A Discontinuous Galerkin Finite Element Method Solution of One-Dimensional Richards’ Equation

THESIS

Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University

Yilong Xiao

Graduate Program in Civil Engineering

The Ohio State University

2016

Master's Examination Committee:

Dr. Gajan Sivandran, Advisor

Dr. Ethan J. Kubatko

Dr. Yu-Ping Chin

Yilong Xiao

2016

Abstract

Unsaturated flows in porous media, attributing to hydraulic conductivity, capillary

pressure and gravity, are governed by the Richards’ equation. Due to high non-linearity

in the relations among it variables, the Richards’ equation does not have a general closed-

form solution. Past numerical Richards’ equation solvers were predominantly predicated

on conforming finite element methods or finite different methods, of which the

requirement on continuity in solutions occasionally led to convergence failure when

abrupt change occurs in boundary conditions or in hydraulic properties of the porous

media or both. Moreover, past infiltration schemes influenced by different user-defined vertical discretization could produce significantly different runoff partitioning results even with a same set of parameters. In light of such challenges, hereby presented is a one- dimensional numerical Richards’ equation solver based on the discontinuous Galerkin finite element method (DG-FEM) – a class of finite element method that operates on piecewise continuous trial spaces and allows discontinuity in solutions. Such characteristics make DG-FEM solutions fit for high-intensity moisture inputs and natural heterogeneities. This thesis outlines the steps of DG-FEM formulation on a one- dimensional Richards’ equation. Soil moisture readings under laboratory settings were

ii obtained to validate accuracy, while a long term precipitation dataset was used to contrast the capability of this DG-FEM model to that of a continuous Galerkin FEM model in

handling extreme rainfall events and partitioning runoff.

iii

Acknowledgments

Words cannot express my gratitude enough to Gaj, my advisor, for broadening my horizon in the realm of hydrology, dragging me through dungeons and dens.

Learning DG from Ethan has been a joyful journey of stumbling and bruising.

The way you put up with such a numerical modeling noob in me is beyond reckoning.

It is to my greatest delight that Yo accepted my sudden request to be on the committee. Your perspectives as a scientist have always been refreshing and educating.

Vita

2014...... B.S. Environmental Engineering, The Ohio

State University

2015 – Present ...... Graduate Teaching Assistant, Department of

Engineering Education, The Ohio State

University

Fields of Study

Major Field: Civil Engineering

Table of Contents

Abstract ...... ii

Acknowledgments...... iv

Vita ...... v

Table of Contents ...... vi

List of Tables ...... ix

List of Figures ...... x

Chapter 1: Introduction ...... 1

1.1 Motivation and Objective ...... 2

Chapter 2: Literature Review ...... 5

2.1 The Green-Ampt Equation ...... 5

2.2 The Richards’ Equation ...... 9

2.2.1 Solution of the Richards’ Equation – Analytical ...... 12

2.2.2 Solution of the Richards’ Equation – Numerical ...... 14

2.3 Discontinuous Galerkin Finite Element Method ...... 15

Chapter 3: Discontinuous Galerkin Formulation ...... 17

vi 3.1 Spatial Discretization ...... 17

3.1.1 Weak Formulation ...... 17

3.1.2 Numerical Flux ...... 19

3.1.3 Basis Polynomials ...... 21

3.1.4 Master Element Transformation ...... 26

3.1.5 Numerical Integration ...... 28

3.1.6 Global Equations ...... 31

3.1.7 Hydraulic Conductivity and Diffusivity ...... 33

3.1.8 Numerical Representation of Diffusivity ...... 35

3.1.9 Equation Summary ...... 39

3.2 Temporal Discretization ...... 40

3.2.1 Initial Condition ...... 41

3.3 Boundary Conditions and Sink/Source Terms ...... 43

Chapter 4: Materials and Methods ...... 45

4.1 Data Collection ...... 45

4.2 Model Comparison ...... 48

Chapter 5: Results and Discussion ...... 50

5.1 Parameter Fitting ...... 50

5.2 Rainfall Simulation ...... 56

vii 5.2.1 Times Series ...... 56

5.2.2 Point-to-Point ...... 66

5.2.3 Mass Conservation and Sensitivity Study...... 70

5.3 Model Comparison: DG vs. CG ...... 77

Chapter 6: Conclusion...... 79

References ...... 81

Appendix A: Setup ...... 89

Appendix B: Procedures for Determining Soil Parameters ...... 91

Appendix C: Pseudo-Code ...... 95

viii

List of Tables

Table 1: Numerical Fluxes ...... 21

Table 2: Default Settings of Simulation I ...... 49

Table 3: Default Settings of Simulation II ...... 49

Table 4: Soil Parameters for Play Sand ...... 51

Table 5: Rainfall/Hiatus Regime ...... 56

Table 6: Coefficients of Determination ...... 68

Table 7: Comparison of R2 and Error ...... 72

Table 8: Summary of Simulation II ...... 78

Table 9: Location of Sensors in Sand Column ...... 90

List of Figures

Figure 1: Scenario of the Green-Ampt Equation ...... 6

Figure 2: Concept of Numerical Fluxes ...... 20

Figure 3: Concept of the Master Element ...... 26

Figure 4: Vertical Discretization and Moisture Content ...... 42

Figure 5: Soil Depth vs. Moisture Content at Saturation ...... 53

Figure 6: Dry-down Time Series of Layer 2 ...... 53

Figure 7: Dry-down Times Series of All Layers ...... 54

Figure 8: Moisture Time Series of Pulse Inputs ...... 55

Figure 9: Rainfall Times Series of Layer 1 ...... 59

Figure 10: Rainfall Time Series of Layer 2 ...... 60

Figure 11: Rainfall Time Series of Layer 3 ...... 61

Figure 12: Rainfall Time Series of Layer 4 ...... 62

Figure 13: Rainfall Time Series of Layer 5...... 63

Figure 14: Rainfall Time Series of Layer 6 ...... 64

Figure 15: Rainfall Time Series of Layer 1 (with Evaporation) ...... 65

Figure 16: Point-to-point Comparison between Measurement and Simulation ...... 69

Figure 17: Comparison of R2 (without Evaporation) ...... 73

Figure 18: Comparison of Absolute Error (without Evaporation) ...... 74

x Figure 19: Comparison of R2 (with Evaporation) ...... 75

Figure 20: Comparison of Absolute Error (with Evaporation) ...... 76

Figure 21: Rain Barrel Configuration ...... 89

Chapter 1: Introduction

Soil moisture is fundamentally important in the exchange of water and energy

between the land and the atmosphere despite its relatively small proportion in the

hydrologic cycle. It is a critical state variable in runoff estimation, flood control,

irrigation, weather forecasting and many other environmental practices. Moreover, soil

moisture acts as a solvent and transporter of chemicals and nutrients in support of

chemical and biological activities in soil environments. Understanding soil moisture is

thus important in many engineering practices.

The temporal and spatial evolution of soil moisture content is attributed to multiple physical processes, one of which is flow in the vadose zone (or unsaturated

flow). Governed by gravity and capillary forces, unsaturated flow fundamentally adheres

to Darcy’ Law which indicates that flux of water in a porous medium is proportional to

hydraulic gradient by a factor related to characteristics of the porous medium 22. Based on

Darcy’s Law, L. A. Richards derived the famous Richards’ equation that describes the

relations among hydraulic conductivity, capillary pressure and gravity 51. The Richards’

equation has since been the state-of-art governing equation for unsaturated flow. In particular, the one-dimensional Richards’ equation is widely applicable because unsaturated flow is generally a vertically downward process due to the predominant

1 effect of gravity 52. As the intrinsic properties of water are presumably stable throughout

a particular soil environment unless otherwise specified, hydraulic properties of the

porous media are of primary concerns and consideration in soil moisture studies 18,51.

It remains a challenge to accurately solve the Richard’s equation or to well

represent unsaturated flow due to hysteresis in the relation between soil moisture content

and capillary pressure induced by multiphase flows (i.e. water and air) during wetting and

drying cycles. The combined effect of pore space irregularity, air entrapment,

shrinking/swelling of soil, etc. may result in distinct soil retention curves ( - relations)

even for the same studied soil, adding complexity to representing hydraulic𝜓𝜓 conductivity𝜃𝜃

as a function of soil moisture content and/or capillary pressure 19. Consequently, the

Richards’ equation is not readily solvable using basic techniques for partial differential equations and is therefore devoid of a general closed-form analytical solution. This sets the stage for development of numerical solutions of the Richards’ equation. Over the past decades, different numerical methods have been exploited and compared in an effort to optimize accuracy and computational effort 21,42; adaptive discretization schemes have

been added to account for natural complexity such as soil heterogeneity 29,57,58.

1.1 Motivation and Objective

While considerable progress has been made since the birth of Richards’ equation,

development of its numerical solvers continues. When soil moisture undergoes drastic

changes in event of, for example, extreme rainfall intensities or massive spill and where soil hydraulic properties alter significantly due to subsurface heterogeneity, finite element solvers that enforce continuity in solutions tend to significantly sacrifice computational 2 speed by reducing time step sizes to allow for convergence 35. Some finite difference

models may fail to fulfill the Courant-Friedrichs-Lewy condition should decent efficiency be maintained. Considering that an increase in occurrence of extreme rainfall events has

been implied in meteorological reports globally 27,69, having more adaptive and efficient

Richards’ equation solvers is imperative.

The discontinuous Galerkin finite element method (DG-FEM) is hereby

introduced. Its rich implementation includes solving problems in computational

structures, mechanics, energy transfer, as well as computational fluid mechanics. In contrast of conforming finite element methods, the discontinuous Galerkin finite element method allow spatial and/or temporal discontinuity in solutions, making it highly capable

of handling shock and dealing with complex geometry.

In light of the success of DG-FEM in countless other problems, the objective of this thesis is to present a DG-FEM solution of the one-dimensional Richards’

Equation. Rainfall experiment was designed to validate model accuracy; comparison between this model and a conforming continuous Galerkin model was made to evaluate computational speed and application. The remainder of this thesis is organized as follows:

. Chapter 2 reviews literatures on past developments on infiltration models;

. Chapter 3 outlines step-by-step the DG-FEM formulation procedures on a one-

dimensional Richards’ equation;

. Chapter 4 describes experiment design and model testing procedures;

. Chapter 5 discusses the performance of the model based on experiment results;

. Conclusion sums up the thesis and proposes potential future work.

