Analysis and Robust Preconditioning for Numerical Implementations of Richards' Equations in Groundwater Flow
Open Access
- Author:
- Batista, Juan
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- October 04, 2019
- Committee Members:
- Ludmil Tomov Zikatanov, Dissertation Advisor/Co-Advisor
Ludmil Tomov Zikatanov, Committee Chair/Co-Chair
Xiantao Li, Committee Member
Corina Stefania Drapaca, Outside Member
Anna L Mazzucato, Dissertation Advisor/Co-Advisor
Anna L Mazzucato, Committee Chair/Co-Chair
Mark Levi, Program Head/Chair - Keywords:
- Hydrology
Numerical Simulation
Porous Media
Nonlinear PDE
Applied Mathematics
Preconditioners
Auxiliary Space Method - Abstract:
- This thesis serves as a mathematical and numerical exploration of Richards’ equation, a quasilinear partial differential equation modeling the flow of nearly incompressible fluid through unsaturated porous media, with degeneracies at full saturation. This physical model can be seen as a reduction of a full two phase (wetting/air phase) flow model, where the air phase is assumed to have constant atmospheric pressure. In this case, the air pressure only affects the pressure of the wetting phase via the hydraulic conductivity of the porous matrix through capillary action, which is in turn modeled as a function of saturation of the water phase. We discuss various formulations of Richards’ equation, and popular models for the water content θ and hydraulic conductivity K as functions of the pressure head. For the ubiquitous VGM model, we describe some analytic properties of the physical parameters. Due to the nonlinear nature of the problem, closed-form solutions do not exist, except in some special cases. As such, numerical treatment is required to approximate solutions for physically relevant problems. In this thesis we consider several linearization schemes used to treat the nonlinearities, including the Picard, Newton-Raphson, modified Picard, and the L-scheme. For a time-continuous Picard linerization, we were able to prove that under a technical assumption on the behavior of the nonlinearities, the sequence of solutions to the linearized problem is a contractive sequence, thus guaranteeing convergence of the iterates to the solution of the nonlinear problem. The need for control of K and θ in the analysis is confirmed by various numerical results reported in the literature in which the Picard linearization of Richards’ equation fail to converge due to the nonlinearities. For the resulting sequence of parabolic problems, we discretize with the implicit Euler method in time, and a mixed finite element discretization in space (lowest order Raviart- Thomas elements). For the efficient solution of the resulting linear systems during the iterations we introduce a combined preconditioner: an inexact Uzawa iteration paired with an auxiliary space preconditioner using the standard linear continuous Lagrange finite element space. We prove that the preconditioned system has a uniformly bounded condition number. The combined preconditioner for the symmetric linearizations is robust with respect to discretization parameters, and jumps in the conductivity, though the convergence theory of these linear schemes to a weak solution for problems with layered media is a more complicated matter, and was not the focus of this work. We present several numerical tests veryfying the theoretical results. Additionally, we present numerical results for the nonsymmetric linear problems arising from the Newton-Raphson linearization, and in some cases we observe that the preconditioners are robust with respect to discretization parameters and the nonlinear physical parameters K and θ.