(untitled) (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 of non-uniform meshes, i.e., triangular meshes and h-adaptive rectangular meshes. The TCIs are composed of two parts: one is to create the stencils for troubled-cell indication that are composed of computational cells in a local region, and the other is to detect the troubled cells stencil by stencil. Compared with the uniform meshes, the creation of stencils for non-uniform meshes is no longer trivial. We develop new stencil creation approaches specifically tailored to triangular meshes and h-adaptive rectangular meshes, respectively. Another contribution of this work is a new classification criterion in the troubled-cell indication part which is used to determine if there exist troubled cells 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 capture the shocks precisely and produce nonoscillatory solutions, but also work well with multiple indication variables and in a TCI-based h-adaptive scheme. These results demonstrate the accuracy and robustness of the TCIs on non-uniform meshes.