3 . Appendix includes supplementary materials.

Chapter 2: Literature Review

This section reviews past development of infiltration and unsaturated-flow models. While the fundamentals of unsaturated flow in porous media are embedded in the

Richards’ equation, its application requires a reasonable amount of knowledge of the complex relations among its variables and parameters as well as modeling techniques.

For simplicity, infiltration models based on the Green-Ampt equation are occasionally used as alternatives. The Green-Ampt equation primarily differs from the Richards’ equation as the former estimates infiltration rate and volume dictated by a user-defined top soil layer (i.e. only the top boundary) while the latter describes changes in soil moisture or capillary pressure within a domain of interest over time (i.e. the entire soil column). It is plausible to incorporate a Green-Ampt model in a Richards’ equation solver to handle the latter’s top boundary condition.

2.1 The Green-Ampt Equation

Developed by W. H. Green and G. A. Ampt, the Green-Ampt equation was one of the earliest physically-based means to describe infiltration 20. In its application, the soil column is assumed to be homogeneous, of a uniform initial moisture profile, constantly

5 ponded on its surface, and saturated from its surface to a sharp wetting front, as illustrated in Fig. 1:

Figure 1: Scenario of the Green-Ampt Equation. is the initial soil moisture content, is soil moisture content at saturation. The surface is assumed to be always ponded as the sharp wetting front progresses vertically downward in the soil layer. 𝜃𝜃0 𝜃𝜃𝑠𝑠

The Green-Ampt equation is as follows 49:

= 1 + (1) 𝑛𝑛 𝜓𝜓𝑓𝑓 𝑓𝑓 𝐾𝐾 � � where 𝐹𝐹

infiltration rate [L T-1];

𝑓𝑓 cumulative depth of infiltration [L];

𝐹𝐹 effective porosity minus initial soil water content [L3 L-3];

𝑛𝑛 6 wetting front capillary pressure head [L];

𝑓𝑓 𝜓𝜓 hydraulic conductivity [L T-1];

Additionally,𝐾𝐾 in its integrated form,

ln 1 + = ,

𝑓𝑓 𝐹𝐹 𝐹𝐹 − 𝑛𝑛𝜓𝜓 � 𝑓𝑓� 𝐾𝐾𝐾𝐾 where is time. 𝑛𝑛𝜓𝜓

𝑡𝑡A list of Green-Ampt parameters ranging from sandy to clayey soils was

compiled by Rawls et al. 49. Although these parameters are physically based (i.e.

measurable), the Green-Ampt equation is not sophisticated enough to adequately

represent complicated soil environments (e.g. soil heterogeneity, multi-phase flows) and

infiltration characteristics (e.g. the relations among rainfall rate, infiltration capacity and

hydraulic conductivity) due to the assumptions it bases on. In addition, the Green-Ampt

equation considers only a user-defined top soil layer: all rain water would be “pushed”

into the soil layer if the rainfall rate were smaller than soil hydraulic conductivity,

whereas runoff would occur should infiltration capacity be smaller than the rainfall rate.

In other words, even if a same set of parameters were used, runoff partitioning could vary

greatly based on how this “top soil layer” is defined – a deeper “top layer” leads to a

larger infiltration capacity and likely a lower runoff volume; conversely, a shallower “top

layer” likely results in a larger runoff volume if there is any.

In an effort to reconcile the Green-Ampt equation with the Richards’ Equation,

Aggelides and Youngs deduced multiples sets of pressure head values based on

infiltration experiments in a soil column under different antecedent conditions 1. After comparing their results with those estimated by a Richards’ equation solver, Aggelides

7 and Youngs discovered that for all antecedent conditions, equating to the air-entry

𝑓𝑓 pressure value ( ) would return values within a reasonable range.𝜓𝜓 Although this

𝑏𝑏 reflects the easiness𝜓𝜓 in implementing𝐹𝐹 the Green-Ampt equation, accuracy of results could

be jeopardized. For example, Charbeneau and Asgian studied a hydrologic model in which the Green-Ampt equation was incorporated to estimate the potential infiltration rate 10. Their profiles of average soil water content vs. soil depth showed close agreement

with those of a presumably accurate model except near the soil surface (depth < 2 m)

where noticeably discrepancies in plots were visible. van Mullem estimated runoff

volumes and peak discharges of 99 storm events in 12 watersheds using the Green-Ampt

equation, which eventually underestimated runoff in almost all events 67.

Nevertheless, the Green-Ampt equation can still be effectively applied with restrictions and modifications. An explicit form of the Green-Ampt equation was developed by Salvucci and Entekhabi with modification on ponding time and successfully implemented in some models because of its straightforwardness and accuracy as compared to the basic integrated form 47,55. Philip examined the effects of

variable ponding depth on infiltration rate with application of the Green-Ampt equation only up to the instant when the asymptotely-decreasing infiltration capacity became smaller than rainfall rate (i.e. when all ponded water infiltrated) without considering runoff 44. King et al. simulated runoff using the SCS Curve Number method (an empirical method that relates runoff to land use and soil types; not physically based) and the Green-

Ampt Mein-Larson method which determines infiltration by the Green-Ampt model and

calculates ponding time by that of Mein and Larson 13,30,38,65. Comparison between the two showed that the Green-Ampt Mein-Larson method returned consistently more

8 accurate results than that of the Curve Number method in the particular area of study.

Additionally, King et al. highlighted that physically based methods like Green-Ampt

Mein-Larson would be more advantageous than entirely empirical models should rainfall

intensity and duration be considered. Some other attempts made to overcome the deficit

of the basic Green-Ampt equation include using a surface-condition indicator for periods without surface ponding during unsteady rainfall events 61, relating the hydraulic conductivity to soil depth with an exponential relationship for vertically non-uniform soils, applying an explicit redistribution scheme to predict soil surface water content in events of multiple drying and ponding phases, and adding trigonometric terms to represent hill slopes 4,11,41.

2.2 The Richards’ Equation

The Richards’ Equation (RE) is the state-of-art equation that describes flow in the unsaturated zone induced by capillary and gravitational forces 51:

+ + = (2) 2 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝐾𝐾∇ 𝜓𝜓 ∇𝐾𝐾 ⋅ ∇𝜓𝜓 𝑔𝑔 −𝜌𝜌𝜌𝜌 where 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

hydraulic conductivity [L T-1];

𝐾𝐾 capillary pressure head [L];

𝜓𝜓 gravitational acceleration [L T-2];

𝑔𝑔 depth of porous medium [L];

𝑧𝑧 density of fluid [M L-3].

𝜌𝜌 rate of change of the fluid content with respect to [M L-1]

𝐴𝐴 𝜓𝜓 9 time [T]

Based on Darcy’s𝑡𝑡 law and 1D scalar conservation law, several conventional forms of the

1D RE are derived:

The moisture-based ( -based) form:

𝜃𝜃 = ( ) ( ) (3)

𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕𝜕𝜕 �𝐷𝐷 𝜃𝜃 − 𝐾𝐾 𝜃𝜃 � The head-based ( -based)𝜕𝜕𝜕𝜕 form:𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

𝜓𝜓 = = ( ) 1 (4) 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕𝜕𝜕 � � 𝐶𝐶 �𝐾𝐾 𝜓𝜓 � − �� The mixed form: 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

= ( ) 1 (5) 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕𝜕𝜕 �𝐾𝐾 𝜓𝜓 � − �� where 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

hydraulic conductivity [L T-1];

𝐾𝐾 diffusivity [L2 T-1];

𝐷𝐷 volumetric soil moisture content [L3 L-3]

𝜃𝜃 capillary pressure head [L]

𝜓𝜓 soil depth [L];

𝑧𝑧 time [T].

As a side note,𝑡𝑡 the “ ” sign in 1D RE sometimes appears as “+”. This is solely dependent on which vertical− orientation is chosen to be positive by the user. Also, occasionally in literatures and research articles the English alphabet “ ” is used to represent pressure head. In this thesis, instead the Greek letter “ ” is used,ℎ so as to distinguish pressure head ( ) from hydraulic head ( ) as is the convention𝜓𝜓 in many fluid

𝜓𝜓 10 ℎ mechanics and groundwater hydraulics literatures 17. In fact, hydraulic head is the sum of

pressure head and elevation head.

The choice of which form to implement in numerical models deserves some

contemplation because, interestingly, Eq. (3), (4) and (5) are not entirely equivalent to

one another. The issue lies in the hysteretic relations among , and as was hypothesized by L. A. Richards in his original work and supported𝐾𝐾 𝜓𝜓by later𝜃𝜃 findings

23,24,51. It has been argued that the head-based RE would be more desirable in numerical

modeling as capillary pressure head stays continuous regardless of soil heterogeneities or

changes in moisture content gradient 18,21,40. In addition, a disadvantage of the -based

form can be illustrated from a mathematical standpoint: when diffusivity is formulated𝜃𝜃 as

= 𝑑𝑑𝑑𝑑 𝐷𝐷 𝐾𝐾 under the assumption that the - relationship 𝑑𝑑is𝑑𝑑 unique, if pressure head exceeds the air- entry pressure near saturation 𝜓𝜓(or𝜃𝜃 when ponding with a certain free water depth occurs),

would numerically approach infinity as approach zero 21,47. Conversely, however, it

𝐷𝐷was also demonstrated that the head-based𝑑𝑑 𝑑𝑑RE could result in more mass balance error

during time stepping, for is a function of and thus should vary in time and space with

9,40. Although this can 𝐶𝐶be resolved by applying𝜓𝜓 the mixed form which conserves mass

𝜓𝜓in time while maintaining spatial continuity in , some less advanced models and solvers

may not be able to handle two state variables together𝜓𝜓 46,64. Measuring capillary pressure

also tends to be more difficult than taking moisture readings 37. The choice of form may

very well be situational.

