A study on iterative methods for solving Richards` equation

TL;DR

This paper compares various linearization methods for solving Richards' equation, introduces a new robust scheme, and evaluates their performance through theoretical analysis and numerical experiments.

Contribution

It presents a comparative study of existing linearization schemes for Richards' equation and proposes a new, more robust and quadratically convergent method.

Findings

The Lscheme's convergence is theoretically proven.

The new Lscheme/Newton method shows improved robustness.

Numerical tests demonstrate the effectiveness of the proposed scheme.

Abstract

This work concerns linearization methods for efficiently solving the Richards` equation,a degenerate elliptic-parabolic equation which models flow in saturated/unsaturated porous media.The discretization of Richards` equation is based on backward Euler in time and Galerkin finite el-ements in space. The most valuable linearization schemes for Richards` equation, i.e. the Newtonmethod, the Picard method, the Picard/Newton method and theLscheme are presented and theirperformance is comparatively studied. The convergence, the computational time and the conditionnumbers for the underlying linear systems are recorded. The convergence of theLscheme is theo-retically proved and the convergence of the other methods is discussed. A new scheme is proposed,theLscheme/Newton method which is more robust and quadratically convergent. The linearizationmethods are tested on illustrative numerical examples.