An optimized spectral difference scheme for CAA problems

In the implementation of spectral difference (SD) method, the conserved variables at the flux points are calculated from the solution points using extrapolation or interpolation schemes. The errors incurred in using extrapolation and interpolation would result in instability. On the other hand, the difference between the left and right conserved variables at the edge interface will introduce dissipation to the SD method when applying a Riemann solver to compute the flux at the element interface. In this paper, an optimization of the extrapolation and interpolation schemes for the fourth order SD method on quadrilateral element is carried out in the wavenumber space through minimizing their dispersion error over a selected band of wavenumbers. The optimized coefficients of the extrapolation and interpolation are presented. And the dispersion error of the original and optimized schemes is plotted and compared. An improvement of the dispersion error over the resolvable wavenumber range of SD method is obtained. The stability of the optimized fourth order SD scheme is analyzed. It is found that the stability of the 4th order scheme with Chebyshev–Gauss–Lobatto flux points, which is originally weakly unstable, has been improved through the optimization. The weak instability is eliminated completely if an additional second order filter is applied on selected flux points. One and two dimensional linear wave propagation analyses are carried out for the optimized scheme. It is found that in the resolvable wavenumber range the new SD scheme is less dispersive and less dissipative than the original scheme, and the new scheme is less anisotropic for 2D wave propagation. The optimized SD solver is validated with four computational aeroacoustics (CAA) workshop benchmark problems. The numerical results with optimized schemes agree much better with the analytical data than those with the original schemes.