11 2.2.1 Solution of the Richards’ Equation – Analytical

The Richards’ equation (RE) is highly non-linear and does not have a general

closed-form analytical solution. To give a glimpse of its complexity, as follows is the

famous van Genuchten soil-water retention model 66:

2 1 𝑚𝑚 = 𝑙𝑙 1 1 , 𝑚𝑚 𝜃𝜃 − 𝜃𝜃𝑟𝑟 𝜃𝜃 − 𝜃𝜃𝑟𝑟 𝐾𝐾 𝐾𝐾𝑠𝑠 ⋅ � � ⋅ � − � − � � � � 𝜃𝜃𝑠𝑠 − 𝜃𝜃𝑟𝑟 𝜃𝜃𝑠𝑠 − 𝜃𝜃𝑟𝑟 ( ) = + , (1 + | | ) 𝜃𝜃𝑠𝑠 − 𝜃𝜃𝑟𝑟 𝜃𝜃 𝜓𝜓 𝜃𝜃𝑟𝑟 𝑛𝑛 𝑚𝑚 where , , and are soil-dependent parameters.𝛼𝛼𝛼𝛼 Inevitably, mathematical

modifications𝑙𝑙 𝑚𝑚 would𝑛𝑛 be 𝛼𝛼necessary to obtain closed-form solutions. One of the widely

implemented methods in linearization of variables in RE is the Kirchhoff transform

2,21,53,54. When performed on in 1D RE, the Kirchhoff Transform results in the following: 𝐾𝐾

( ) = 𝜃𝜃 ( ) or ( ) = 𝜓𝜓 ( )

𝑈𝑈 𝜃𝜃 � 𝐾𝐾 𝜃𝜃 𝑑𝑑𝑑𝑑 𝑈𝑈 𝜓𝜓 � 𝐾𝐾 𝜓𝜓 𝑑𝑑𝑑𝑑 𝜃𝜃0 𝜓𝜓0 where and are arbitrary lower limits and is an auxiliary variable that replaces the

0 0 non-linear𝜃𝜃 , so𝜓𝜓 that the second derivative term𝑈𝑈 (i.e. diffusivity) on the right-hand side of

RE is linearized.𝐾𝐾 Taking the head-based RE as an example, the transformed RE would be

= 2 𝜕𝜕𝜕𝜕 𝜕𝜕 𝑈𝑈 𝜕𝜕𝜕𝜕 𝐹𝐹 2 − 𝐺𝐺 where 𝜕𝜕𝜕𝜕 𝜕𝜕𝑧𝑧 𝜕𝜕𝜕𝜕

( ) 1 = and = ( ) ( ) 𝐶𝐶 𝜓𝜓 𝑑𝑑𝑑𝑑 𝐹𝐹 𝐺𝐺 ⋅ 𝐾𝐾 𝜓𝜓 𝐾𝐾 𝜓𝜓 𝑑𝑑𝑑𝑑 12 Other linearization methods include the re-formulation of diffusivity by Knight and

Philip in the one-dimensional non-linear diffusion equation,

= ( ) 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕𝜕𝜕 �𝑫𝑫 𝜽𝜽 � which produced 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

( ) = ( ) (6) −2 and was claimed to hold for non-hysteretic𝐷𝐷 𝜃𝜃 𝑎𝑎water𝑏𝑏 − movement𝜃𝜃 in unsaturated soil 31. Eq. (6)

was adopted by Broadbridge and White in deriving an empirical expression of ( ), and

subsequently in developing an infiltration model for constant-rate rainfall 6𝐾𝐾. The𝜃𝜃 test

results of Broadbridge and White showed that for certain soils their model could

accurately predict the soil moisture profiles, but how generalizable their model is remains

untested 6,70. Following the method of Broadbridge and White, Warrick et al. derived an

analytical solution of the RE for deep drainage under a constant top-boundary flux via a

series of heavy algebraic transformation 68. Despite that a closed-form solution was obtained, it could be difficult to reproduce the method of Warrick et al. as the algebraic transformation involved may not be applicable to all formulations of diffusivity and hydraulic conductivity. Alternatively, instead of the complex van Genuchten model,

Tracy linearized the mixed-form RE using the simpler Gardner’s exponential model,

( ) = exp( ),

𝑠𝑠 𝐺𝐺 via Fourier transformation based𝐾𝐾 𝜓𝜓on specific𝐾𝐾 ⋅ boundary𝛼𝛼 𝜓𝜓 conditions, resulting in clean

analytical solutions 63. It should be noted, however, that the goal of Tracy was to test performance levels of models rather than solving a real problem. Specifications made by

Tracy were leaning toward mathematically simplifying the derivation processes.

13 2.2.2 Solution of the Richards’ Equation – Numerical

Numerical methods commonly applied on RE include finite difference methods

(FDMs), finite volume methods (FVMs) and finite element methods (FEMs) in

conjunction with time-stepping schemes such as the Euler methods 9,28,39,40,58. As error is

inevitably introduced due to truncation of series (e.g. Taylor, Fourier), fulfillment of

convergence criteria (e.g. user-defined acceptable increment in degrees of freedom over

time), etc., the development of numerical models has always been aiming at striking a

balance between accuracy and computational effort.

Efficiency of different time-stepping strategies has long been studied. Caviedes-

Voullième et al. evaluated the performance of a finite difference method applied to the mixed form and head-based form of RE using both the forward (explicit) and the backward (implicit) Euler’s methods 8. Caviedes-Voullième et al. found that the explicit time-stepping scheme could result in discontinuity near saturation for both forms of RE, which was primarily due to that approaches zero toward saturation; the explicit

scheme required less computation per𝑑𝑑𝑑𝑑 time step but experienced more restriction on time

step size than the implicit and was overall less efficient. In retrospect, Choi and Chung

mentioned that a constant time-step size could result in non-linear distribution of

acceleration in time and thereby discretization error, revealing the importance of time step variability 12. Some adaptive time-stepping schemes include setting an allowance

range for the absolute relative error associated with the next time step, computing a safety

factor per time step and vary the sizes of the upcoming steps accordingly, and simply

multiplying the time step size by a user-imposed factor within an error constraint 28,29.

14 On the other hand, computational effort within each time step is determined by spatial discretization. Considerations need to be made based on the nature of the problem.

For example, in light of stability issue arisen from sharp wetting front where the value of

/ escalates, Solin and Kuraz solved the RE with -FEM which is adaptive in both

𝜕𝜕space𝜕𝜕 𝜕𝜕𝜕𝜕 ( -) and the degrees of polynomial basis ( -) 62ℎ. 𝑝𝑝Their error analysis showed that for a sameℎ number of degrees of freedom and simulation𝑝𝑝 period, -FEM could achieve an error several orders of magnitude smaller than that of -FEMs.ℎ𝑝𝑝 Adaptivity of mesh was also explored in the work of Kuraz et al., in which an ℎalgorithm is used to augment neighboring mesh clusters where the gradient of hydraulic conductivity across is small to form coarser meshes in certain parts of the domain 35. This allows more computational resources to be assigned to where drastic changes occur with finer meshes, thus balancing overall efficiency.

2.3 Discontinuous Galerkin Finite Element Method

The application of Discontinuous Galerkin (DG) methods traces back to the work of Reed and Hill in which a solution for the hyperbolic neutron transport equation is presented 50. Since then, DG methods have been actively implemented in solving ordinary differential equations and time-dependent partial differential equations 15. In general, DG methods are a class of finite element methods (FEMs) utilizing piecewise- continuous polynomials to construct approximate solutions. Discontinuity in solutions across element boundaries and in time is allowed, enabling data communications to be local in subdomains and within time steps. As a result, DG-FEMs are advantageous at handling complex geometry and advection-dominated problems where shocks and 15 sudden shifts in natural conditions prevail. For more details on DG-FEM applications, please refer to the work of Cockburn and Shu and their references 15.

Chapter 3: Discontinuous Galerkin Formulation

Outlined in this chapter are steps of DG formulation on the -based 1D RE.

Spatial and temporal discretization of the problem is explained, followed𝜃𝜃 by specification of the initial and boundary conditions.

To be specific, the DG method presented in this thesis is the Local Discontinuous

Galerkin (LDG) method of Cockburn and Shu which allows spatial discontinuity with an implicit time scheme 15. Similar to Cockburn and Shu’s method, an auxiliary variable “ ” is introduced to rewrite the -based RE as a system of first-order differential equations:𝑞𝑞

𝜃𝜃 + = (7) 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 (8) = 𝜕𝜕𝜕𝜕 𝑞𝑞 𝐷𝐷 Formulation will be performed first on Eq. (7) and𝜕𝜕𝜕𝜕 then on Eq. (8).

3.1 Spatial Discretization

3.1.1 Weak Formulation

A “weak” form of the original partial differential equation is first obtained to reduce strong requirement on smoothness of solutions. The domain of interest ( ) is partitioned into subdomains or “elements” ( , = 1,2, … , ). In this chapter, theΩ -th

17Ω 𝑖𝑖 𝑖𝑖 𝑛𝑛 𝑗𝑗 element is used as a generic example. Multiply both sides of Eq. (7) by an arbitrary,

smooth test function, , and integrate on both sides over the -th element ( , between

𝑗𝑗 and ): 𝑣𝑣 𝑗𝑗 Ω

𝑧𝑧𝑗𝑗−1 𝑧𝑧𝑗𝑗 + = (9)

𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 j 𝜕𝜕𝜕𝜕 j �Ωj � � 𝑣𝑣 𝑑𝑑Ω �Ωj � 𝑣𝑣� 𝑑𝑑Ω 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 (10) + = 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 � � 𝑣𝑣� 𝑑𝑑Ωj � � 𝑣𝑣� 𝑑𝑑Ωj � � 𝑣𝑣� 𝑑𝑑Ωj Ωj 𝜕𝜕𝜕𝜕 Ωj 𝜕𝜕𝜕𝜕 Ωj 𝜕𝜕𝜕𝜕 Next, apply the formula of integration by parts on spatial terms ( -derivatives):

𝑧𝑧 + = (11) 𝑧𝑧𝑗𝑗 𝑧𝑧𝑗𝑗 𝜕𝜕𝜕𝜕 j 𝑧𝑧𝑗𝑗−1 𝜕𝜕𝜕𝜕 j 𝑧𝑧𝑗𝑗−1 𝜕𝜕𝜕𝜕 j �Ωj � 𝑣𝑣� 𝑑𝑑Ω �𝐾𝐾𝐾𝐾│ − �Ωj �𝐾𝐾 � 𝑑𝑑Ω � �𝑞𝑞𝑞𝑞│ − �Ωj �𝑞𝑞 � 𝑑𝑑Ω � Requirement𝜕𝜕𝜕𝜕 on smoothness of the original𝜕𝜕𝜕𝜕 partial differential equation𝜕𝜕𝜕𝜕 is now

