A new convergence criterion for the modified Picard iteration method to solve the variably saturated flow equation

Abstract

Solutions of the Richards equation for water flow in variably saturated porous media are increasingly being used in water resources evaluation and environmental management. Besides the accuracy of solution, also of concern is the required computational effort, especially when highly nonlinear soil hydraulic properties and dry initial conditions are involved. In this paper we evaluate the performance of different convergence criteria when the modified Picard iteration method is used for solving the mixed-form Richards equation. Results are compared in terms of computer processing (CPU) time and number of iterations. A new nonlinear convergence criterion derived using a Taylor series expansion of the water content was implemented in the mixed-form numerical algorithm. The computational efficiency of the new criterion was evaluated against two widely used convergence criteria for different soil types, boundary conditions, initial conditions, and layered soils. Whereas all three criteria produced nearly identical results in terms of calculated water content, pressure head, and waser flux distributions, all with negligible mass balance errors, the required CPU times were significantly different. In general, the new nonlinear convergence criterion was found to be computationally much more efficient than the other two criteria. The new criterion was also more robust (i.e. the solution remained convergent) for highly nonlinear flow problems for which the other two convergence criteria failed. Results of this study indicate that the new convergence criterion, when implemented in the modified Picard solution of the mixed-form Richards equation, produces a very efficient and accurate method for simulating variably saturated water flow in soils.