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Due to complex dynamics inherent in the physical models, numerical formulation of subsurface and overland flow coupling can be challenging to solve. ParFlow is a subsurface flow code that utilizes a structured grid discretization in order to benefit from fast and efficient structured solvers. Implicit coupling between subsurface and overland flow modes in ParFlow is obtained by prescribing an overland boundary condition at the top surface of the computational domain. This form of implicit coupling leads to the activation and deactivation of the overland boundary condition during simulations where ponding or drying events occur. This results in a discontinuity in the discrete system that can be challenging to resolve. Furthermore, the coupling relies on unstructured connectivities between the subsurface and surface components of the discrete system, which makes it challenging to use structured solvers to effectively capture the dynamics of the coupled flow. We present a formulation of the discretized algebraic system that enables the use of an analytic form of the Jacobian for the Newton–Krylov solver, while preserving the structured properties of the discretization. An effective multigrid preconditioner is extracted from the analytic Jacobian and used to precondition the Jacobian linear system solver. We compare the performance of the new solver against one that uses a finite difference approximation to the Jacobian within the Newton–Krylov approach, previously used in the literature. Numerical results explores the effectiveness of using the analytic Jacobian for the Newton–Krylov solver, and highlights the performance of the new preconditioner and its cost. The results indicate that the new solver is robust and generally outperforms the solver that is based on the finite difference approximation to the Jacobian, for problems where the overland boundary condition is activated and deactivated during the simulation. A parallel weak scaling study highlights the efficiency of the new solver.

In this paper, we investigate the effectiveness of the Anderson acceleration method applied to modified Picard iteration for nonlinear problems arising in variably saturated flow modeling. While many authors have studied the relative merits of Newton’s method and modified Picard iteration in this context, the combination of Anderson acceleration and modified Picard iteration has not been investigated for these problems until recently. Since modified Picard iteration can be slow to converge, we investigate the use of Anderson acceleration to provide faster convergence while maintaining the robustness and lower memory requirements of modified Picard iteration relative to Newton’s method. Results indicate that Anderson acceleration significantly improves not only convergence speed but also robustness of modified Picard iteration and can often provide faster solutions than Newton’s method without the need for derivative computations.

In this paper, we present a class of solvers developed for the parallel solution of Richards' equation, a model used in variably saturated flow simulations. These solvers take advantage of the fast, robust convergence of globalized Newton methods as well as the parallel scalability of multigrid preconditioners. We compare two multigrid methods. The methods differ primarily in their handling of discontinuous and anisotropic permeability fields, with one method invoking a simple pointwise smoothing technique and the other a more expensive plane smoother. Computational results are presented to show the effectiveness of the entire nonlinear solution procedure, to demonstrate the effect of discontinuities and anisotropies, and to explore parallel efficiencies.