Convergence Properties of Iteratively Coupled Surface-Subsurface Models

TL;DR

This work addresses the convergence of partitioned, iteratively coupled surface-subsurface models that pair Richards' equation in the subsurface with surface flow models. By deriving a linearized 1D-0D model and applying continuous and fully discrete analyses, the authors obtain explicit convergence factors and optimal relaxation parameters that depend on material properties ($c$, $K$) and discretization parameters ($\,\Delta t$, $\Delta z$, $L$). They validate the theory against nonlinear 2D-1D benchmark simulations using the DUNE framework and preCICE, finding that the linear analysis explains the observed fast convergence in practice while outlining regimes where nonlinear effects and discretization dominate. The results show that convergence is primarily governed by vertical Richards dynamics, and that, for typical materials and grid choices, relaxation is often unnecessary, providing practical guidance for efficiently coupling surface and subsurface solvers.

Abstract

Surface-subsurface flow models for hydrological applications solve a coupled multiphysics problem. This usually consists of some form of the Richards and shallow water equations. A typical setup couples these two nonlinear partial differential equations in a partitioned approach via boundary conditions. Full interaction between the subsolvers is ensured by an iterative coupling procedure. This can be accelerated using relaxation. In this paper, we apply continuous and fully discrete linear analysis techniques to study an idealized, linear, 1D-0D version of a surface-subsurface model. These result in explicit expressions for the convergence factor and an optimal relaxation parameter, depending on material and discretization parameters. We test our analysis results numerically for fully nonlinear 2D-1D experiments based on existing benchmark problems. The linear analysis can explain fast convergence of iterations observed in practice for different materials and test cases, even though we are not able to capture various nonlinear effects.

Figures (10)

• Figure 1: The spatial domain in the 2D-1D coupling problem.
• Figure 2: Verification of fully discrete analysis: Comparison of the theoretical and experimental convergence factors for varying grid and time step sizes.
• Figure 3: Convergence factor $|S|$ for varying hydraulic conductivity $K$, hydraulic capacity $c$, and grid sizes. The white contour line for $|S|=1$ marks the border between converging and diverging coupling iterations.
• Figure 4: $\omega_\mathrm{opt}$ for varying hydraulic conductivity $K$, hydraulic capacity $c$, and grid sizes.
• Figure 5: Geometry for the drainage trench experiments. The dashed line marks the initial groundwater height.