A Recovery-Assisted Discontinuous Galerkin Method for the Compressible Navier-Stokes Equations*

The Discontinuous Galerkin (DG) method is a promising approach for high-fidelity simulations of turbulent flows in complex geometries, given the method's capability for arbitrarily high orders of accuracy on unstructured meshes. We propose a DG approach for advection-diffusion equations (such as the Navier-Stokes equations) based on the principle of Recovery, whereby the underlying solution between two adjacent computational elements is "recovered" into a smooth, more accurate polynomial representation. By combining a biased approach to Recovery for advection with a compact gradient recovery for diffusion, we achieve $2p+2$ order of accuracy in the cell-average error on a nearest-neighbors stencil, where $p$ is the degree of the polynomial basis, and demonstrate higher efficiency than other state-of-the-art DG approaches. Several test cases, including compressible turbulence, are presented to support our method.