Multiwavelet troubled-cell indicator for discontinuity detection of discontinuous Galerkin schemes

In this paper, we introduce a new global troubled-cell indicator for the discontinuous Galerkin (DG) method in one- and two-dimensions. This is done by taking advantage of the global expression of the DG method and re-expanding it in terms of a multiwavelet basis, which is a sum of the global average and finer details on different levels. Examining the higher level difference coefficients acts as a troubled-cell indicator, thus avoiding unnecessary increased computational cost of a new expansion. In two-dimensions the multiwavelet decomposition uses combinations of scaling functions and multiwavelets in the $x-$ and $y-$directions for improved troubled-cell indication. By using such a troubled-cell indicator, we are able to reduce the computational cost by avoiding limiting in smooth regions. We present numerical examples in one- and two-dimensions and compare our troubled-cell indicator to the subcell resolution technique of Harten (1989) and the shock detector of Krivodonova, Xin, Remacle, Chevaugeon, and Flaherty (2004), which were previously investigated by Qiu and Shu (2005).