A stable Spectral Difference approach for computations with triangular and hybrid grids up to the 6th order of accuracy

Abstract

In the present paper, a stable Spectral Difference formulation on triangles is defined using a flux polynomial expressed in the Raviart-Thomas basis up to the sixth-order of accuracy. Compared to the literature on the Spectral Difference approach, the present work increases the order of accuracy that the stable formulation can deal with. The proposed scheme is based on a set of flux points defined in the paper. The sets of points leading to a stable formulation are determined using a Fourier stability analysis of the linear advection equation coupled with an optimization process. The proposed Spectral Difference formulation differs from the Flux Reconstruction method on hybrid grids: the distinction between the two approaches is highlighted through the definition of the number of interior flux points. Validation starts from a convergence study using Euler equations and continues with simulations of laminar viscous flows over the NACA0012 airfoil using quadratic triangles and around a cylinder using a hybrid grid.