Improving Newton's Method Performance by Parametrization: The Case of the Richards Equation

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 the Richards equation, a strategy that consists of 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 the 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 nondegeneracy conditions on the parametrization that Newton's method converges locally and quadratically. Finally, we provide numerical evidences of the efficiency of our approach.