“weakened” as the partial derivative with respect to have been transferred from and to the test function, . 𝑧𝑧 𝐾𝐾 𝑞𝑞

Next, Eq. (11)𝑣𝑣 is modified by adding an “ ” subscript to and . Terms with an

“ ” subscript are approximations belonging to polynomialℎ spaces𝜃𝜃 of degree𝑣𝑣 (denoted

byℎ ). More details on polynomial spaces will be given in Chapter 3.1.3. For𝑝𝑝 now, the 𝑝𝑝 problemℙ becomes finding approximate solutions for , such that 𝑝𝑝 𝜃𝜃ℎ 𝑣𝑣ℎ ∈ ℙ + ( ) ( ) 𝑗𝑗 𝜕𝜕𝜃𝜃ℎ 𝑧𝑧 𝜕𝜕𝑣𝑣ℎ � � 𝑣𝑣ℎ� 𝑑𝑑Ωj �𝐾𝐾 𝜃𝜃 𝑣𝑣│𝑧𝑧𝑗𝑗−1 − � �𝐾𝐾 𝜃𝜃ℎ � 𝑑𝑑Ωj� (12) Ωj 𝜕𝜕𝜕𝜕 Ωj 𝜕𝜕𝜕𝜕 = ( ) ( ) 𝑗𝑗 𝑧𝑧 𝜕𝜕𝑣𝑣ℎ �𝑞𝑞 𝜃𝜃 𝑣𝑣│𝑧𝑧𝑗𝑗−1 − � �𝑞𝑞 𝜃𝜃ℎ � 𝑑𝑑Ωj� Ωj 𝜕𝜕𝜕𝜕 , 𝑝𝑝 ℎ j To simplify representation, the following∀𝑣𝑣 ∈ ℙ terms ∀areΩ introduced:∈ Ω

18 ( , ) = (13) 𝜕𝜕𝜃𝜃ℎ 𝑀𝑀𝑗𝑗 𝜃𝜃ℎ 𝑣𝑣ℎ � � 𝑣𝑣ℎ� 𝑑𝑑Ωj Ωj 𝜕𝜕𝜕𝜕 ( ), = ( ) (14) 𝑧𝑧𝑗𝑗 ⟨𝐾𝐾 𝜃𝜃ℎ 𝑣𝑣ℎ⟩𝑗𝑗 𝐾𝐾 𝜃𝜃 𝑣𝑣│𝑧𝑧𝑗𝑗−1 ( ), = ( ) (15) 𝑧𝑧𝑗𝑗 ⟨𝑞𝑞 𝜃𝜃ℎ 𝑣𝑣ℎ⟩𝑗𝑗 𝑞𝑞 𝜃𝜃 𝑣𝑣│𝑧𝑧𝑗𝑗−1 ( , ) = ( ) (16) ′ 𝜕𝜕𝑣𝑣ℎ 𝐹𝐹𝑗𝑗 𝜃𝜃ℎ 𝑣𝑣ℎ � �𝐾𝐾 𝜃𝜃ℎ � 𝑑𝑑Ωj Ωj 𝜕𝜕𝜕𝜕 ( , ) = ( ) (17) ′ ℎ 𝑗𝑗 ℎ ℎ ℎ 𝜕𝜕𝑣𝑣 j 𝑄𝑄 𝜃𝜃 𝑣𝑣 �Ωj �𝑞𝑞 𝜃𝜃 � 𝑑𝑑Ω Eq. (12) is therefore equivalent to 𝜕𝜕𝜕𝜕

( , ) + ( ), ( , ) = ( ), ( , ) (18) ′ ′ 𝑗𝑗 ℎ ℎ ℎ ℎ 𝑗𝑗 𝑗𝑗 ℎ ℎ ℎ ℎ 𝑗𝑗 𝑗𝑗 ℎ ℎ in which 𝑀𝑀( 𝜃𝜃 , 𝑣𝑣 ) is linear⟨𝐾𝐾 𝜃𝜃 in 𝑣𝑣both⟩ − 𝐹𝐹 and𝜃𝜃 𝑣𝑣 whereas⟨𝑞𝑞 𝜃𝜃 the𝑣𝑣 remaining⟩ − 𝑄𝑄 𝜃𝜃 terms𝑣𝑣 are linear

𝑗𝑗 ℎ ℎ ℎ ℎ only in 𝑀𝑀 . 𝜃𝜃Eq.𝑣𝑣 (18) represents the𝜃𝜃 -th 𝑣𝑣element equation that requires further

ℎ transformation𝑣𝑣 to be numerically solved. The𝑗𝑗 next step is to represent the boundary terms,

( ), and ( ), , with numerical fluxes.

ℎ ℎ 𝑗𝑗 ℎ ℎ 𝑗𝑗 ⟨𝐾𝐾 𝜃𝜃 𝑣𝑣 ⟩ ⟨𝑞𝑞 𝜃𝜃 𝑣𝑣 ⟩

3.1.2 Numerical Flux

As discontinuity in solution across element boundaries is allowed in DG methods,

the boundary terms, ( ), and ( ), , may be dual-valued due to different

ℎ ℎ 𝑗𝑗 ℎ ℎ 𝑗𝑗 left and right limits of⟨𝐾𝐾 𝜃𝜃 and𝑣𝑣 ⟩ across⟨𝑞𝑞 adjacent𝜃𝜃 𝑣𝑣 ⟩ elements. This concept is illustrated in

ℎ ℎ Fig. 2 and is what distinguishes𝜃𝜃 𝑣𝑣 DG-FEMs from conforming FEMs.

Figure 2: Concept of Numerical Fluxes. A generic flux is represented by ; degrees of freedom are represented by ; is the domain of interest 32,33,34. 𝒇𝒇 𝒖𝒖𝒉𝒉 𝒙𝒙

At element boundaries, and will be replaced by single-valued numerical fluxes

denoted by and . The𝐾𝐾 numerical𝑞𝑞 fluxes are generally dependent on the left and right

limits of 𝐾𝐾at� the 𝑞𝑞-�th node, i.e.

ℎ 𝜃𝜃 𝑗𝑗 , , , . − + − + �𝑗𝑗 � 𝑗𝑗 𝑗𝑗 𝑗𝑗 𝑗𝑗 𝑗𝑗 The boundary terms of𝐾𝐾 the≡ 𝐾𝐾-�th𝜃𝜃 element𝜃𝜃 � equation can𝑞𝑞� now≡ 𝑞𝑞 �be�𝜃𝜃 expanded:𝜃𝜃 �

( 𝑗𝑗), . + − ℎ ℎ 𝑗𝑗 �𝑗𝑗−1 ℎ 𝑗𝑗−1 �𝑗𝑗 ℎ 𝑗𝑗 ⟨𝐾𝐾(𝜃𝜃 ), 𝑣𝑣 ⟩ ≡ 𝐾𝐾 ⋅ 𝑣𝑣 �𝑧𝑧 � − 𝐾𝐾 ⋅ 𝑣𝑣 �𝑧𝑧 � + − ℎ ℎ 𝑗𝑗 𝑗𝑗−1 ℎ 𝑗𝑗−1 𝑗𝑗 ℎ 𝑗𝑗 And the element equation⟨𝑞𝑞 𝜃𝜃 is rewritten𝑣𝑣 ⟩ ≡ 𝑞𝑞�as: ⋅ 𝑣𝑣 �𝑧𝑧 � − 𝑞𝑞� ⋅ 𝑣𝑣 �𝑧𝑧 �

( , ) + ( , ) + − ′ (19) 𝑗𝑗 ℎ ℎ �𝑗𝑗−1 ℎ 𝑗𝑗−1 �𝑗𝑗 ℎ 𝑗𝑗 𝑗𝑗 ℎ ℎ 𝑀𝑀 𝜃𝜃 𝑣𝑣 �𝐾𝐾= ⋅ 𝑣𝑣 �𝑧𝑧 � − 𝐾𝐾 ⋅ 𝑣𝑣 �𝑧𝑧 �� − 𝐹𝐹 𝜃𝜃 (𝑣𝑣 , ). + − ′ 𝑗𝑗−1 ℎ 𝑗𝑗−1 𝑗𝑗 ℎ 𝑗𝑗 𝑗𝑗 ℎ ℎ Comprehensive descriptions of� 𝑞𝑞�different⋅ 𝑣𝑣 � types𝑧𝑧 � of− 𝑞𝑞numerical� ⋅ 𝑣𝑣 �𝑧𝑧 �fluxes� − 𝑄𝑄 are𝜃𝜃 available𝑣𝑣 3,45,59. In

this model, the Lax-Friedrichs flux is selected for advection and the Bassi-Rebay flux for diffusivity.

