A Functional Iteration Technique for Solving the Richards Equation Applied to Two‐Dimensional Infiltration Problems

Abstract

In the solution of nonlinear parabolic partial differential equations, such as the Richards equation, classical implicit schemes often oscillate and fail to converge. A fully implicit scheme has been developed along with a functional iteration method for solving the system of nonlinear difference equations. Newton's iteration technique is mathematically the most preferable of all functional iteration methods because of its quadratic convergence. The Richards equation, Newton‐linearized with respect to relative permeability and saturation as functions of capillary pressure, is particularly aided by this new approach for problems in which saturations vary rapidly with time (infiltration fronts, cone of depression near a well bore, and so forth). Although the computing time is almost twice as long for a time step with Newton's iteration scheme, the smaller time truncation than that of classical implicit schemes and the stability in cases in which classical schemes are unstable permit the use of much larger time steps. To demonstrate the method, heterogeneous (layered) soil systems are used to simulate sharp infiltration fronts caused by ponding at the soil surface.