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 induce dissipation to the SD method when applying a Riemann solver to compute the common flux at the element interface. In this paper, an optimization of the extrapolation and interpolation schemes for the fourth order SD method 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 standard and optimized schemes is plotted and compared. An improvement of the dispersion error over the resolvable wavenumber range of SD method is obtained. The optimized SD solver is validated with four CAA workshop benchmark problems. The numerical results with optimized schemes agree much better with the analytical data than those with standard schemes. An improvement of the accuracy of the 4th order SD method for CAA problems is obtained.