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This article discusses the effectiveness of the theoretical detection of troubled cells using multiwavelet approaches for the discontinuous Galerkin method and compares them with a statistical approach that uses boxplots. This is an important study in order to understand in which regime these tools are valid. Troubled cells are regions in the domain where a shock or discontinuity has developed, leading to spurious oscillations in the numerical approximation. One option for improving the numerical treatment near these artifacts is through the application of a limiter. The elements where such treatment is necessary are referred to as troubled cells. In this article, a new multiwavelet troubled-cell indicator is formed using the cancelation property (Dahmen, J Comput Appl Math 128:133–185, 2001) and the derived thresholding technique (Gerhard et al., J Sci Comput 62:25–52, 2015). We show that the new detector based on the cancelation property is very accurate in asymptotic regimes (very fine meshes). For coarser meshes, it might be better to use a different indicator, such as the ones that were designed by Vuik and Ryan (J Comput Phys 270:138–160, 2014; SIAM J Sci Comput 38:A84–A104, 2016).

Multiwavelet Troubled-Cell Indication: A Comparison of Utilizing Theory Versus Outlier Detection