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

TL;DR

This paper introduces a hybrid SIAC-CNN filter for discontinuous Galerkin solutions to PDEs, combining a classical filter with a data-driven approach to improve accuracy near shocks and in smooth regions.

Contribution

It develops a novel hybrid filter that integrates SIAC and CNN techniques, trained on top-hat functions, to enhance error reduction near discontinuities while maintaining accuracy in smooth areas.

Findings

Reduces errors near shocks in Euler equations

Preserves theoretical accuracy in smooth regions

Effective on Lax, Sod, and Shu-Osher shock-tube problems

Abstract

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.