Abstract

Finite element discretization of the pressure head form of the Richards equation leads to a nonlinear model, which yields numerical convergence difficulties. When the numerical solution to this problem has either an extremely sharp moving front, infiltration into dry soil, flow domains containing materials with spatially varying properties, or involves time-dependent boundary conditions, the corrector iteration used in many time integrators can terminate prematurely, which leads to incorrect results. While the Picard and Newton iteration methods can solve this problem through tightening the tolerances provided to the solvers, there is a more efficient approach to overcome the convergence difficulties. Four tests examples are examined, and each test case is solved with five sufficiently small tolerances to demonstrate the effectiveness of convergence. The numerical results illustrate that the methods greatly improve the convergence and stability. Test experiments show that the Newton method is more complex and expensive on a per iteration basis than the Picard method for simulating variably saturated–unsaturated flow in one spatial dimension. Consequently, it is suggested that the resulting local and global mass balance is exact within the minimum specified accuracy.

Highlights

  • Fluid flow through variably saturated–unsaturated porous media is usually described by the classical Richards equation [1]

  • The numerical results needed to meet the objective of this work; we focus our attention on the analysis of how much the small tolerance will affect the efficiency, accuracy, and convergence properties of the numerical solution to the Richards equation

  • A realistic computational approach is presented that effectively addresses the minimal head tolerance to solve the Richards equation accurately for four difficult test problems in combination with

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Summary

Introduction

Fluid flow through variably saturated–unsaturated porous media is usually described by the classical Richards equation [1]. It is defined by coupling a statement of mass conservation with. It is a highly nonlinear partial differential equation that can be represented in three standard forms, depending on whether pressure, moisture, or both are used as dependent variables. For one-dimensional vertical flow, the pressure head-based Richards equation is written as ∂Ψ ∂ C (Ψ ) = K (Ψ) +1 ∂t ∂z (1)

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