Large Scale Numerical Simulation Of Richards Equation For Variably Saturated Soil Water Flows

Prediction of water movements in variably saturated soil is currently of critical importance for many applications of sciences and engineering,such as water resources,petroleum reservoirs,multiphase flow models and so on.The extensive growing demand on describing,understanding,and predicting the dynamics of the systems with varying degree of complexity has produced persistent requirements on developing state-of-theart mathematical models and solvers with enhanced prediction capabilities.Mathematical modeling of variably saturated soil water flows usually results in systems of time-dependent nonlinear partial differential equations(PDE).Due to its highly nonlinear behaviour arising from the complexity of relevant physical,chemical,and biological flow processes in the geological media,high resolution simulations with advanced numerical methods are efficient to solve this intricate problem.For high fidelity simulations on supercomputers,the accuracy,robustness,and scalability of the parallel solver with respect to the number of processors are critically important.In this thesis,the Richards equation is discretized in space and time based on a fully implicit,cell-centered finite volume scheme,where several different approximate methods are applied to guarantee the stability of the spatial discretization and an adaptive time-stepping technique for the temporal integration is designed to accelerate the simulation progress.A key to the success of a fully implicit method is the solution of a nonlinear algebraic system at every time step.Hence,we propose and study a highly parallel solver based on the framework of Newton-Krylov algorithms to guarantee the nonlinear consistency.In the proposed parallel framework,an inexact Newton method with backtracking is the basic nonlinear iteration and a Krylov subspace iterative method is applied as the linear solver at each Newton step.The success of the overall Newton-Krylov solver depends heavily on substantially reducing the condition number of the corresponding linear system,which signifies the urgent demand of preconditioners to accelerate the convergence of nonlinear and linear iterations.Hence,the main focus is on the design of some efficient overlapping additive Schwarz type domain decomposition methods to build the preconditioner.The domain decomposition method,which follows the nature of divide-and-conquer techniques,is to transform the original problem into more sub-problems of the same or related type,and hence can perfectly match the fundamental characteristic of the massively parallel computing.Finally,our experiments show that the proposed fully implicit solver based on the Newton-Krylov algorithm and the domain decomposition technique is efficient,robust and scalable on a supercomputer platform.We implement the proposed algorithm using the Portable,Extensible Toolkit for Scientific computation(PETSc)library with a variety of test cases and focus on(a)validating the numerical accuracy of the fully implicit solver with respect to the numerical order of convergence;(b)studying the robustness and efficiency of the proposed algorithm for both standard benchmarks and realistic problems;(c)comparing the performance of different approximate methods to calculate the conductivity and the effect of using the adaptively approach;and(d)analyzing the parallel performance of the proposed method with different parameters.