A linear domain decomposition method for partially saturated flow in porous media

TL;DR

This paper introduces a linear domain decomposition method for solving the nonlinear Richards equation in porous media, proving its convergence and demonstrating its stability and efficiency through numerical experiments.

Contribution

It proposes a new linear iterative domain decomposition scheme for Richards equation, with rigorous convergence proof and comparative analysis against Newton and Picard methods.

Findings

The scheme converges unconditionally.

It is more stable than Newton's method.

Comparable computational time without parallelization.

Abstract

The Richards equation is a nonlinear parabolic equation that is commonly used for modelling saturated/unsaturated flow in porous media. We assume that the medium occupies a bounded Lipschitz domain partitioned into two disjoint subdomains separated by a fixed interface $\Gamma$. This leads to two problems defined on the subdomains which are coupled through conditions expressing flux and pressure continuity at $\Gamma$. After an Euler implicit discretisation of the resulting nonlinear subproblems a linear iterative ($L$-type) domain decomposition scheme is proposed. The convergence of the scheme is proved rigorously. In the last part we present numerical results that are in line with the theoretical finding, in particular the unconditional convergence of the scheme. We further compare the scheme to other approaches not making use of a domain decomposition. Namely, we compare to a Newton and a Picard scheme. We show that the proposed scheme is more stable than the Newton scheme while remaining comparable in computational time, even if no parallelisation is being adopted. Finally we present a parametric study that can be used to optimize the proposed scheme.