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

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 $$\mathcal {O}(1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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 $$\ell _2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> and $$\ell _\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> 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.