A flux reconstruction approach is implemented in the spectral difference (SD) framework, which we refer to it as discontinuous spectral element method (DSEM), to improve the accuracy and stability of numerical simulations. A new class of correction functions is developed based on a weighted orthogonality condition, which leads to the introduction of Jacobi correction functions. Jacobi correction functions can construct the well-known DSEM and staggered-grid version of nodal discontinuous Galerkin spectral element method (DGSEM) as well as a broad range of other high-order numerical schemes with a variety of numerical characteristics, such as high numerical dissipation to suppress aliasing-driven errors, super accuracy, and solution boundedness in shock prediction. Three single-parameter families of Jacobi correction functions are recognized, and a von Neumann analysis is performed to acquire their wave propagation properties. The order of accuracy and stability criterion of a broad range of schemes are calculated, and the most super-accurate and stable numerical schemes are identified and later validated through a set of numerical experiments on a one-dimensional advection equation. The solution unboundedness issue of the high-order DSEM scheme is discussed through simulating the Burgers equation. An accuracy test performed on the Burgers equation revealed that the new staggered-grid flux reconstruction approach could acquire a higher accuracy than that of the energy stable flux reconstruction framework on a collocated grid. The two-dimensional extension of the new framework is also examined by simulation of a non-linear Euler system of equations for the isentropic vortex in free-stream flow. The staggered-grid flux reconstruction using the Jacobi correction function can precisely reconstruct the DSEM approach for non-linear two-dimensional hyperbolic equations for all polynomial orders.

在谱差（SD）框架中实现了通量重构方法，我们将其称为不连续谱元素法（DSEM），以提高数值模拟的准确性和稳定性。基于加权正交性条件开发了新一类的校正函数，这导致了Jacobi校正函数的引入。Jacobi校正函数可以构造节点不连续Galerkin谱元法（DGSEM）的著名DSEM和交错网格版本，以及具有各种数值特性（例如高数值耗散）的各种其他高阶数值方案来抑制震动预测中的混叠驱动误差，超高精度和解决方案有界性。识别了三个单参数Jacobi校正函数系列，然后进行冯·诺依曼分析以获取其波传播特性。计算了范围广泛的方案的准确性和稳定性准则的顺序，确定了最超精确和稳定的数值方案，随后通过对一维对流方程的一组数值实验对它们进行了验证。通过模拟Burgers方程，讨论了高阶DSEM方案的解无界问题。对Burgers方程进行的精度测试表明，与并置网格上的能量稳定通量重构框架相比，新的交错网格通量重构方法可以获得更高的精度。还通过模拟自由流中等熵涡旋的非线性Euler方程组，对新框架的二维扩展进行了研究。使用雅可比校正函数的交错网格通量重构可以针对所有多项式的非线性二维双曲型方程精确地重构DSEM方法。