Troubled-Cell Indicators Using K-Means Clustering for RKDG Methods on Triangular Meshes and $h$-Adaptive Rectangular Meshes (Advances in Applied Mathematics and Mechanics, 2025)
We have proposed a novel framework of troubled-cell indicator (TCI) using K-means clustering on uniform meshes in [SIAM J. Sci. Comput., 43 (2021), pp.A3009–A3031]. Based on this framework, we develop TCIs on two typical types ofnon-uniform meshes, i.e., triangular meshes and $h$-adaptive rectangular meshes. TheTCIs are composed of two parts: one is to create the stencils for troubled-cell indicationthat are composed of computational cells in a local region, and the other is to detectthe troubled cells stencil by stencil. Compared with the uniform meshes, the creationof stencils for non-uniform meshes is no longer trivial. We develop new stencil creation approaches specifically tailored to triangular meshes and $h$-adaptive rectangularmeshes, respectively. Another contribution of this work is a new classification criterionin the troubled-cell indication part which is used to determine if there exist troubledcells in a stencil. It contains only one parameter, which leads to a much easier implementation of the TCIs. Numerical results show that the TCIs not only can capturethe shocks precisely and produce nonoscillatory solutions, but also work well withmultiple indication variables and in a TCI-based $h$-adaptive scheme. These resultsdemonstrate the accuracy and robustness of the TCIs on non-uniform meshes.