Stable Spectral Difference Approach Using Raviart-Thomas Elements for 3D Computations on Tetrahedral Grids

Abstract

In this paper, the Spectral Difference approach using Raviart-Thomas elements (SDRT) is formulated for the first time on tetrahedral grids. To determine stable formulations, a Fourier analysis is conducted for different SDRT implementations, i.e. different interior flux points locations. This stability analysis demonstrates that using interior flux points located at the Shunn-Ham quadrature rule points leads to linearly stable SDRT schemes up to the third order. For higher orders of accuracy, a significant impact of the position of flux points located on faces is shown. The Fourier analysis is then extended to the coupled time-space discretization and stability limits are determined. Additionally, a comparison between the number of interior FP required for the SDRT scheme and the Flux Reconstruction method is proposed and shows that the two approaches always differ on the tetrahedron. Unsteady validation test cases include a convergence study using the Euler equations and the simulation of the Taylor-Green vortex.