In this study, convection-pressure split Euler flux functions which contain weakly hyperbolic convective subsystems are analyzed. A system of first-order partial differential equations (PDEs) is said to be weakly hyperbolic if the corresponding flux Jacobian does not contain a complete set of linearly independent (LI) eigenvectors. Thus, the application of existing flux difference splitting (FDS) based schemes, which depend heavily on both eigenvalues and eigenvectors, are non-trivial to such systems. In the case of weakly hyperbolic systems, a required set of LI eigenvectors can be constructed through the addition of generalized eigenvectors by utilizing the theory of Jordan canonical forms. Once this is achieved for a weakly hyperbolic convective subsystem, an upwind solver can be constructed in the splitting framework.

In the present work, the above approach is used for developing two new schemes. The first scheme is based on the Zha–Bilgen type splitting while the second is based on the Toro–Vázquez splitting. Both the schemes are tested on various benchmark problems in one-dimension (1-D) and two-dimensions (2-D). The concept of generalized eigenvectors based on Jordan forms is found to be useful in dealing with the weakly hyperbolic parts of the considered Euler systems.

在这项研究中，对流压力分裂欧拉通量函数进行了分析，该函数包含弱双曲对流子系统。如果相应的通量Jacobian不包含线性独立（LI）特征向量的完整集合，则一阶偏微分方程（PDE）系统被称为弱双曲。因此，在很大程度上依赖于特征值和特征向量的现有基于通量差分裂（FDS）的方案的应用对于这样的系统而言是不平凡的。在弱双曲系统的情况下，可以利用约旦规范形式的理论通过添加广义特征向量来构造所需的LI特征向量集。一旦针对弱双曲对流子系统实现了这一目标，就可以在拆分框架中构造迎风求解器。

在当前的工作中，以上方法用于开发两个新方案。第一种方案基于Zha–Bilgen类型分裂，而第二种方案基于Toro–Vázquez分裂。两种方案都针对一维（1-D）和二维（2-D）的各种基准问题进行了测试。发现基于约旦形式的广义特征向量的概念在处理所考虑的欧拉系统的弱双曲部分时很有用。