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

TL;DR

This paper analyzes the spectral properties of Jacobian matrices in Richards equation discretizations, providing theoretical insights and computational experiments that support the development of scalable preconditioners for large-scale simulations.

Contribution

It offers new spectral analysis results for Jacobians in Richards equation and introduces a software framework for scalable preconditioning at very large computational scales.

Findings

Eigenvalue distribution analysis supports preconditioner effectiveness

New explicit Jacobian matrix expressions derived

Framework demonstrates promising performance on large-scale tests

Abstract

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