A review of troubled cell indicators for discontinuous Galerkin method

TL;DR

This review compares eight troubled cell indicators for discontinuous Galerkin methods applied to hyperbolic conservation laws, evaluating their performance and computational efficiency through extensive simulations.

Contribution

It provides a comprehensive comparison of troubled cell indicators, highlighting the most effective ones for different problem types and mesh configurations.

Findings

Fu and Shu indicator performs well in 1D problems.

ANN indicator shows good performance in 2D but is computationally expensive.

Modified KXRCF indicator is also a strong candidate for 2D applications.

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.