Enhancing the accuracy of the solution to unsaturated flow by a Hermitian Green element model

Our earlier paper demonstrated that the singular boundary element (BE) theory, which hitherto had been considered unsuitable for the highly nonlinear variably saturated transient flow problem, could be applied to the same problem when implemented along the lines of the Green element method (GEM) (Taigbenu, A. E. and Onyejekwe, O. O., Green element simulations of transient nonlinear unsaturated flow equation, Applied Mathematical Modelling, 1995, 19, 675–684). Here, the transient one-dimensional unsaturated flow problem is revisited with a Green element model which incorporates the cubic Hermitian interpolation basis functions to approximate the distribution of the primary variable and the soil constitutive relations. Because the soil parameters vary appreciably by several orders of magnitude over small intervals of soil moisture, approximating those parameters by linear interpolation functions, as earlier done, could be inadequate, and this fact is demonstrated here using two numerical examples of infiltration into vertical soil columns. With the first example, there is good agreement between the solutions of Hermitian GEM and Hermitian finite element method (FEM) which confirms that the discrepancy of less amount of soil moisture in the linear GEM solution, earlier observed, is due to an error from interpolation (Taigbenu, A. E. and Onyejekwe, O. O., Green element simulations of transient nonlinear unsaturated flow equation, Applied Mathematical Modelling, 1995, 19, 675–684). The second example serves to show that comparable accuracy between Hermitian and linear GE models can be achieved using a fewer number of elements in the former model.