We report the development of a discontinuous spectral element flow solver that includes the implementation of both spectral difference and flux reconstruction formulations. With this high order framework, we have constructed a foundation upon which to provide a fair and accurate assessment of these two schemes in terms of accuracy, stability, and performance with special attention to the true spectral difference scheme and the modified spectral difference scheme recovered via the flux reconstruction formulation. Building on previous analysis of the spectral difference and flux reconstruction schemes, we provide a novel nonlinear stability analysis of the spectral difference scheme. Through various numerical experiments, we demonstrate the additional stability afforded by the true, baseline spectral difference scheme without explicit filtering or de-aliasing due to its inherent feature of staggered flux points. This arrangement leads to favorable suppression of aliasing errors and improves stability needed for under-resolved simulations of turbulent flows.

我们报告了不连续谱元素流求解器的开发，该求解器包括实现谱差和通量重建公式。在这个高阶框架的基础上，我们建立了一个基础，可以在准确性，稳定性和性能方面对这两种方案进行公正，准确的评估，并特别注意真实的光谱差方案和经过修正的光谱差方案。通量重建公式。在先前对谱差方案和通量重建方案的分析的基础上，我们提供了对谱差方案的新型非线性稳定性分析。通过各种数值实验，我们证明了真实的，基线谱差方案，由于其固有的交错通量点特征，因此无需显式滤波或去混叠。这种安排可以很好地抑制混叠误差，并提高了湍流模拟不足时所需的稳定性。