The multi-dimensional limiters for discontinuous Galerkin method on unstructured grids

Accuracy-preserving and non-oscillatory shock-capturing technique is one of the bottle necks in the development of discontinuous Galerkin method. In this paper, a new limiter based on the secondary reconstruction and the WENO approach in the characteristic space is developed for the discontinuous Galerkin method. Specifically, an efficient secondary reconstruction technique is proposed which provides the candidate polynomials used in the weighted average procedure of the WENO approach. The secondary reconstructions are performed only on the face neighboring cells to keep the compactness of the discontinuous Galerkin method. Moreover, an improved version of the characteristic limiting procedure is proposed for the nonlinear Euler equations, which is considerably more efficient than the traditional characteristic limiting procedures. The resulting schemes are easy to implement and effective in capturing the shock waves. Some standard cases are computed to validate the accuracy and robustness of the proposed limiters for the DG methods.