Table 1: Numerical Fluxes

Expression Notation

Lax-Friedrichs: 1 : Left limit of solution at the -th = + 2 − node − + − + 𝜃𝜃𝑗𝑗 𝑗𝑗 𝐾𝐾�𝑗𝑗 �𝐾𝐾�𝜃𝜃𝑗𝑗 � 𝐾𝐾�𝜃𝜃𝑗𝑗 � − 𝛼𝛼 ⋅ �𝜃𝜃𝑗𝑗 − 𝜃𝜃𝑗𝑗 �� : Right limit of solution at the - + th node Bassi-Rebay: 𝜃𝜃𝑗𝑗 𝑗𝑗 1 : Maximum propagation speed = + 2 − + 𝛼𝛼 𝑞𝑞�𝑗𝑗 �𝑞𝑞�𝜃𝜃𝑗𝑗 � 𝑞𝑞�𝜃𝜃𝑗𝑗 ��

3.1.3 Basis Polynomials

The approximate solution over each element is constructed by basis polynomials independent of time and of degree 0 or higher. Let denote a set of basis polynomials in the space of as a column vector, and let denote𝚽𝚽 a column vector of arbitrary scalar

ℎ coefficients: 𝑣𝑣 𝜶𝜶

= [ , , , ] (20) T 1 2 𝑛𝑛 𝚽𝚽 = [𝜙𝜙 , 𝜙𝜙 , ⋯, 𝜙𝜙 ] (21) T 1 2 𝑛𝑛 Thus can be expressed as a dot 𝜶𝜶product𝛼𝛼 of𝛼𝛼 the⋯ above𝛼𝛼 vectors:

𝑣𝑣ℎ

= = [ , , , ] 𝜙𝜙1 = 𝑛𝑛 (22) T 𝜙𝜙2 𝑣𝑣ℎ 𝜶𝜶 𝚽𝚽 𝛼𝛼1 𝛼𝛼2 ⋯ 𝛼𝛼𝑛𝑛 � � � 𝑎𝑎𝑖𝑖 ⋅ 𝜙𝜙𝑖𝑖 ⋮ 𝑖𝑖=1 𝑛𝑛 Similarly, can be written as a dot product. Instead𝜙𝜙 of , a vector of (denoted as )

ℎ will be the 𝜃𝜃coefficients: 𝜶𝜶 𝜃𝜃 𝚯𝚯

= = + + + (23) 𝐓𝐓 𝜃𝜃ℎ 𝚯𝚯 𝚽𝚽 𝜃𝜃1𝜙𝜙1 𝜃𝜃2𝜙𝜙2 ⋯ 𝜃𝜃𝑛𝑛𝜙𝜙𝑛𝑛 21 Taking the first derivative on both sides of Eq. (23) with respect to time yields

= ( + + + ) 𝜕𝜕𝜃𝜃ℎ 𝜕𝜕 𝜃𝜃1𝜙𝜙1 𝜃𝜃2𝜙𝜙2 ⋯ 𝜃𝜃𝑛𝑛𝜙𝜙𝑛𝑛 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 = + + + (24) 𝜕𝜕𝜃𝜃1 𝜕𝜕𝜃𝜃2 𝜕𝜕𝜃𝜃𝑛𝑛 𝜙𝜙1 𝜙𝜙2 ⋯ 𝜙𝜙𝑛𝑛 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 = 𝑛𝑛 𝜕𝜕𝜕𝜕𝑖𝑖 � ⋅ 𝜙𝜙𝑖𝑖 𝑖𝑖=1 𝜕𝜕𝜕𝜕 Eq. (22) and (24) can now be substitute into the element equation. The linearity of all terms in Eq. (19) with respect to allows the scalar coefficients to be separated and

ℎ placed in their fronts, i.e.: 𝑣𝑣

( , ) = ( , ) = 𝑛𝑛 ( , ) T 𝑀𝑀𝑗𝑗 𝜃𝜃ℎ 𝑣𝑣ℎ 𝑀𝑀𝑗𝑗 𝜃𝜃ℎ 𝜶𝜶 𝚽𝚽 � 𝛼𝛼𝑖𝑖 ⋅ 𝑀𝑀𝑗𝑗 𝜃𝜃ℎ 𝜙𝜙𝑖𝑖 𝑖𝑖=1

LHS: ( ), = ( ), = 𝑛𝑛 T + − ⟨𝐾𝐾 𝜃𝜃ℎ 𝑣𝑣ℎ⟩𝑗𝑗 ⟨𝐾𝐾 𝜃𝜃ℎ 𝜶𝜶 𝚽𝚽⟩𝑗𝑗 � 𝛼𝛼𝑖𝑖 ⋅ �𝐾𝐾�𝑗𝑗−1 ⋅ 𝜙𝜙𝑖𝑖�𝑧𝑧𝑗𝑗−1� − 𝐾𝐾�𝑗𝑗 ⋅ 𝜙𝜙𝑖𝑖�𝑧𝑧𝑗𝑗 �� 𝑖𝑖=1

( , ) = ( , ) = 𝑛𝑛 ( , ) ′ T ′ ′ 𝐹𝐹𝑗𝑗 𝜃𝜃ℎ 𝑣𝑣ℎ 𝐹𝐹𝑗𝑗 𝜃𝜃ℎ 𝜶𝜶 𝚽𝚽 � 𝛼𝛼𝑖𝑖 ⋅ 𝐹𝐹𝑗𝑗 𝜃𝜃ℎ 𝜙𝜙𝑖𝑖 𝑖𝑖=1

( ), = ( ), = 𝑛𝑛 T + − RHS: ⟨𝑞𝑞 𝜃𝜃ℎ 𝑣𝑣ℎ⟩𝑗𝑗 ⟨𝑞𝑞 𝜃𝜃ℎ 𝜶𝜶 𝚽𝚽⟩𝑗𝑗 � 𝛼𝛼𝑖𝑖 ⋅ �𝑞𝑞�𝑗𝑗−1 ⋅ 𝜙𝜙𝑖𝑖�𝑧𝑧𝑗𝑗−1� − 𝑞𝑞�𝑗𝑗 ⋅ 𝜙𝜙𝑖𝑖�𝑧𝑧𝑗𝑗 �� 𝑖𝑖=1

( , ) = ( , ) = 𝑛𝑛 ( , ) ′ T ′ ′ 𝑄𝑄𝑗𝑗 𝜃𝜃ℎ 𝑣𝑣ℎ 𝑄𝑄𝑗𝑗 𝜃𝜃ℎ 𝜶𝜶 𝚽𝚽 � 𝛼𝛼𝑖𝑖 ⋅ 𝑄𝑄𝑗𝑗 𝜃𝜃ℎ 𝜙𝜙𝑖𝑖 𝑖𝑖=1

22 In order for LHS and RHS to be equal for any given , the following relations should hold:

LHS 𝜶𝜶 RHS

: ( , ) + ( , ) = ( , ) + − ′ + − ′ 1 𝑗𝑗 ℎ 1 �𝑗𝑗−1 1 𝑗𝑗−1 �𝑗𝑗 1 𝑗𝑗 𝑗𝑗 ℎ 1 𝑗𝑗−1 1 𝑗𝑗−1 𝑗𝑗 1 𝑗𝑗 𝑗𝑗 ℎ 1 𝛼𝛼 : 𝑀𝑀 (𝜃𝜃 , 𝜙𝜙 ) + �𝐾𝐾 ⋅ 𝜙𝜙 �𝑧𝑧 � − 𝐾𝐾 ⋅ 𝜙𝜙 �𝑧𝑧 �� − 𝐹𝐹 (𝜃𝜃 , 𝜙𝜙 ) = �𝑞𝑞� ⋅ 𝜙𝜙 �𝑧𝑧 � − 𝑞𝑞� ⋅ 𝜙𝜙 �𝑧𝑧 �� − 𝑄𝑄 (𝜃𝜃 ,𝜙𝜙 ) + − ′ + − ′ 2 𝑗𝑗 ℎ 2 �𝑗𝑗−1 2 𝑗𝑗−1 �𝑗𝑗 2 𝑗𝑗 𝑗𝑗 ℎ 2 𝑗𝑗−1 2 𝑗𝑗−1 𝑗𝑗 2 𝑗𝑗 𝑗𝑗 ℎ 2 𝛼𝛼 𝑀𝑀 𝜃𝜃 𝜙𝜙 �𝐾𝐾 ⋅ 𝜙𝜙 �𝑧𝑧 � − 𝐾𝐾 ⋅ 𝜙𝜙 �𝑧𝑧 �� − 𝐹𝐹 𝜃𝜃 𝜙𝜙 �𝑞𝑞� ⋅ 𝜙𝜙 �𝑧𝑧 � − 𝑞𝑞� ⋅ 𝜙𝜙 �𝑧𝑧 �� − 𝑄𝑄 𝜃𝜃 𝜙𝜙

⋮ ( , ) + ( ⋮, ) = ( , ) + − ′ + − ′ 𝑛𝑛 𝑗𝑗 ℎ 𝑛𝑛 �𝑗𝑗−1 𝑛𝑛 𝑗𝑗−1 �𝑗𝑗 𝑛𝑛 𝑗𝑗 𝑗𝑗 ℎ 1 𝑗𝑗−1 𝑛𝑛 𝑗𝑗−1 𝑗𝑗 𝑛𝑛 𝑗𝑗 𝑗𝑗 ℎ 𝑛𝑛 𝛼𝛼 𝑀𝑀 𝜃𝜃 𝜙𝜙 �𝐾𝐾 ⋅ 𝜙𝜙 �𝑧𝑧 � − 𝐾𝐾 ⋅ 𝜙𝜙 �𝑧𝑧 �� − 𝐹𝐹 𝜃𝜃 𝜙𝜙 �𝑞𝑞� ⋅ 𝜙𝜙 �𝑧𝑧 � − 𝑞𝑞� ⋅ 𝜙𝜙 �𝑧𝑧 �� − 𝑄𝑄 𝜃𝜃 𝜙𝜙

23 Recall that is also linear in . Substituting Eq. (24) into the LHS leads to further transformation:

𝑀𝑀𝑗𝑗 𝜃𝜃ℎ : ( , ) = ( , ) + ( , ) + + ( , ) 𝜕𝜕𝜃𝜃1 𝜕𝜕𝜃𝜃2 𝜕𝜕𝜃𝜃𝑛𝑛 𝛼𝛼1 𝑀𝑀𝑗𝑗 𝜃𝜃ℎ 𝜙𝜙1 𝑀𝑀𝑗𝑗 𝜙𝜙1 𝜙𝜙1 𝑀𝑀𝑗𝑗 𝜙𝜙2 𝜙𝜙1 ⋯ 𝑀𝑀𝑗𝑗 𝜙𝜙𝑛𝑛 𝜙𝜙1 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 : ( , ) = ( , ) + ( , ) + + ( , ) 𝜕𝜕𝜃𝜃1 𝜕𝜕𝜃𝜃2 𝜕𝜕𝜃𝜃𝑛𝑛 𝛼𝛼2 𝑀𝑀𝑗𝑗 𝜃𝜃ℎ 𝜙𝜙2 𝑀𝑀𝑗𝑗 𝜙𝜙1 𝜙𝜙2 𝑀𝑀𝑗𝑗 𝜙𝜙2 𝜙𝜙2 ⋯ 𝑀𝑀𝑗𝑗 𝜙𝜙𝑛𝑛 𝜙𝜙2 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

