Fast-secant algorithms for the non-linear Richards equation

Abstract

SUMMARYGroundwater¯ow in partially saturated porous media is modelled by using the non-linear Richards equation, which is discretized in the present work by using linear mixed-hybrid ®nite elements.The discretization produces an algebraic non-linear system, which can be solved by an iterative ®xed-point algorithm, the Picard method. The convergence rate is linear, and may be too poor for practical applications. A superlinear convergence rate is obtained by considering a Broyden-type approach, based on the Shermann±Morrison formula.The local character of the Broyden method can be overcome by an accurate estimate of the initial solution, that is by appropriately initializing the computation via some (relaxed) Picard iterations. This strategy needs a convergence criterion to decide when switching from the Picard to the quasi-Newton method, which is crucial for the eectiveness of the scheme, as illustrated by some numerical experiments.We also consider the non-linear algebraic problem from a dierent viewpoint. Instead of applying the quasi-Newton method directly to such a non-linear system, we applied it to the non-linear function tied to the Picard scheme. Each function evaluation requested by such an algorithm corresponds to a local step of the Picard method, which is then used to compute a Broyden displacement. The present technique can be seen as an accelerated Picard algorithm.We compare the performances of these algorithms when applied to a stationary and a time-dependent benchmark problem. #

A Nested Newton-Type Algorithm for Finite Volume Methods Solving Richards' Equation in Mixed Form

“…The ψ-based form allows for both unsaturated and saturated conditions. In highly nonlinear problems, however, numerical methods based on the ψ-based form can suffer from large mass balance errors (see, e.g., [7,9,11]). …”

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

“…The solution of the resulting large nonlinear algebraic systems that arise in implicit numerical discretizations of the Richards equation has been the subject of significant research, and hence, one needs efficient and robust linearization techniques that maintain not only the accuracy of the solution, but also its mass conservation property. Typical linearization methods, such as the Picard, Newton, fast-secant and relaxation methods, as well as non-iterative methods, such as the implicit factored schemes and three-level Lees schemes, have been proposed [4,11,[16][17][18].…”

“…On the other hand, a pressure head formulation of Richards equation is continuous in both saturated and unsaturated zones, and can be used for homogeneous and non-homogeneous soils. However, numerical approaches based on this form can suffer large mass balance errors [2][3][4]. Small step size coupled with mass lumping effectively ensures the improvement of the mass balance of the pressure head-based Richards equation [2].…”

An investigation of temporal adaptive solution of Richards’ equation for sharp front problems

“…To resolve the nonlinearities, there are some iterative schemes have been proposed [10,13,14], e.g., Picard and Newton iteration methods, fast secant, and relaxation methods as well as non-iterative methods (e.g., the implicit factored scheme). In practice, the Picard method is prevalent due to its simple formulation and satisfactory performance [15].…”