The discontinuous Galerkin spectral element methods for compressible flows on two-dimensional mixed grids

This paper presents a practical algorithm for constructing high order discontinuous Galerkin spectral element methods (DGSEM) on mixed triangular and quadrilateral grids. The traditional DGSEM belongs to the collocation-type nodal discontinuous Galerkin method which is computationally efficient on one-dimensional and tensor-product grids. This work generalizes DGSEM to triangular grids using symmetric quadrature points in sphere close packed arrangement. The dispersion and dissipation analysis shows that the scheme is stable for linear problems on triangles. A problem independent limiting procedure is proposed for capturing the shock waves. The main components are a modified KXRCF shock detector, a smoothness indicator and a high-resolution WENO limiter based on candidate polynomials constructed by newly developed projection approach. The implementation of the DGSEM on the mixed grids is also presented. Numerical examples on triangular and mixed grids verify the scheme's high order accuracy and robustness in computing inviscid and viscous compressible flows.