The conservation laws governing the flow of liquids in porous media are often non-linear and have steep fronts that require resolution in time. It is one of the aims of this work to analyse the use of higher order spatial weighting schemes and temporal methods for reducing numerical dispersion. Another important ingredient in the development of an efficient simulator is the treatment of the non-linear system that results from the discrete analogue of the conservation law. In this work a vertex-centered finite volume method is used for discretising a representative conservation law in one-dimension and two non-linear iterative methods, an inexact full Newton method and the modified Shamanskii method, are scrutinised. Two case studies are chosen to highlight the performance of the chosen numerical techniques. At first, the focus is on the accuracy and efficiency of the spatial weighting methods for a linear advection-dispersion equation and then, a two-phase flow problem is analysed to gauge the performance of the non-linear solvers. In both cases, comparisons with exact solutions are presented.