⋮ ⋮ : ( , ) = ( , ) + ( , ) + + ( , ) 𝜕𝜕𝜃𝜃1 𝜕𝜕𝜃𝜃2 𝜕𝜕𝜃𝜃𝑛𝑛 𝛼𝛼𝑛𝑛 𝑀𝑀𝑗𝑗 𝜃𝜃ℎ 𝜙𝜙𝑛𝑛 𝑀𝑀𝑗𝑗 𝜙𝜙1 𝜙𝜙𝑛𝑛 𝑀𝑀𝑗𝑗 𝜙𝜙2 𝜙𝜙𝑛𝑛 ⋯ 𝑀𝑀𝑗𝑗 𝜙𝜙𝑛𝑛 𝜙𝜙𝑛𝑛 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

23 Notice that the ( , ) terms can be conveniently expressed as a matrix product, i.e.:

𝑗𝑗 ℎ 𝑖𝑖 𝑀𝑀 𝜃𝜃 𝜙𝜙 ( , ) ( , ) ( , ) /

𝑗𝑗( 1, 1) 𝑗𝑗( 2, 1) 𝑗𝑗( 𝑛𝑛, 1) 𝑀𝑀 𝜙𝜙 𝜙𝜙 𝑀𝑀 𝜙𝜙 𝜙𝜙 ⋯ 𝑀𝑀 𝜙𝜙 𝜙𝜙 𝜕𝜕𝜃𝜃1/𝜕𝜕𝜕𝜕 𝑛𝑛 ( , ) = ⎡ ⎤ ⎡ ⎤ ⎢𝑀𝑀𝑗𝑗 𝜙𝜙1 𝜙𝜙2 𝑀𝑀𝑗𝑗 𝜙𝜙2 𝜙𝜙2 ⋯ 𝑀𝑀𝑗𝑗 𝜙𝜙𝑛𝑛 𝜙𝜙2 ⎥ ⎢𝜕𝜕𝜃𝜃2 𝜕𝜕𝜕𝜕⎥ (25) 𝑗𝑗 ℎ 𝑖𝑖 � 𝑀𝑀 𝜃𝜃 𝜙𝜙 ⎢ ( , ) ( , ) ( , )⎥ ⎢ ⎥ 𝑖𝑖=1 ⎢ ⋮ ⋮ ⋱ ⋮ ⎥ ⎢ ⋮/ ⎥ ⎢ ⎥ ⎢ ⎥ 𝑗𝑗 1 𝑛𝑛 𝑗𝑗 2 𝑛𝑛 𝑗𝑗 𝑛𝑛 𝑛𝑛 𝑛𝑛 �⎣𝑀𝑀��𝜙𝜙��𝜙𝜙��𝑀𝑀��𝜙𝜙��𝜙𝜙��⋯��𝑀𝑀��𝜙𝜙��𝜙𝜙��⎦ �⎣𝜕𝜕��𝜃𝜃��𝜕𝜕�𝜕𝜕⎦ ≡ 𝑴𝑴𝒋𝒋 ̇𝒋𝒋 is called the “mass matrix” and is the “vector of degrees of freedom” (i.e. the solution vector).≡ 𝜽𝜽 𝒋𝒋 ̇𝒋𝒋 𝑴𝑴On the other hand, the remaining terms𝜽𝜽 can be expressed as vectors:

24 ( ) ( )

+ − 1( ) 1( ) 𝑛𝑛 = 𝜙𝜙 𝑧𝑧𝑗𝑗−1 𝜙𝜙 𝑧𝑧𝑗𝑗 ⎡ + ⎤ − + − 2 𝑗𝑗−1 ⎡ 2 𝑗𝑗 ⎤ (26) �𝑗𝑗−1 𝑖𝑖 𝑗𝑗−1 �𝑗𝑗 𝑖𝑖 𝑗𝑗 �𝑗𝑗−1 ⎢𝜙𝜙 (𝑧𝑧 )⎥ �𝑗𝑗 ⎢𝜙𝜙 (𝑧𝑧 )⎥ ��𝐾𝐾 ⋅ 𝜙𝜙 �𝑧𝑧 � − 𝐾𝐾 ⋅ 𝜙𝜙 �𝑧𝑧 �� 𝐾𝐾 ⋅ ⎢ ⎥ − 𝐾𝐾 ⋅ 𝑖𝑖=1 ⋮ + ⎢ ⋮ − ⎥ ⎢ 𝑛𝑛 ⎥ 𝑛𝑛 𝑗𝑗 ⎣𝜙𝜙 𝑧𝑧𝑗𝑗−1 ⎦ �⎣𝜙𝜙��𝑧𝑧��⎦ ��+�� − 𝑗𝑗−1 𝑗𝑗 ( , ) ≡ 𝚽𝚽�𝑧𝑧 � ≡ 𝚽𝚽�𝑧𝑧 � ( , ′ ) 𝑛𝑛 ( , ) = 𝐹𝐹𝑗𝑗 𝜃𝜃ℎ 𝜙𝜙1 ⎡ ′ ⎤ (27) ′ 𝐹𝐹𝑗𝑗 𝜃𝜃ℎ 𝜙𝜙2 � 𝐹𝐹𝑗𝑗 𝜃𝜃ℎ 𝜙𝜙𝑖𝑖 ⎢ ( , )⎥ 𝑖𝑖=1 ⎢ ⋮ ′ ⎥ 𝑗𝑗 ℎ 𝑛𝑛 �⎣𝐹𝐹��𝜃𝜃��𝜙𝜙��⎦ ≡ 𝑭𝑭𝒋𝒋

24 ( ) ( ) + − 1( ) 1( ) 𝑛𝑛 = 𝜙𝜙 𝑧𝑧𝑗𝑗−1 𝜙𝜙 𝑧𝑧𝑗𝑗 ⎡ + ⎤ − + − 2 𝑗𝑗−1 ⎡ 2 𝑗𝑗 ⎤ (28) 𝑗𝑗−1 𝑖𝑖 𝑗𝑗−1 𝑗𝑗 𝑖𝑖 𝑗𝑗 𝑗𝑗−1 ⎢𝜙𝜙 (𝑧𝑧 )⎥ 𝑗𝑗 ⎢𝜙𝜙 (𝑧𝑧 )⎥ ��𝑞𝑞� ⋅ 𝜙𝜙 �𝑧𝑧 � − 𝑞𝑞� ⋅ 𝜙𝜙 �𝑧𝑧 �� 𝑞𝑞� ⋅ ⎢ ⎥ − 𝑞𝑞� ⋅ 𝑖𝑖=1 ⋮ + ⎢ ⋮ − ⎥ ⎢ 𝑛𝑛 ⎥ 𝑛𝑛 𝑗𝑗 ⎣𝜙𝜙 𝑧𝑧𝑗𝑗−1 ⎦ �⎣𝜙𝜙��𝑧𝑧��⎦ ��+�� − 𝑗𝑗−1 𝑗𝑗 ( , ) ≡ 𝚽𝚽�𝑧𝑧 � ≡ 𝚽𝚽�𝑧𝑧 � ( , ′ ) 𝑛𝑛 ( , ) = 𝐹𝐹𝑗𝑗 𝜃𝜃ℎ 𝜙𝜙1 ⎡ ′ ⎤ (29) ′ 𝐹𝐹𝑗𝑗 𝜃𝜃ℎ 𝜙𝜙2 � 𝑄𝑄𝑗𝑗 𝜃𝜃ℎ 𝜙𝜙𝑖𝑖 ⎢ ( , )⎥ 𝑖𝑖=1 ⎢ ⋮ ′ ⎥ 𝑗𝑗 ℎ 𝑛𝑛 �⎣𝐹𝐹��𝜃𝜃��𝜙𝜙��⎦ 𝒋𝒋 Therefore, the element equation for the -the element≡ 𝑸𝑸 is: 25

+ 𝑗𝑗 = (30) + − + − 𝒋𝒋 ̇ 𝒋𝒋 �𝑗𝑗−1 𝒋𝒋−𝟏𝟏 �𝑗𝑗 𝒋𝒋 𝒋𝒋 𝑗𝑗−1 𝒋𝒋−𝟏𝟏 𝑗𝑗 𝒋𝒋 𝒋𝒋 The next step is to numerically𝑴𝑴 𝜽𝜽 �represent𝐾𝐾 ⋅ 𝚽𝚽� terms𝒛𝒛 � in− Eq.𝐾𝐾 ⋅ (30)𝚽𝚽�𝒛𝒛, which�� − 𝑭𝑭 begins�𝑞𝑞� with⋅ 𝚽𝚽the� 𝒛𝒛introduction� − 𝑞𝑞� ⋅ 𝚽𝚽 of� 𝒛𝒛the� �master− 𝑸𝑸 element.

25 3.1.4 Master Element Transformation

While it is feasible to compute every entry in the element equation independently,

having a master element would generally increase computational efficiency. The idea is

to introduce a reference element over a fixed domain and map it to every element in the

mesh via linear transformation, allowing the same terms be used for all elements. This

concept is illustrated in Fig. 3. 𝜙𝜙

Figure 3: Concept of the Master Element. Coordinates of the master element ( ) are mapped to those of every element ( ) in the actual domain. In one dimension, the transform follows the idea of 𝟎𝟎 the two-point form of a line 32,33,34. 𝛀𝛀 𝛀𝛀𝐣𝐣

The domain of the master element can be arbitrary. For ease of computation, it is set to

[ 1, 1] where is the master element coordinate (as is the coordinate of the actual domain).𝜉𝜉 ∈ − One-dimensionally,𝜉𝜉 it can be easily derived 𝑧𝑧using two-point rule that the transformation from to is:

𝑧𝑧 𝜉𝜉 + = + 2 2 𝑧𝑧𝑗𝑗 𝑧𝑧𝑗𝑗−1 𝑧𝑧𝑗𝑗 − 𝑧𝑧𝑗𝑗−1 𝑧𝑧 ↦ 𝑍𝑍𝑗𝑗 ⋅ 𝜉𝜉 And the first derivative of with respect to is:

𝑍𝑍𝑗𝑗 𝜉𝜉 = 2 2 𝑑𝑑𝑍𝑍𝑗𝑗 𝑧𝑧𝑗𝑗 − 𝑧𝑧𝑗𝑗−1 Δ𝑧𝑧𝑗𝑗 ≡ where denotes the length of the𝑑𝑑𝑑𝑑 -th element. Hence, for example, the entry in the -

