Mass types, element orders and solution schemes for the richards equation

Abstract

The Richards equation for water movement in unsaturated soils is highly nonlinear and generally cannot be solved analytically. An alternative is to solve the equation numerically with finite-element methods. Conventionally, it has been accepted that the consistent mass scheme with higher-order elements is superior for solving complex nonlinear physical problems. However, results from other studies indicate that a finite-element model based on the consistent mass with higher-order elements would cause numerical oscillation problems. In this study, lumped mass, consistent mass, linear elements, and quadratic/cubic elements were evaluated to determine the most efficient method to solve the Richards equation with finite-element models. Results demonstrated that, when using consistent mass schemes or quadratic/cubic elements, the time step cannot be arbitrarily reduced to achieve the convergence; the mesh size must first be reduced in order to avoid numerical oscillations. In a time-dependent problem with large and complex domain, the minimum mesh size allowed is often unknown a priori . As a result, the intrinsic necessity of constantly adjusting the mesh size and hence rearranging the mesh structure is not efficient. On the other hand, in the lumped mass scheme with linear elements, one can arbitrarily reduce the time step at any time during the simulation to obtain a stable and consistent solution without changing the mesh structure. For nonlinear and time-dependent problems with large mesh domain, most of the computer time is used in algorithms to solve the linearized matrix equations. Results from this study indicate that, to solve the Richards equation, the conjugate-gradient methods are more efficient than the sky-line decomposition method. The three preconditioning schemes investigated do not have any significant difference in computer memory and time required.