The exchange of ground- and surface water plays a crucial role in a variety of practically relevant processes ranging from flood protection measures to preservation of ecosystem health in natural and human-impacted water resources systems.

Commonly accepted models are based on the shallow water equations for overland flow and the Richards equation for saturated–unsaturated subsurface flow with suitable coupling conditions. Continuity of mass flow across the interface is natural, because it directly follows from mass conservation. Continuity of pressure is typically imposed for simplicity. Mathematically, this makes sense for sufficiently smooth height of surface water as occurring, e.g., in filtration processes. Here we impose Robin-type coupling conditions modelling a thin, nearly impermeable
layer at the bottom of the river bed that may cause pressure discontinuities; an effect which is known in hydrology as clogging. From a mathematical perspective, clogging can be regarded as a kind of regularization, because, in contrast to Dirichlet conditions, Robin conditions can be straightforwardly formu lated in a weak sense.

Existence and uniqueness results for the Richards equation and the shallow water equations are rare and hard to obtain, and nothing seems to be known about solvability of coupled problems. Extending the general framework of heterogeneous Steklov–Poincaré formulations and iterative substructuring to time-dependent problems, we introduce a Robin–Neumann iteration for the continuous coupled problem and motivate its feasibility by well-known existence results for the linear case. As surface and subsurface flow are only weakly coupled by clogging and continuity of mass flux, different discretizations with different time steps and different meshes can be used in a natural way. This is absolutely necessary, to resolve the vastly different time and length scales of surface and subsurface flow. Discrete mass conservation can be proved in a straightforward way.

Finally, we illustrate our considerations by coupling a finite element discretization of the Richards equation based on Kirchhoff transformation with a simple upwind discretization of surface flow. Numerical experiments confirm discrete mass conservation and show fast convergence of the Robin–Neumann iteration for real-life soil data.