𝑗𝑗 th row Δand𝑧𝑧 -th column in the mass matrix𝑗𝑗 can be expressed as: 𝑘𝑘

𝑙𝑙 26 ( , ) ( ) ( ) 𝑧𝑧𝑗𝑗 𝑗𝑗 𝑘𝑘 𝑙𝑙 𝑘𝑘 𝑙𝑙 𝑀𝑀 𝜙𝜙 𝜙𝜙 ≡ �𝑧𝑧𝑗𝑗−1𝜙𝜙 𝑧𝑧 ⋅ 𝜙𝜙 𝑧𝑧 𝑑𝑑𝑑𝑑 = ( ) ( ) 1 𝑗𝑗 𝑑𝑑𝑍𝑍 � 𝜙𝜙𝑘𝑘 𝜉𝜉 ⋅ 𝜙𝜙𝑙𝑙 𝜉𝜉 𝑑𝑑𝑑𝑑 −1 𝑑𝑑𝑑𝑑 = ( ) ( ) 1 2𝑗𝑗 Δ𝑧𝑧 𝑘𝑘 𝑙𝑙 ⋅ �−1𝜙𝜙 𝜉𝜉 ⋅ 𝜙𝜙 𝜉𝜉 𝑑𝑑𝑑𝑑 ( ( ), ( )) 2 Δ𝑧𝑧𝑗𝑗 ≡ ⋅ 𝜙𝜙𝑘𝑘 𝜉𝜉 𝜙𝜙𝑙𝑙 𝜉𝜉 0 Applying the above to every entry in the mass matrix results in the following:

( ( ), ( )) ( ( ), ( )) ( ( ), ( )) ( ) ( ) ( ) 𝜙𝜙1(𝜉𝜉), 𝜙𝜙1(𝜉𝜉) 0 𝜙𝜙2(𝜉𝜉), 𝜙𝜙1(𝜉𝜉) 0 ⋯ 𝜙𝜙𝑛𝑛(𝜉𝜉), 𝜙𝜙1(𝜉𝜉) 0 (31) 2 ⎡ ⎤ 𝑗𝑗 ⎢ 𝜙𝜙1 𝜉𝜉 𝜙𝜙2 𝜉𝜉 0 𝜙𝜙2 𝜉𝜉 𝜙𝜙2 𝜉𝜉 0 ⋯ 𝜙𝜙𝑛𝑛 𝜉𝜉 𝜙𝜙2 𝜉𝜉 0⎥ 𝒋𝒋 Δ𝑧𝑧 𝑴𝑴 ↦ ⎢( ) ( ) ( ) ⎥ ⎢ ( ),⋮ ( ) ( ),⋮ ( ) ⋱ ( ),⋮ ( ) ⎥ ⎢ ⎥ 1 𝑛𝑛 0 2 𝑛𝑛 0 𝑛𝑛 𝑛𝑛 0 Similarly, the master⎣ element𝜙𝜙 𝜉𝜉 𝜙𝜙 concept𝜉𝜉 can𝜙𝜙 be𝜉𝜉 applied𝜙𝜙 𝜉𝜉 to ⋯and 𝜙𝜙: 𝜉𝜉 𝜙𝜙 𝜉𝜉 ⎦

𝑭𝑭𝒋𝒋 𝑸𝑸𝒋𝒋 ( ) ( ) ( , ) 1 ( ) ( ) ′ ′ ℎ 1 ′ 𝑗𝑗( ℎ, 1) ⎡� 𝐾𝐾�𝜃𝜃 𝜉𝜉 � ⋅ 𝜙𝜙 𝜉𝜉 𝑑𝑑𝑑𝑑⎤ ( ) 1( ) 𝐹𝐹 𝜃𝜃 𝜙𝜙 −1 ( ) ( ) �𝐾𝐾 𝜉𝜉 ⋅ 𝜙𝜙 𝜉𝜉 �0 = ⎡ ′ ⎤ ⎢ 1 ⎥ ⎡ ′ ⎤ (32) 𝑗𝑗 ℎ 2 2𝑗𝑗 ⎢ ′ ⎥ 2𝑗𝑗 2 ⎢𝐹𝐹 𝜃𝜃 𝜙𝜙 ⎥ Δ𝑧𝑧 ℎ 2 Δ𝑧𝑧 ⎢�𝐾𝐾 𝜉𝜉 ⋅ 𝜙𝜙 𝜉𝜉 �0⎥ 𝒋𝒋 ⎢�−1𝐾𝐾�𝜃𝜃 𝜉𝜉 � ⋅ 𝜙𝜙 𝜉𝜉 𝑑𝑑𝑑𝑑⎥ 𝑭𝑭 ⎢ ( , )⎥ ↦ ≡ ⎢ ( ) ( ) ⎥ ⎢ ⋮ ⎥ ⎢ ( ) ( ) ⎥ ⎢ ⋮ ⎥ ⎢ ′ ⎥ 1 ⋮ ′ 𝑗𝑗 ℎ 𝑛𝑛 ⎢ ⎥ ⎢ 𝑛𝑛 ⎥ ⎣𝐹𝐹 𝜃𝜃 𝜙𝜙 ⎦ ′ ⎣�𝐾𝐾 𝜉𝜉 ⋅ 𝜙𝜙 𝜉𝜉 �0⎦ ⎢� 𝐾𝐾�𝜃𝜃ℎ 𝜉𝜉 � ⋅ 𝜙𝜙𝑛𝑛 𝜉𝜉 𝑑𝑑𝑑𝑑⎥ ⎣ −1 ⎦ ( ) ( ) ( , ) 1 ( ) ( ) ′ ′ ℎ 1 ′ 𝑗𝑗( ℎ, 1) ⎡� 𝑞𝑞�𝜃𝜃 𝜉𝜉 � ⋅ 𝜙𝜙 𝜉𝜉 𝑑𝑑𝑑𝑑⎤ ( ) 1( ) 𝑄𝑄 𝜃𝜃 𝜙𝜙 −1 ( ) ( ) �𝑞𝑞 𝜉𝜉 ⋅ 𝜙𝜙 𝜉𝜉 �0 = ⎡ ′ ⎤ ⎢ 1 ⎥ ⎡ ′ ⎤ (33) 𝑗𝑗 ℎ 2 2𝑗𝑗 ⎢ ′ ⎥ 2𝑗𝑗 2 ⎢𝑄𝑄 𝜃𝜃 𝜙𝜙 ⎥ Δ𝑧𝑧 ℎ 2 Δ𝑧𝑧 ⎢�𝑞𝑞 𝜉𝜉 ⋅ 𝜙𝜙 𝜉𝜉 �0⎥ 𝒋𝒋 ⎢�−1𝑞𝑞�𝜃𝜃 𝜉𝜉 � ⋅ 𝜙𝜙 𝜉𝜉 𝑑𝑑𝑑𝑑⎥ 𝑸𝑸 ⎢ ( , )⎥ ↦ ≡ ⎢ ( ) ( ) ⎥ ⎢ ⋮ ⎥ ⎢ ( ) ( ) ⎥ ⎢ ⋮ ⎥ ⎢ ′ ⎥ 1 ⋮ ′ 𝑗𝑗 ℎ 𝑛𝑛 ⎢ ⎥ ⎢ 𝑛𝑛 ⎥ ⎣𝑄𝑄 𝜃𝜃 𝜙𝜙 ⎦ ′ ⎣�𝑞𝑞 𝜉𝜉 ⋅ 𝜙𝜙 𝜉𝜉 �0⎦ ⎢� 𝑞𝑞�𝜃𝜃ℎ 𝜉𝜉 � ⋅ 𝜙𝜙𝑛𝑛 𝜉𝜉 𝑑𝑑𝑑𝑑⎥ ⎣ −1 ⎦

27 3.1.5 Numerical Integration

In numerical analysis, definite integrals (such as entries in , and ) can be

𝒋𝒋 𝒋𝒋 𝒋𝒋 computed by numerical integration (aka “quadrature”) via basic𝑴𝑴 𝑭𝑭linear 𝑸𝑸algebraic

operations. This also takes advantage of MATLAB’s high capability in matrix operation.

Suppose ( ) is some generic, known function over the master element defined in

Chapter 3.1.4. The𝑓𝑓 𝜉𝜉 integral of ( ) over the domain of the master element can be

approximated by a Riemann sum 𝑓𝑓as 𝜉𝜉follows:

= ( ) ( ) 1 𝑛𝑛 (34) 𝐼𝐼 � 𝑓𝑓 𝜉𝜉 𝑑𝑑𝑑𝑑 ≈ � 𝑤𝑤𝑖𝑖 ⋅ 𝑓𝑓 𝜉𝜉𝑖𝑖 −1 𝑖𝑖=1 in which is the weight term and is the -th sampling point of the -point integration

𝑖𝑖 𝑖𝑖 formula. In𝑤𝑤 this thesis, the Legendre𝜉𝜉-Gauss q𝑖𝑖uadrature is used. To integrate𝑛𝑛 a -th degree polynomial exactly (i.e. to have the Riemann sum equal the integral), = (𝑘𝑘 + 1)/2

Gauss points are required. For example, if ( ) were a second-degree 𝑛𝑛polynomial,⌈ 𝑘𝑘 then⌉ the required number of Gauss points to exactly𝑓𝑓 𝜉𝜉 integrate ( ) would be:

= (2 + 1)/2 = 1.5 =𝑓𝑓 2𝜉𝜉

Additionally, if ( ) and ( 𝑛𝑛) are⌈ some known⌉ function⌈ ⌉ s over the master element, the exact integral of 𝑓𝑓their𝜉𝜉 product𝑔𝑔 𝜉𝜉 would be

( ) ( ) = ( ) ( ) = 1 𝑛𝑛 (35) T � 𝑓𝑓 𝜉𝜉 ⋅ 𝑔𝑔 𝜉𝜉 𝑑𝑑𝑑𝑑 � 𝑤𝑤𝑖𝑖 ⋅ 𝑓𝑓 𝜉𝜉𝑖𝑖 ⋅ 𝑔𝑔 𝜉𝜉𝑖𝑖 𝒇𝒇𝒏𝒏𝑾𝑾𝒏𝒏𝒈𝒈𝒏𝒏 −1 𝑖𝑖=1 where

( ) 0 0 ( ) ( ) 0 0 ( ) = 𝑓𝑓 𝜉𝜉1 , = 𝑤𝑤1 ⋯ , = 𝑔𝑔 𝜉𝜉1 . 𝑓𝑓 𝜉𝜉2 0 𝑤𝑤02 ⋯ 𝑔𝑔 𝜉𝜉2 𝒇𝒇𝒏𝒏 � ( )� 𝑾𝑾𝒏𝒏 � � 𝒈𝒈𝒏𝒏 � ( )� ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ 𝑓𝑓 𝜉𝜉𝑛𝑛 ⋯ 𝑤𝑤𝑛𝑛 𝑔𝑔 𝜉𝜉𝑛𝑛 28 Note that in this case, the degree of the polynomial [ ( ) ( )] is the sum of the

degrees of ( ) and ( ). Tabulated Gauss points and associated𝑓𝑓 𝜉𝜉 ⋅ 𝑔𝑔 weights𝜉𝜉 can be found in literatures and𝑓𝑓 𝜉𝜉 reference𝑔𝑔 𝜉𝜉 materials 5,43.

