An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible Navier–Stokes equations

Discontinuous Galerkin (DG) methods have proved to be well suited for the construction of robust high-order numerical schemes on unstructured and possibly nonconforming grids for a variety of problems. Their application to the incompressible Navier–Stokes (INS) equations has also been recently considered, although the subject is far from being fully explored. In this work, we propose a new approach for the DG numerical solution of the INS equations written in conservation form. The inviscid numerical fluxes both in the continuity and in the momentum equation are computed using the values of velocity and pressure provided by the (exact) solution of the Riemann problem associated with a local artificial compressibility perturbation of the equations. Unlike in most of the existing methods, artificial compressibility is here introduced only at the interface flux level, therefore resulting in a consistent discretization of the INS equations irrespectively of the amount of artificial compressibility introduced. The discretization of the viscous term follows the well-established DG scheme named BR2. The performance and the accuracy of the method are demonstrated by computing the Kovasznay flow and the two-dimensional lid-driven cavity flow for a wide range of Reynolds numbers and for various degrees of polynomial approximation.