A Comparison of various nodal Discontinuous Galerkin Methods for the 3D Euler equations

Abstract

Recent research has indicated that collocation-type Discontinuous Galerkin Spectral Element Methods (DGSEM) represent a more efficient alternative to the standard modal or nodal DG approaches. In this paper, we compare two collocation-type nodal DGSEM and a standard nodal DG approach in the context of the three-dimensional Euler equations. The nodal DG schemes for hexahedral elements are based on the polynomial interpolation of the unknown solution using tensor product Lagrange basis functions and the use of Gaussian quadrature for integration. In the standard nodal DG approach, we employ uniform interpolation nodes and Legendre-Gauss (LG) quadrature points. The two collocated DGSEM schemes arise from using either LG or Legendre-Gauss-Lobatto (LGL) points as both interpolation and integration nodes. The resulting diagonal mass matrices and the ability to compute the fluxes directly from the solution nodes give rise to highly efficient schemes. The results of the numerical convergence studies highlight, especially at high approximation orders, the performance improvement of the DGSEM schemes compared to the standard DG scheme. Although having advantages in the evaluation of the boundary values over the LG-DGSEM, the lower degree of precision of the LGL quadrature negates this benefit. In addition, without the application of filtering techniques or over-integration, the lower integration accuracy of the LGL-DGSEM leads to numerical instabilities at stagnation points. Hence, the LG-DGSEM is found to be the most efficient scheme as it is more accurate and robust for the considered test cases.