Reply to “Comments on An Adaptive Finite Difference Scheme for the One‐Dimensional Water Flow Equation”

We recently read with interest the article by J. H. Dane and F. H. Mathis (Soil Sci. Soc. Am. J. 45:1048-1054) on an adaptive finite difference scheme for the solution of the one-dimensional water flow equation. Our researches have also been devoted to the application of adaptive grid techniques to the solution of this and similar equations and, consequently, we felt that a brief description of our findings would complement the aforementioned article. The method we have employed differs from that used by Dane and Mathis in two important aspects. Firstly, we chose to use the finite element method as the basis for our numerical solution and secondly, perhaps more significantly, we introduced additional grid points into the computational mesh in those regions of the solution domain where additional spatial resolution was required. This contrasts with Dane and Mathis' technique in which the total number of grid points remained constant with existing nodes being translated to those regions requiring high spatial resolution. The prime objective of adaptive grid methods is to improve the accuracy of the solution for a given computational cost. The fact that Dane and Mathis have achieved this aim is amply demonstrated by their Tables 2 and 4 and Fig. 1 to 3. However, there comes a point in any numerical solution based on a uniform mesh, where a further increase in the number of nodes leads to no significant improvement in the accuracy of the solution. As the most time-consuming part of any numerical scheme is the assembly and solution of the discretised equations it is desirable to reduce the total number of degrees of freedom as much as possible whilst maintaining the desired accuracy. Consequently, we feel that the total number of nodes as well as their distribution should be determined on the basis of the current solution. In this way the required accuracy can be maintained for a minimum cost. To illustrate these remarks, our equivalent to Table 2 of Dane and Mathis is shown below as Table 1. The solutions leading to these results were based on a ten element, Galerkin finite element discretisation in space using linear trial and test functions with additional nodes being introduced into those regions requiring greater spatial resolution. The time integration scheme used a two-level predictor-corrector algorithm with the required criterion of A0 stability. A variable time step was available as an option, the time step being adjusted to maintain the number of corrections required per time step constant. It can be seen from Table 1 that the adaptive mesh solution is similar in cost to that having an equal number of uniformly spaced grid points for much improved accuracy. One notable feature of these solutions was that equivalent accuracy to the results of Dane and Mathis was obtained from only one-third the number of grid points. This is attributed to the property of the finite element method of integrating the water capacity and hydraulic conductivity functions over each element rather than using representative single-point values as in the finite difference solution. In conclusion, we feel that an adaptive grid scheme for the solution of problems involving sharp fronts is best implemented by varying the number of grid points in the solution rather than by rearranging existing nodes. In this way the number of degrees of freedom and hence the computational cost can be kept