A linear instability analysis was conducted firstly on the interface of a stratified gas-liquid two-phase flow in a circular piper employing a two-fluid model. The constitutive equations simulation technique was discussed, and the dispersive equation of interfacial waves was derived. The effects of flow rates of gas and liquid, liquid viscosity, surface tension and tube inclination on the stability of interface were investigated. A set of non-linear hyperbolic governing equations was deduced from the complete two-fluid model equation by omitting the effect of the surface tension and assuming a quasi-steady-state for the gas phase. Using characteristic line and finite difference, the propagation and growth of the interfacial disturbances were investigated in terms of gas and liquid superficial velocities. Then the results of the non-linear stability analysis were compared with those obtained by the linear stability analysis and experimental data. The non-linear stability analysis not only confirms the conclusions reached by the linear instability analysis, but also gives an insight into the growth and propagation of the interfacial disturbances on the interface of a gas-liquid two-phase flow.