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A hybrid SIAC: data-driven post-processing filter for discontinuities in solutions to numerical PDEs.

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Title: A hybrid SIAC: data-driven post-processing filter for discontinuities in solutions to numerical PDEs.
Authors: Terrab, Soraya1 (AUTHOR) sterrab@mines.edu, Wu Fung, Samy1,2 (AUTHOR), Ryan, Jennifer K.3 (AUTHOR)
Source: Journal of Engineering Mathematics. 11/12/2025, Vol. 155 Issue 1, p1-32. 32p.
Subjects: Shock waves, Numerical solutions to partial differential equations, Scientific computing, Convolutional neural networks, Galerkin methods
Abstract: We present a post-processing 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 for approximation with reduced stabilization applied during time-evolution. This filter is designed for discontinuous Galerkin approximations to PDEs and combines a rigorous moment-based Smoothness-Increasing Accuracy-Conserving (SIAC) filter with a consistent data-driven Convolutional-Neural-Network (CNN) filter. While SIAC improves accuracy in smooth regions, it fails to reduce the 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 ℓ 2 and ℓ ∞ errors near discontinuities and preserving theoretical accuracy in smooth regions. We demonstrate the effectiveness of the hybrid filter on the Euler equations for the Lax, Sod, and Shu–Osher shock-tube problems. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Engineering Mathematics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
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  Data: <searchLink fieldCode="JN" term="%22Journal+of+Engineering+Mathematics%22">Journal of Engineering Mathematics</searchLink>. 11/12/2025, Vol. 155 Issue 1, p1-32. 32p.
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  Data: <searchLink fieldCode="DE" term="%22Shock+waves%22">Shock waves</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+solutions+to+partial+differential+equations%22">Numerical solutions to partial differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Scientific+computing%22">Scientific computing</searchLink><br /><searchLink fieldCode="DE" term="%22Convolutional+neural+networks%22">Convolutional neural networks</searchLink><br /><searchLink fieldCode="DE" term="%22Galerkin+methods%22">Galerkin methods</searchLink>
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  Data: We present a post-processing 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 for approximation with reduced stabilization applied during time-evolution. This filter is designed for discontinuous Galerkin approximations to PDEs and combines a rigorous moment-based Smoothness-Increasing Accuracy-Conserving (SIAC) filter with a consistent data-driven Convolutional-Neural-Network (CNN) filter. While SIAC improves accuracy in smooth regions, it fails to reduce the 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 ℓ 2 and ℓ ∞ errors near discontinuities and preserving theoretical accuracy in smooth regions. We demonstrate the effectiveness of the hybrid filter on the Euler equations for the Lax, Sod, and Shu–Osher shock-tube problems. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Group: Ab
  Data: <i>Copyright of Journal of Engineering Mathematics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1007/s10665-025-10490-3
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        Text: English
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        Type: general
      – SubjectFull: Numerical solutions to partial differential equations
        Type: general
      – SubjectFull: Scientific computing
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      – SubjectFull: Convolutional neural networks
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      – SubjectFull: Galerkin methods
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              Text: 11/12/2025
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