We consider the simulation of steady-state variably saturated groundwater flow using RichardsÕ equation (RE). The difficulties associated with solving RE numerically are well known. Most discretization approaches for RE lead to nonlinear systems that are large and difficult to solve. The solution of nonlinear systems for steady-state problems can be particularly challenging, since a good initial guess for the steady-state solution is often hard to obtain, and the resulting linear systems may be poorly scaled. Common approaches like Picard iteration or variations of NewtonÕs method have their advantages but perform poorly with standard globalization techniques under certain conditions.
Pseudo-transient continuation has been used in computational fluid dynamics for some time to obtain steady-state solutions for problems in which NewtonÕs method with standard line-search strategies fails. Here, we examine the use of pseudo-transient continuation as well as NewtonÕs method combined with standard globalization techniques for steady-state problems in heterogeneous domains. We investigate the methodsÕ performance with direct and preconditioned Krylov iterative linear solvers. We then make recommendations for robust and efficient approaches to obtain steady-state solutions for RE under a range of conditions.