Improving Newton's method performance by parametrization: the case of Richards equation

TL;DR

This paper introduces a parametrization strategy for Richards equation to improve Newton's method convergence, demonstrating local quadratic convergence and numerical efficiency through a stabilized formulation and finite volume discretization.

Contribution

It proposes a novel parametrization approach that stabilizes Newton's method for Richards equation, with theoretical convergence proof and numerical validation.

Findings

Newton's method converges quadratically under non-degeneracy conditions.

The parametrized scheme is well-posed and numerically efficient.

Numerical experiments confirm improved convergence and stability.

Abstract

The nonlinear systems obtained by discretizing degenerate parabolic equations may be hard to solve, especially with Newton's method. In this paper, we apply to Richards equation a strategy that consists in defining a new primary unknown for the continuous equation in order to stabilize Newton's method by parametrizing the graph linking the pressure and the saturation. The resulting form of Richards equation is then discretized thanks to a monotone Finite Volume scheme. We prove the well-posedness of the numerical scheme. Then we show under appropriate non-degeneracy conditions on the parametrization that Newton\^as method converges locally and quadratically. Finally, we provide numerical evidences of the efficiency of our approach.