A Compact Recovery-Assisted DG Method for Advection-Diffusion Problems

We propose an improved discontinuous Galerkin (DG) spatial discretization for advection-diffusion problems. Our approach makes use of our improved schemes for diffusive (Compact Gradient Recovery, CGR) and advective (Interface-Centered Binary reconstruction, ICB) fluxes. Compared to conventional DG methods, our new approach offers improved accuracy and larger allowable timestep sizes when explicit time integration is employed. However, unlike other accuracy-enhancing schemes for DG, our approach maintains a compact, nearest-neighbors stencil. Superior performance is facilitated via careful use of the Recovery concept: the CGR method benefits from the accuracy of the Recovery concept without requiring differentiation of the recovered polynomial, while the ICB method makes use of biased recovered solutions at each interface to maintain stability. We will further demonstrate how the recovered solution at an element-element interface can be recast as a set of derivative-based penalty terms. Fourier analysis and compressible Navier-Stokes test problems will be presented to demonstrate our new DG approach.