Slip flow boundary conditions in discontinuous Galerkin discretizations of the Euler equations of gas dynamics

Discontinuous Galerkin discretizations of the slip flow boundary condition at curved walls in in- viscid gas dynamics are not very accurate when linear basis functions are combined with elements with straight edges at the boundary. This is particularly true when the boundary integrals are com- puted with a Gauss quadrature rule, but also occurs when the more accurate Taylor quadrature rule is used. The error at the solid surface results in a boundary layer which can significantly pollute the numerical solution. In this paper sources of these problems are analyzed and demonstrated for the subsonic flow about a circular cylinder. It is shown that the use of the recently developed Taylor quadrature rule for the flux integrals in combination with superparametric elements results in more than a factor three reduction in total pressure loss at the wall in comparison with isopara- metric elements and Gauss quadrature. The effects of boundary curvature can also be removed using mesh adaptation. Local mesh refinement of linear isoparametric elements is very effective in reducing the error at slip flow boundaries and provides a good alternative to the use of super- parametric elements. This is possible because it is demonstrated that it is not necessary to use a higher-order boundary representation.