A numerical investigation of a spectral-type nodal collocation discontinuous Galerkin approximation of the Euler and Navier-Stokes equations

This paper focuses on the applicability of spectral-type collocation discontinuous Galerkin methods to the steady state numerical solution of the inviscid and viscous Navier-Stokes equations on meshes consisting of curved quadrilateral elements. The solution is approximated with piecewise Lagrange polynomials based on both Legendre-Gauss and Legendre-Gauss-Lobatto interpolation nodes. For the sake of computational efficiency, the interpolation nodes can be used also as quadrature points. In this case, however, the effect of the nonlinearities in the equations and/or curved elements leads to aliasing and/or commutation errors which may result in inaccurate or unstable computations. By a thorough numerical testing on a set of well known test cases available in the literature, it is here shown that the two sets of nodes behave very differently, with a clear advantage of the Legendre-Gauss nodes which always displayed an accurate and robust behaviour in all the test cases considered.