The aim of the thesis is to find a method, in conjunction with the ordinary differential equation (ODE) based method of lines (MOL) solution of Richards’ equation, to model the steep wetting front infiltration in very dry soils, accurately and efficiently. Due to the steep pressure head or steep water volumetric content gradients, highly nonlinear soil hydraulic properties and the rapid movement of the wetting front, accurate solutions for infiltration into a dry soil are usually difficult to obtain. Additionally, such problems often require very small time steps and large computation times. As an enhancement to the used ODE/MOL approach, Higher Order Finite Differencing, Varying Order Finite Differencing, Vertical Scaling, Adaptive Schemes and Non-uniform Stretching Techniques have been implemented and tested in this thesis. Success has been found in the ability of Vertical Scaling to simulate very steep moving front solution for the Burgers’ equation. Unfortunately, the results also show that Vertical Scaling needs significant research and improvement before their full potential in routine applications for difficult nonlinear problems, such as Richard’s equation with very steep moving front solution, can be realized. However, we have also shown that the use of the composed form of RE and a 2nd order finite differencing for the first order derivative approximation is conducive for modelling steep moving front problem in a very dry soil. Additionally, with the combination of an optimal influx value at the edges of the inlet, the ODE/MOL approach is able to model a 2-D infiltration in very dry soils, effectively and accurately. Furthermore, one of the strengths of this thesis is the use of a MATLAB PDE template. Implementing the ODE/MOL approach via a MATLAB PDE template has shown to be most suitable for modelling of partial differential equations. The plug and play mode of modifying the PDE template for solving time-dependent partial differential equations is user-friendly and easy, as compared to more conventional approaches using Pascal, Fortran, C or C++. The template offers greater modularity, flexibility, versatility, and efficiency for solving PDE problems in both 1-D and 2-D spatial dimensions. Moreover, the 2-D PDE template has been extended for irregular shaped domains.