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

TL;DR

This work tackles the persistent challenge of accurately resolving shocks while maintaining high-order accuracy in smooth regions for discontinuous Galerkin solutions of PDEs. It introduces a hybrid post-processing filter that combines a moment-based SIAC kernel with a data-driven CNN, trained on top-hat discontinuities, and applied only at the final time. The filter uses a discontinuity window to apply a consistency-only SIAC globally and a CNN-based correction at the discontinuity cell, with adjacent cells updated by Hermite interpolation; Euler data are post-processed from Conserved to Primitive variables. Numerical results on Lax, Sod, and Shu-Osher shock-tube problems show reduced both $\ell_2$ and $\ell_\infty$ errors near discontinuities while preserving SIAC accuracy in smooth regions, highlighting the method’s potential for robust shock resolution in DG schemes.

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.

Figures (21)

• Figure 1: The Euler exact reference solutions for the Lax, Sod, and sine-entropy (Shu-Osher) test problems at their respective benchmark final times, $T_f=\{1.3, 2, 1.8\}$. Note the different data ranges for each of the problems.
• Figure 2: Comparison of filtering approaches: global SIAC (red dotted) with full kernel support and order-2 B-splines, adapted SIAC for discontinuities (blue) with a single order-1 B-spline, and the data-driven filter (green dashed). The left plots show the approximated contact and shock discontinuities for a DG degree-3 sample, while the right plots display the corresponding pointwise errors.
• Figure 3: Demonstration of the discontinuity window about a shock discontinuity with an unfiltered DG approximation (gray) and its exact solution (black), where the following are shown: troubled cells (red squares at the left boundary of the cell/element), the grouping of all troubled cells into set $S_i$ (yellow), and the padding of $d=4$ elements (green) to the left and right of set $S_i$, and the cell where the discontinuity is located (black diamond).
• Figure 4: Tophat samples with varying DG degree $p$ and wave speed $a$ generated for jump discontinuity data (DG approximations $u_h$ versus their exact solutions $u$).
• Figure 5: (Left) Window about troubled cell data for cells $[TC_i-4,TC_i+4]$ flagged from samples in Fig. \ref{['fig:training_samples']}; (right) Sorted and normalized discontinuity data for NN training.