A transversal method of lines for the numerical modeling of vertical infiltration into the vadose zone

Here some issues are studied, related to the numerical solution of Richards' equation in a one dimensional spatial domain by a technique based on the Transversal Method of Lines (TMoL). The core idea of TMoL approach is to semi-discretize the time derivative of Richards' equation: afterward a system of second order differential equations in the space variable is derived as an initial value problem. The computational framework of this method requires both Dirichlet and Neumann boundary conditions at the top of the column. The practical motivation for choosing such a condition is argued. We will show that, with the choice of the aforementioned initial conditions, our TMoL approach brings to solutions comparable with the ones obtained by the classical Methods of Lines (hereafter referred to as MoL) with corresponding standard boundary conditions: in particular, an appropriate norm is introduced for effectively comparing numerical tests obtained by MoL and TMoL approach and a sensitivity analysis between the two methods is performed by means of a mass balance point of view. A further algorithm is introduced for deducing in a self-sustaining way the gradient boundary condition on top in the TMoL context.