An Efficient Multiple-Dimensional Finite Element Solution for Water Flow in Variably Saturated Soils

Multiple-dimensional water flow in variably saturated soils plays an important role in ecological systems such as irrigation and water uptake by plant roots; its quantitative description is usually based on the Richards' equation. Because of the nonlinearity of the Richards' equation and the complexity of natural soils, most practical simulations rely on numerical solutions with the nonlinearity solved by iterations. The commonly used iterations for solving the nonlinearity are Picard and Newton methods with the former converging at first-order rate and the later at second-order rate. A recent theoretical analysis by the authors, however, revealed that for solving the diffusive flow, the classical Picard method is actually a chord-Newton method, converging at a rate faster than first order; its linear convergence rate is due to the treatment of the gravity term. To improve computational efficiency, a similar chord-Newton method as for solving the diffusive term was proposed to solve the gravity term. Testing examples for one-dimensional flow showed significant improvement. The core of this method is to produce a diagonally dominant matrix in the linear system so as to improve the iteration-to-iteration stability and hence the convergence. In this paper, we develop a similar method for multiple-dimensional flow and compare its performance with the classical Picard and Newton methods for water flow in soils characterised by a wide range of van Genuchten parameters.