An Enriched Discontinuous Galerkin Method for Representing High-Gradient Features in Fluid Dynamics

Discontinuous Galerkin (DG) methods have difficulty resolving high gradient solutions, such as shocks and boundary layers, without stabilization to remove spurious oscillations or significant mesh refinement in regions of high-gradient feature. We present a DG scheme that enriches the standard basis polynomials with analytical solutions to canonical problems. This strategy allows one to capture large gradient features without spurious oscillations or significant mesh refinement, while also retaining the higher order accuracy of the scheme. We discuss the procedure to choose the analytical enrichment functions and integrate them into the DG framework. The method is evaluated in application to fluid dynamics problems with high gradient solutions, such as boundary layers and transient shock-flow interaction.