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An optimized spectral difference scheme for CAA problems

机译：针对CAA问题的优化光谱差异方案

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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.
机译：在实施光谱差（SD）方法时，使用外推或插值方案从解点中计算出通量点处的守恒变量。使用外推法和内插法所引起的错误将导致不稳定。另一方面，当应用Riemann求解器来计算单元界面处的通量时，边缘界面处左右保守变量之间的差异将向SD方法引入耗散。在本文中，通过在波数空间中最小化四阶SD方法在选定波数带上的色散误差，对四边形元素进行外插和内插方案的优化。给出了外插和内插的优化系数。并绘制了原始方案和优化方案的色散误差并进行了比较。获得了在SD方法的可分辨波数范围内色散误差的改善。分析了优化的四阶SD方案的稳定性。发现通过优化可以改善带有Chebyshev-Gauss-Lobatto通量点的四阶方案的稳定性，该稳定性本来是弱不稳定的。如果在选定的磁通点上附加了一个二阶滤波器，则可以完全消除弱的不稳定性。为优化方案进行了一维和二维线性波传播分析。结果发现，在可分辨波数范围内，新的SD方案与原始方案相比具有较小的分散性和较小的耗散性，并且新方案对于2D波传播具有较小的各向异性。经过优化的SD解算器已通过四个计算航空声学（CAA）车间基准问题进行了验证。优化方案的数值结果与原始方案的分析数据吻合得更好。

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