A COMPARISON OF PICARD AND NEWTON ITERATION IN THE NUMERICAL-SOLUTIONOF MULTIDIMENSIONAL VARIABLY SATURATED FLOW PROBLEMS

Picard iteration is a widely used procedure for solving the nonlinear equation governing flow in variably saturated porous media. The method is simple to code and computationally cheap, but has been known to fa il or converge slowly. The Newton method is more complex and expensive (on a per-iteration basis) than Picard, and as such has not received very much attention. Its robustness and higher rate of convergence, ho wever, make it an attractive alternative to the Picard method, particu larly for strongly nonlinear problems. In this paper the Picard and Ne wton schemes are implemented and compared in one-, two-, and three-dim ensional finite element simulations involving both steady state and tr ansient flow. The eight test cases presented highlight different aspec ts of the performance of the two iterative methods and the different f actors that can affect their convergence and efficiency, including pro blem size, spatial and temporal discretization, initial solution estim ates, convergence error norm, mass lumping, time weighting, conductivi ty and moisture content characteristics, boundary;conditions, seepage fades, and the extent of fully saturated zones in the soil. Previous s trategies for enhancing the performance of the Picard and Newton schem es are revisited, and new ones are suggested. The strategies include c hord slope approximations for the derivatives of the characteristic eq uations, relaxing convergence requirements along seepage faces, dynami c time step control, nonlinear relaxation, and a mixed Picard-Newton a pproach. The tests show that the Picard or relaxed Picard schemes are often adequate for solving Richards' equation, but that in cases where these fail to converge or converge slowly, the Newton method should b e used. The mixed Picard-Newton approach can effectively overcome the Newton scheme's sensitivity to initial solution estimates, while compa ratively poor performance is reported for the various chord slope appr oximations. Finally,given the reliability and efficiency of current co njugate gradient-like methods for solving linear nonsymmetric systems; the only real drawback of using Newton rather than Picard iteration i s the algebraic complexity and computational cost of assembling the de rivative terms of the Jacobian matrix, and it is suggested that both m ethods can be effectively implemented and used in numerical models of Richards' equation.