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

High order limiters remain one of the main challenges for discontinuous Galerkin (DG) methods in solving hyperbolic conservation laws. This paper proposes an efficient limiting procedure for the DG method. The key feature is to construct additional polynomials from the solutions on neighboring cells by means of secondary reconstruction. Then the limited solution on current cell can be obtained using WENO or other limiting procedures. This limiting procedure uses only the face-neighbor information and thus is compact and easy to generalize to multi-dimensions. The numerical experiments show that the limiter can achieve high order accuracy in smooth region and also capture the strong discontinuities without oscillations.