A COMPARISON OF VARIOUS NODAL DISCONTINUOUS GALERKIN METHODS FOR THE 3D EULER EQUATIONS

摘要

Recent research has indicated that collocation-type Discontinuous Galerkin SpectraludElement Methods (DGSEM) represent a more efficient alternative to the standard modal orudnodal DG approaches. In this paper, we compare two collocation-type nodal DGSEM and audstandard nodal DG approach in the context of the three-dimensional Euler equations. The nodaludDG schemes for hexahedral elements are based on the polynomial interpolation of the unknownudsolution using tensor product Lagrange basis functions and the use of Gaussian quadratureudfor integration. In the standard nodal DG approach, we employ uniform interpolation nodesudand Legendre-Gauss (LG) quadrature points. The two collocated DGSEM schemes arise fromudusing either LG or Legendre-Gauss-Lobatto (LGL) points as both interpolation and integrationudnodes. The resulting diagonal mass matrices and the ability to compute the fluxes directly fromudthe solution nodes give rise to highly efficient schemes.udThe results of the numerical convergence studies highlight, especially at high approximationudorders, the performance improvement of the DGSEM schemes compared to the standardudDG scheme. Although having advantages in the evaluation of the boundary values over theudLG-DGSEM, the lower degree of precision of the LGL quadrature negates this benefit. In addition,udwithout the application of filtering techniques or over-integration, the lower integrationudaccuracy of the LGL-DGSEM leads to numerical instabilities at stagnation points. Hence, theudLG-DGSEM is found to be the most efficient scheme as it is more accurate and robust for theudconsidered test cases.