Troubled-Cell Indication Using K-means Clustering with Unified Parameters

In Zhu et al. (SIAM J Sci Comput 43: A3009–A3031, 2021), we proposed a new framework of troubled-cell indicator (TCI) using K-means clustering for the discontinuous Galerkin (DG) methods. However, there are two user-tunable parameters in the framework that depend on the polynomial degree of the solution space, the indication variable and even the test problem, which circumscribe the application of the framework. To overcome this drawback, we introduce two simple techniques in this paper: one is to modify the indication variables and the other is to apply a statistical normalization to the modified values. Coupled with four different indication variables, the modified framework is tested via the classical benchmark problems and produces close results under the same setting of the parameters. The discontinuities are overall well captured and the solutions are free of spurious oscillations. The numerical results demonstrate the effectiveness and flexibility of the modified framework and the success in unifying the parameters. Existing TCIs/limiters for the DG methods can be easily implemented into this framework.