Shu-Osher shock-tube problems. |
| title | A hybrid SIAC -- data-driven post-processing filter for discontinuities in solutions to numerical PDEs |
| topic | Numerical Analysis 65M99 |
| url | https://arxiv.org/abs/2408.05193 |