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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]
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Database: Engineering Source
Description
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]
ISSN:00220833
DOI:10.1007/s10665-025-10490-3