Several noniterative procedures for solving the nonlinear Richards equation are introduced and compared to the conventional Newton and Picard iteration methods. Noniterative strategies for the numerical solution of transient, nonlinear equations arise from explicit or linear time discretizations, or they can be obtained by linearizing an implicit differencing scheme. We present two first order accurate linearization methods, a second order accurate two-level “implicit factored” scheme, and a second order accurate three-level “Lees” method. The accuracy and computational efficiency of these four schemes and of the Newton and Picard methods are evaluated for a series of test problems simulating one-dimensional flow processes in unsaturated porous media. The results indicate that first order accurate schemes are inefficient compared to second order accurate methods; that second order accurate noniterative schemes can be quite competitive with the iterative Newton and Picard methods; and that the Newton scheme is no less efficient than the Picard method, and for strongly nonlinear problems can outperform the Picard scheme. The two second order accurate noniterative schemes appear to be attractive alternatives to the iterative methods, although there are concerns regarding the stability behavior of the three-level scheme which need to be resolved. We conclude that of the four noniterative strategies presented, the implicit factored scheme is the most promising, and we suggest improved formulations of the method.

Numerical evaluation of iterative and noniterative methods for the solution of the nonlinear richards equation

The interaction of a wetting front with an impervious layer is described by adding a reflected solution to the incoming solution for a semi-infinite medium. It is shown and checked by comparison with a numerical solution that the result is accurate during the early times of the interaction between the front and the impervious surface. This superposition principle is quite general and should prove especially useful to initiate numerical schemes by this analytical approximation as in the early times singularities are difficult to describe numerically.

/pdf/superposition-principle-for-short-term-solutions-of-richards-3kcz5t9m3t.pdf

Superposition principle for short-term solutions of Richards' equation : application to the interaction of wetting fronts with an impervious surface

Analysis of movement of water in unsaturated soils is a basis for water management and for understanding the environment in which soil mechanical, chemical, and biological processes occur. This commentary presents further examples of quasianalytic and analytic approaches to unsaturated flow. Three approaches to one-dimensional problems are discussed in some detail: (1) approximate methods subject to integral constraints, (2) methods based on transformations of nonlinear equations to linear equations, and (3) solutions for vertical flows dominated by gravity. Also, some further examples of quasilinear analysis of steady multidimensional flows are given, with emphasis on flows from shallow circular ponds.

Quasianalytic and Analytic Approaches to Unsaturated Flow: Commentary

Preferential flow in vertically oriented, unsaturated soil layers

Water movement in a finite layer : influence of gravity for constant water content at the surface

An accurate, analytical formulation describes very simply the outflow of water from a soil layer when the potentials (water contents) are maintained constant at both surfaces. The analytical formulation is adapted from a previously developed approach to the present problem. The accuracy of the results is checked by comparison with a precise numerical simulation which is greatly simplified for a Brooks and Corey type diffusivity. It is suggested that a laboratory experiment using this configuration would yield the soil properties relevant for adsorption.

Water movement in a finite layer: absorption for constant water content at the surfaces

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Water movement in a finite layer: absorption for constant water content at the surfaces

Water movement in a finite layer : influence of gravity for constant water content at the surface