High-Order Entropy Correction with SIAC Filters

This article considers the application of Smoothness-Increasing Accuracy-Conserving (SIAC) filtering for the non-linear stabilization of discontinuous Galerkin (DG) discretizations via entropy correction. Upon constructing discrete filters from continuous convolution SIAC kernels, the schemes are made to be conservative and are then appended to the DG method in a semi-discrete fashion. Performance of these tunable SIAC filters is compared to the local averaging typically employed in the entropy correction of finite element methods, and their capabilities are demonstrated for energy conservation as well as a shock regularization strategy based on an artificial viscosity estimate. Relaxation Runge-Kutta time integration methods are further employed in order to ensure a fully-discrete energy preserving procedure, with impacts of the overall solution accuracy being investigated for calculations of the one- and two-dimensional Burgers equation.