An apparatus and method are provided for solving a non-linear S-shaped function F=f(S) which is representative of a property S in a physical system, such saturation in a reservoir simulation. A Newton iteration (T) is performed on the function f(S) at Sv to determine a next iterative value Sv+1. It is then determined whether Sv+1 is located on the opposite side of the inflection point Sc from Sv. If Sv+1 is located on the opposite side of the inflection point from Sv, then Sv+1 is set to Sl, a modified new estimate. The modified new estimate, Sl, is preferably set to either the inflection point, Sc, or to an average value between Sv and Sv+1, i.e., Sl=0.5(Sv+Sv+1). The above steps are repeated until Sv+1 is within the predetermined convergence criteria. Also, solution algorithms are described for two-phase and three-phase flow with gravity and capillary pressure.