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We consider here a cell-centered finite difference approximation of theRichards equation in three dimensions, averaging for interface values thehydraulic conductivity $K=K(p)$, a highly nonlinear function, by arithmetic,upstream, and harmonic means. The nonlinearities in the equation can lead tochanges in soil conductivity over several orders of magnitude anddiscretizations with respect to space variables often produce stiff systems ofdifferential equations. A fully implicit time discretization is provided by\emph{backward Euler} one-step formula; the resulting nonlinear algebraicsystem is solved by an inexact Newton Armijo-Goldstein algorithm, requiring thesolution of a sequence of linear systems involving Jacobian matrices. We provesome new results concerning the distribution of the Jacobians eigenvalues andthe explicit expression of their entries. Moreover, we explore some connectionsbetween the saturation of the soil and the ill-conditioning of the Jacobians.The information on eigenvalues justifies the effectiveness of somepreconditioner approaches which are widely used in the solution of Richardsequation. We also propose a new software framework to experiment with scalableand robust preconditioners suitable for efficient parallel simulations at verylarge scales. Performance results on a literature test case show that ourframework is very promising in the advance towards realistic simulations atextreme scale.

Why diffusion-based preconditioning of Richards equation works: spectral analysis and computational experiments at very large scale