A collocated-grid spectral difference method for compressible flows. (15th January 2020) - Chen, Wenqian; Ju, Yaping; Zhang, Chuhua
Highlights: A collocated-grid spectral difference (CGSD) method is proposed. The solution points and only the interface flux points are used in the CGSD method. A blending of a SD-like and a DGSEM-like scheme is used for spatial discretization. The CGSD method bridges SD method and DGSEM, and promises a higher efficiency. Abstract: Among various high-order numerical methods for compressible flows, the spectral difference method is very promising and attractive due to its simple formulation, easy implementation and high efficiency, but most of the so far developed spectral difference methods are based on a staggered grid arrangement and thus suffer from extra interpolation operations between solution points and flux points. In order to further improve computational efficiency, we propose a collocated-grid spectral difference method on unstructured hexahedral grids by removing the interior flux points. The proposed method holds all the advantages of spectral difference methods while involving the solution points and only the interface flux points at each element. In the proposed method, the spatial terms are calculated by a weighted average of two proposed schemes: a spectral difference-like scheme and a discontinuous Galerkin spectral element-like scheme. The collocated-grid spectral difference method is firstly analysed with a Fourier technique in terms of dissipation, dispersion and stability for the one-dimensional linear advection equation. Then, the proposed method is further verified with analytic solutions of Euler vortex propagation problem. The numerical accuracy of order P + 1 is verified for a reconstruction polynomial of order P and the best weights are obtained for the enhanced computational efficiency. Finally, the capability of the proposed method in handling curved boundary and turbulent flow is demonstrated with two typical problems: flow over NACA0012 airfoil and decay of Taylor Green vortex. The proposed method bridges the spectral difference and discontinuous Galerkin spectral element method in the framework of differential formulation, and promises a higher computational efficiency over them.