Propagation of saturation overshoots for two-phase flow of immiscible and incompressible fluids in porous media is analyzed using different computational methods. In particular, it is investigated under which conditions a given saturation overshoot remains stable while moving through a porous medium. Two standard formulations are employed in this investigation, a fractional flow formulation and a pressure-saturation formulation. Neumann boundary conditions for pressure are shown to emulate flux boundary conditions in homogeneous media. Gravity driven flows with Dirichlet boundary conditions for pressure that model infiltration into heterogeneous media with position-dependent permeability are found to exhibit pronounced saturation overshoots very similar to those seen in experiment.