Convergence Properties of Iteratively Coupled Surface-Subsurface Models

TL;DR

This paper analyzes the convergence of iterative coupling methods for surface-subsurface flow models, providing explicit formulas for convergence factors and optimal relaxation parameters through linear analysis, and validating findings with nonlinear numerical experiments.

Contribution

It introduces a linear analysis framework to derive explicit convergence metrics for coupled flow models and validates these with nonlinear simulations.

Findings

Linear analysis explains fast convergence in practice.

Explicit expressions for convergence factors and relaxation parameters.

Numerical experiments confirm the analysis results.

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.