An optimal switching mechanism for a combined Picard- Newton method for the solution of Richards´equation

Richards' equation, describing flow in partially saturated porous media, contains strong nonlinearities arising from pressure head dependencies in soil moisture and hydraulic conductivity. Additionally, the time- dependent nature of boundary conditions can alter the nonlinear characteristics of equation during a transient simulation. Various iterative methods are used for solving this nonlinear equation, most commonly the quadratically convergent Newton – Raphson technique and the simpler but only linearly convergent Picard method (successive approximation). The initial solution estimate can have a large influence on the behavior of these iterative schemes, and we have observed through many applications of our numerical subsurface flow models that the Newton scheme is more sensitive to the initial solution than the Picard scheme is used to calculate improved initial guess for the Newton iteration. This scheme should achieve quadratic convergence while improving the global behavior of the iteration at less cost and complexity than alternative globalization techniques such as line search and trust region methods. In this work the combined Picard – Newton method is investigated via a theoretical analysis, based on a Taylor – Frechét expansion of the nonlinear