A Simple and Robust Shock-Capturing Approach for Discontinuous Galerkin Discretizations

Abstract

The discontinuous Galerkin (DG) method has become popular in Computational Fluid Dynamics mainly due to its ability to achieve high-order solution accuracy on arbitrary grids, its high arithmetic intensity (measured as the ratio of the number of floating point operations to memory references), and the use of a local stencil that makes scalable parallel solutions possible. Despite its advantages, several difficulties hinder widespread use of the DG method, especially in industrial applications. One of the major challenges remaining is the capturing of discontinuities in a robust and accurate way. In our previous work, we have proposed a simple shock detector to identify discontinuities within a flow solution. The detector only utilizes local information to sense a shock/discontinuity ensuring that one of the key advantages of DG methods, their data locality, is not lost in transonic and supersonic flows. In this work, we reexamine the shock detector capabilities to distinguish between smooth and discontinuous solutions. Furthermore, we optimize the functional relationships between the shock detector and the filter strength, and present it in detail for others to use. By utilizing the shock detector and the corresponding filtering-strength relationships, one can robustly and accurately capture discontinuities ranging from very weak to strong shocks. Our method is demonstrated in a number of two-dimensional canonical examples.