An accuracy preserving limiter for the high-order discontinuous Galerkin method on unstructured grids

This paper develops a new limiter and applies it to the high-order discontinuous Galerkin (DG) method on unstructured grids. We extend the original second-order Van Albada limiter to the third-order accuracy based on the vector integral theorem. In order to achieve the high-order accuracy, we have derived the expressions for the second-order derivatives of the flow variables on the cell center. The developed limiter is differentiable, simple to be implemented and has a good property of convergence. Several numerical cases demonstrate that the new limiter can preserve the high-order numerical accuracy in smooth regions, and effectively control the nonphysical oscillations near the shock waves.