A Simple and Effective High-Order Shock-Capturing Limiter for Discontinuous Galerkin Methods

The discontinuous Galerkin (DG) finite element method when applied to hyperbolic conservation laws requires the use of shock-capturing limiters in order to suppress unphysical oscillations near large solution gradients. In this work we develop a novel shock-capturing limiter that combines key ideas from the limiter of Barth and Jespersen [AIAA-89-0366 (1989)] and the maximum principle preserving (MPP) framework of Zhang and Shu [Proc. R. Soc. A, 467 (2011), pp. 2752--2776]. The limiting strategy is based on traversing the mesh element-by-element in order to (1) find local upper and lower bounds on user-defined variables by sampling these variables on neighboring elements, and (2) to then enforce these local bounds by minimally damping the high-order corrections. The main advantages of this limiting strategy is that it is simple to implement, effective at shock capturing, and retains high-order accuracy of the solution in smooth regimes. The resulting numerical scheme is applied to several standard numerical tests in both one and two-dimensions and on both Cartesian and unstructured grids. These tests are used as benchmarks to verify and assess the accuracy and robustness of the method.