  In this study, we analyze convection-pressure split Euler flux functions
which contain genuine weakly hyperbolic convection subsystems. A system is said
to be a genuine weakly hyperbolic if all eigenvalues are real with no complete
set of linearly independent (LI) eigenvectors. To construct an upwind solver
based on flux difference splitting (FDS) framework, we require to generate
complete set of LI eigenvectors. This can be done through addition of
generalized eigenvectors which can be computed from theory of Jordan canonical
forms. Once we have complete set of LI generalized eigenvectors, we construct
upwind solvers in convection-pressure splitting framework. Since generalized
eigenvectors are not unique, we take extra care to ensure no direct
contribution of generalized eigenvectors in the final formulation of both the
newly developed numerical schemes. First scheme is based on Zha and Bilgen type
splitting approach, while second is based on Toro and V\'azquez splitting. Both
the schemes are tested on several bench-mark test problems on 1-D and one of
them is tested on some typical 2-D test problems which involve shock
instabilities. The concept of generalized eigenvector based on Jordan forms is
found to be useful in dealing with the genuine weakly hyperbolic parts of the
considered Euler systems.
