An in-house, fully parallel compressible Navier-Stokes solver was developed based on an implicit, non-dissipative, energy conserving, finite-volume algorithm. PETSc software was utilized for this purpose. To be able to handle occasional instances of slow convergence due to possible oscillating pressure corrections on successive iterations in time, a fixing procedure was adopted. To demonstrate the algorithms ability to evolve a linear perturbation into nonlinear hydrodynamic turbulence, temporal Kelvin-Helmholtz Instability problem is studied. KHI occurs when a perturbation is introduced into a system with a velocity shear. The theory can be used to predict the onset of instability and transition to turbulence in fluids moving at various speeds. In this study, growth rate of the instability was compared to predictions from linear theory using a single mode perturbation in the linear regime. Effect of various factors on growth rate was also discussed. Compressible KHI is most unstable in subsonic/transonic regime. High Reynolds number (low viscosity) allows perturbations to develop easily, in consistent with the nature of Kill. Higher wave numbers (shorter wavelengths) also grow faster. These results match with the findings of stability analysis, as well as other results presented in the literature.