Two numerical methods are developed and analyzed for studying two-phase jet flows. The first numerical method solves the eigenvalue problem for the matrix system that is constructed from the pseudo-spectral discretization of the 3D linearized, incompressible, perturbed Navier-Stokes (N-S) equations for two-phase flows. This first numerical method will be denoted as LSA for “linear stability analysis.” The second numerical method solves the 3D (nonlinear) N-S equations for incompressible, two-phase flows. The second numerical method will be denoted as DNS for “direct numerical simulation.” In this thesis, predictions of jet-stability using the LSA method are compared with the predictions using DNS. Researchers have not previously compared LSA with DNS for the co-flowing two-phase jet problem. Researchers have only recently validated LSA with DNS for the simpler Rayleigh-Capillary stability problem [77] [20] [103] [26]. In this thesis, a DNS method has been developed for cylindrical coordinate systems. Researchers have not previously simulated 3D, two-phase, jet flow, in cylindrical coordinate systems. The numerical predictions for jet flow are compared: (1) LSA with DNS (2) DNS-CLSVOF with DNS-LS, and (3) 3D rectangular with 3D cylindrical. “DNS-CLSVOF” denotes the coupled level set and volume-of-fluid method for computing solutions to incompressible two-phase flows [99]. “DNS-LS” denotes a novel hybrid level set and volume constraint method for simulating incompressible two-phase flows [89]. The following discoveries have been made in this thesis: (1) the DNS-CLSVOF method and the DNS-LS method both converge under grid refinement to the same results for predicting the break-up of a liquid jet before and after break-up; (2) computing jet break-up in 3D cylindrical coordinate systems is more efficient than computing jet breakup in 3D rectangular coordinate systems; and (3) the LSA method agrees with the DNS method for the initial growth of instabilities (comparison method made for classical Rayleigh-Capillary problem and co-flowing jet problem). It is found that for the classical Rayleigh-Capillary stability problem, the LSA prediction differs from the DNS prediction at later times.