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.

翻译：水文应用中的地表-地下水流模型求解的是一个耦合多物理场问题。该问题通常由某种形式的Richards方程和浅水方程构成。典型的建模方法通过边界条件以分区方式耦合这两个非线性偏微分方程。通过迭代耦合程序可确保子求解器间的完全相互作用，并可采用松弛法加速此过程。本文应用连续及全离散线性分析技术，研究一个理想化、线性的一维-零维地表-地下模型。由此推导出收敛因子与最优松弛参数的显式表达式，这些表达式取决于材料参数和离散化参数。我们基于现有基准问题，在完全非线性的二维-一维实验中进行了数值验证。线性分析能够解释实践中观察到的不同材料和测试案例下迭代快速收敛的现象，尽管该方法尚无法捕捉各类非线性效应。