Nonlinear Effects on the Convergence of Picard and Newton Iteration Methods in the Numerical Solution of One-Dimensional Variably Saturated–Unsaturated Flow Problems

Abstract

Finite element discretization of the pressure head form of the Richards equation leads to a nonlinear model, which yields numerical convergence difficulties. When the numerical solution to this problem has either an extremely sharp moving front, infiltration into dry soil, flow domains containing materials with spatially varying properties, or involves time-dependent boundary conditions, the corrector iteration used in many time integrators can terminate prematurely, which leads to incorrect results. While the Picard and Newton iteration methods can solve this problem through tightening the tolerances provided to the solvers, there is a more efficient approach to overcome the convergence difficulties. Four tests examples are examined, and each test case is solved with five sufficiently small tolerances to demonstrate the effectiveness of convergence. The numerical results illustrate that the methods greatly improve the convergence and stability. Test experiments show that the Newton method is more complex and expensive on a per iteration basis than the Picard method for simulating variably saturated–unsaturated flow in one spatial dimension. Consequently, it is suggested that the resulting local and global mass balance is exact within the minimum specified accuracy.