Spectral p-multigrid discontinuous Galerkin solution of the Navier-Stokes equations

Discontinuous Galerkin (DG) methods are very well suited for the construction of very high-order approximations of the Euler and Navier–Stokes equations on unstructured and possibly nonconforming grids, but are rather demanding in terms of computational resources. In order to improve the computational efficiency of this class of methods, a high-order spectral element DG approximation of the Navier–Stokes equations coupled with a p-multigrid solution strategy based on a semi-implicit Runge–Kutta smoother is considered here. The effectiveness of the proposed approach in the solution of compressible shockless flow problems is demonstrated on 2D inviscid and viscous test cases by comparison with both a p-multigrid scheme with non-spectral elements and a spectral element DG approach with an implicit time integration scheme.