A review of troubled cell indicators for discontinuous Galerkin method

TL;DR

This paper surveys eight troubled-cell indicators for discontinuous Galerkin methods with a WENO limiter applied to nonlinear hyperbolic conservation laws, tested on both structured and unstructured meshes. The indicators include the Persson–Peraire PP, Gelb–Tadmor SJ, Fu–Shu FS1/FS2, Li et al. LPR, and neural-network based RH indicators, evaluated on 1D problems (e.g., Sod, Lax, Shu-Osher, Blast) and 2D problems (Double Mach Reflection, 2D Riemann). The study reports per-cell testing time, flagged-cell percentages, and solution accuracy, showing that FS1, LPR, and the ANN RH indicator are strong candidates, with ANN offering high accuracy at the expense of training cost. The results guide practitioners in selecting robust, efficient troubled-cell indicators for DGM with WENO limiters in large-scale, high-order simulations.

Abstract

In this paper, eight different troubled cell indicators (shock detectors) are reviewed for the solution of nonlinear hyperbolic conservation laws using discontinuous Galerkin (DG) method and a WENO limiter on both structured and unstructured meshes. Extensive simulations using one-dimensional and two-dimensional problems (2D Riemann problem and the double Mach reflection) for various orders on the hyperbolic system of Euler equations are used to compare these troubled cell indicators. They are evaluated based on the percentage of cells flagged as troubled cells for various orders and various grid sizes. CPU time taken to test a single cell for discontinuity is also compared. For one-dimensional problems, the performance of Fu and Shu indicator and the modified KXRCF indicator is better than other indicators. For two-dimensional problems, the performance of the artificial neural network (ANN) indicator of Ray and Hesthaven is quite good and the Fu and Shu and the modified KXRCF indicators are also good. These three indicators are suitable candidates for applications of DGM using WENO limiters though it should be noted that the ANN indicator is quite expensive and requires a lot of training.

Figures (17)

• Figure 1: The stencil S = $\{ \Omega_{0}, \Omega_{1}, \Omega_{2}, \Omega_{3}\}$ for Fu and Shu troubled cell indicator fs1
• Figure 2: Comparison of density solutions of single moving contact discontinuity at $t=3.0$ using 200 elements obtained with the $P^{1}$, $P^{2}$ and $P^{3}$ based DGM and the exact solution on [2.5,3.5]. Density error ($|\rho-\rho_{exact}|$) for $P^{1}$, $P^{2}$ and $P^{3}$ based DGM is also plotted
• Figure 3: The time history of flagged troubled cells of the single contact discontinuity problem for the one-dimensional Euler equations, simulated until $t=3.0$ with 200 elements and $P^{1}$ based DGM
• Figure 4: Comparison of density solutions of Sod Problem at $t=0.2$ using 200 elements obtained with the $P^{1}$, $P^{2}$ and $P^{3}$ based DGM and the exact solution. Density error ($|\rho-\rho_{exact}|$) for $P^{1}$, $P^{2}$ and $P^{3}$ based DGM is also plotted. Figure \ref{['fig:SodSolution']} also includes a zoomed in portion of the solution for better comparison
• Figure 5: The time history of flagged troubled cells of the Sod test problem for the one-dimensional Euler equations, simulated until $t=0.2$ with 200 elements and $P^{1}$ based DGM