The isobaric-isothermal phase equilibrium (PT Flash) calculation has been an active research topic of thermodynamics for decades. In this work, the conventional framework of the PT Flash calculation, consisting of stability analysis and phase-split calculation, is briefly reviewed by giving the key working equations of the first- and second-order methods. With different type of equations of state, the numerical aspects of the PT Flash calculation have been systematically investigated for various systems over a wide range of conditions: the significance of the first-order methods, volume based versus pressure based second-order methods, a safe-unstable criterion in stability analysis, comparisons of different models and modeling approaches, as well as the converged volume as an initial guess in the volume root solver. Moreover, the same numerical algorithm is used in the second-order methods for both volume and pressure based stability analysis as well as pressure based phase-split calculation for fair comparisons to the largest possible extent. The results reveal that a few iterations of the pressure based first-order method will significantly improve the efficiency of stability analysis, and it is not more efficient to use a volume based second-order method from an overall point of view. A volume based second-order method can improve the efficiency of phase-split calculation, of which the extent depends on the systems and models. This study also shows that the efficiency deterioration of using association models compared to cubic ones is moderate.