Adaptive discontinuous Galerkin methods with shock-capturing for the compressible Navier-Stokes equations

Abstract

We present the Interior Penalty discontinuous Galerkin method for the compressible Navier-Stokes equations. Shock-capturing is used to reduce over-shoots at discontinuities and sharp gradients. This stabilization introduces artificial viscosity at places of large local residuals, but preserves conservation and Galerkin orthogonality of the DG method. Based on this discretization we derive <em>a posteriori</em> error estimates for the error measured in terms of arbitrary target functionals, like e.g. the drag and lift coefficients of an airfoil immersed in a viscous or inviscid fluid. The performance of the nonlinear solution process, the <em>a posteriori</em> error estimation and an adaptive mesh refinement specially tailored for the accurate computation of the force coefficients are demonstrated for supersonic laminar flows around the NACA0012 airfoil.