A simple a posteriori indicator for discontinuous Galerkin method on unstructured grids

A simple and efficient troubled-cell indicator based on a posteriori limiting paradigm is proposed for the discontinuous Galerkin (DG) method on the triangular grids. The developed methodology utilizes discrete solution from different time levels in the von Neumann neighborhood to maintain the compactness of the DG schemes. Effective technique is suggested to provide further information about the troubled cells. Different limitation can then be applied to the resulting troubled cells. Favorable numerical characteristic including positivity-preserving and oscillation-suppressing can be achieved. The present indicator has been implemented with both the simple limiter such as the total variation bounded (TVB) limiter and more sophisticated limiter such as the Hermite weighted essentially non-oscillatory (HWENO) limiter. The resulting limiting strategy, compared with the minmod based TVB indicator, has been examined for DG schemes of up to fourth order of accuracy in solving the two dimensional Euler equations on the unstructured grids. Numerical results demonstrate the effectiveness and robustness of the current a posteriori indication method.