Improving Newton's method performance by parametrization: the case of Richards equation
Résumé
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’s
method converges locally and quadratically. Finally, we provide numerical
evidences of the efficiency of our approach.
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