Applying Eq. (35) to every entry in , and (which is a product of two

𝒋𝒋 𝒋𝒋 𝒋𝒋 functions) generates the following: 𝑴𝑴 𝑭𝑭 𝑸𝑸

( ) 0 0 ( ) ( ) T 0 0 ( ) ( ( ), ( )) = 𝑛𝑛 ( ) ( ) = 𝜙𝜙𝑘𝑘 𝜉𝜉1 𝑤𝑤1 ⋯ 𝜙𝜙𝑙𝑙 𝜉𝜉1 𝜙𝜙𝑘𝑘 𝜉𝜉2 0 𝑤𝑤02 ⋯ 𝜙𝜙𝑙𝑙 𝜉𝜉2 𝜙𝜙𝑘𝑘 𝜉𝜉 𝜙𝜙𝑙𝑙 𝜉𝜉 0 � 𝑤𝑤𝑖𝑖 ⋅ 𝜙𝜙𝑘𝑘 𝜉𝜉𝑖𝑖 ⋅ 𝜙𝜙𝑙𝑙 𝜉𝜉𝑖𝑖 � ( )� � � � ( )� 𝑖𝑖=1 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ 𝑘𝑘 𝑛𝑛 𝑛𝑛 𝑙𝑙 𝑛𝑛 𝜙𝜙 ( 𝜉𝜉) 0 ⋯ 0𝑤𝑤 𝜙𝜙( 𝜉𝜉) T ( ) ( ) 0 0 𝑗𝑗 1 ( ) ( ) = 𝑛𝑛 ( ) ( ) = 𝐾𝐾 𝜉𝜉1 𝑤𝑤1 ⋯ 𝜙𝜙 𝜉𝜉 ′ 2 2 ⎡ 𝑗𝑗 2 ⎤ 𝑗𝑗 𝑖𝑖 𝑖𝑖 𝑗𝑗 𝑖𝑖 𝐾𝐾(𝜉𝜉 ) 0 𝑤𝑤0 ⋯ 𝜙𝜙 (𝜉𝜉 ) �𝐾𝐾 𝜉𝜉 ⋅ 𝜙𝜙 𝜉𝜉 �0 � 𝑤𝑤 ⋅ 𝐾𝐾 𝜉𝜉 ⋅ 𝜙𝜙 𝜉𝜉 � � � � ⎢ ⎥ 𝑖𝑖=1 ⋮ ⋮ ⋮ ⋱ ⋮ ⎢ ⋮ ⎥ 𝑛𝑛 ⋯ 𝑤𝑤𝑛𝑛 𝑗𝑗 𝑛𝑛 𝐾𝐾( 𝜉𝜉 ) 0 0 ⎣𝜙𝜙 ( 𝜉𝜉 ) ⎦ T ( ) ( ) 0 0 𝑗𝑗 1 ( ) ( ) = 𝑛𝑛 ( ) ( ) = 𝑞𝑞 𝜉𝜉1 𝑤𝑤1 ⋯ 𝜙𝜙 𝜉𝜉 ′ 2 2 ⎡ 𝑗𝑗 2 ⎤ 𝑗𝑗 𝑖𝑖 𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑞𝑞(𝜉𝜉 ) 0 𝑤𝑤0 ⋯ 𝜙𝜙 (𝜉𝜉 ) �𝑞𝑞 𝜉𝜉 ⋅ 𝜙𝜙 𝜉𝜉 �0 � 𝑤𝑤 ⋅ 𝑞𝑞 𝜉𝜉 ⋅ 𝜙𝜙 𝜉𝜉 � � � � ⎢ ⎥ 𝑖𝑖=1 ⋮ ⋮ ⋮ ⋱ ⋮ ⎢ ⋮ ⎥ 𝑛𝑛 𝑛𝑛 𝑗𝑗 𝑛𝑛 Hence, in quadrature form, , and can be𝑞𝑞 exp𝜉𝜉 ressed as follows⋯: 𝑤𝑤 ⎣𝜙𝜙 𝜉𝜉 ⎦

𝑴𝑴𝒋𝒋 𝑭𝑭𝒋𝒋 𝑸𝑸𝒋𝒋 = [ ] (36) 2 Δ𝑧𝑧𝑗𝑗 T 𝑴𝑴𝒋𝒋 𝚽𝚽 𝑾𝑾𝒏𝒏𝚽𝚽 = [( ) ] ( ) (37) 2 Δ𝑧𝑧𝑗𝑗 ′ T ′ 𝑭𝑭𝒋𝒋 𝚽𝚽 𝑾𝑾𝒏𝒏 𝐾𝐾 𝜽𝜽𝒋𝒋𝚽𝚽 = [( ) ] ( ) (38) 2 Δ𝑧𝑧𝑗𝑗 ′ T ′ 𝑸𝑸𝒋𝒋 𝚽𝚽 𝑾𝑾𝒏𝒏 𝑞𝑞 𝜽𝜽𝒋𝒋𝚽𝚽 It is noteworthy that the terms inside the square brackets remain constant for any element

thanks to the master element, allowing computation to be made once and for all. Each

side of the element equation is now as follows:

29 [ ] + [( ) ] LHS: 2 2 Δ𝑧𝑧𝑗𝑗 T " + " − Δ𝑧𝑧𝑗𝑗 ′ T ′ 𝚽𝚽 𝑾𝑾𝒏𝒏𝚽𝚽 𝜽𝜽̇ 𝒋𝒋 𝐾𝐾�𝑗𝑗−1 ⋅ 𝚽𝚽�𝒛𝒛𝒋𝒋−𝟏𝟏� − 𝐾𝐾�𝑗𝑗 ⋅ 𝚽𝚽�𝒛𝒛𝒋𝒋 � − 𝚽𝚽 𝑾𝑾𝒏𝒏 𝐾𝐾�𝜽𝜽𝒋𝒋𝚽𝚽 � �� ≡ 𝑴𝑴𝒋𝒋 𝐾𝐾 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 ≡ 𝑭𝑭𝒋𝒋

[( ) ] RHS: 2 " + " − Δ𝑧𝑧𝑗𝑗 ′ T ′ 𝑞𝑞�𝑗𝑗−1 ⋅ 𝚽𝚽�𝒛𝒛𝒋𝒋−𝟏𝟏� − 𝑞𝑞�𝑗𝑗 ⋅ 𝚽𝚽�𝒛𝒛𝒋𝒋 � − 𝚽𝚽 𝑾𝑾𝒏𝒏 𝑞𝑞�𝜽𝜽𝒋𝒋𝚽𝚽 � �� 𝑞𝑞 𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝒋𝒋 ≡ 𝑸𝑸

By isolating on one side, the element equation is rearranged into ̇ 𝒋𝒋 =𝜽𝜽 + (39) ′ ′ + − 𝜽𝜽̇ 𝒋𝒋 −𝑨𝑨𝒋𝒋� 𝑞𝑞�𝜽𝜽𝒋𝒋𝚽𝚽 � − 𝐾𝐾�𝜽𝜽𝒋𝒋𝚽𝚽 �� 𝑩𝑩𝒋𝒋 ⋅ �𝑞𝑞�𝑗𝑗−1 − 𝐾𝐾�𝑗𝑗−1� − 𝑩𝑩𝒋𝒋 ⋅ �𝑞𝑞�𝑗𝑗 − 𝐾𝐾�𝑗𝑗� where

= [ ] [( ) ] T −1 ′ T 𝑨𝑨𝒋𝒋 𝚽𝚽 𝑾𝑾𝒏𝒏𝚽𝚽 ⋅ 𝚽𝚽 𝑾𝑾𝒏𝒏 = [ ] 2 −1 + Δ𝑧𝑧𝑗𝑗 T + 𝑩𝑩𝒋𝒋 � 𝚽𝚽 𝑾𝑾𝒏𝒏𝚽𝚽 � ⋅ 𝚽𝚽�𝒛𝒛𝒋𝒋−𝟏𝟏� = [ ] 2 −1 − Δ𝑧𝑧𝑗𝑗 T − 𝑩𝑩𝒋𝒋 � 𝚽𝚽 𝑾𝑾𝒏𝒏𝚽𝚽 � ⋅ 𝚽𝚽�𝒛𝒛𝒋𝒋 � Eq. (39) is the numerical form of the -th element equation. The global equation and

matrices will be discussed in the next sec𝑗𝑗tion.

30 3.1.6 Global Equations

First, element equations of three consecutive elements are written out as follows:

= + ′ ′ + − ̇ 𝒋𝒋−𝟏𝟏 𝒋𝒋−𝟏𝟏 𝒋𝒋−𝟏𝟏 𝒋𝒋−𝟏𝟏 𝒋𝒋−𝟏𝟏 𝑗𝑗−2 �𝑗𝑗−2 𝒋𝒋−𝟏𝟏 𝑗𝑗−1 �𝑗𝑗−1 𝜽𝜽 = −𝑨𝑨 � 𝑞𝑞 �𝜽𝜽 𝚽𝚽 � − 𝐾𝐾�𝜽𝜽 𝚽𝚽 �� 𝑩𝑩 ⋅ �𝑞𝑞� − 𝐾𝐾 � −𝑩𝑩+ ⋅ �𝑞𝑞� − 𝐾𝐾 � ′ ′ + − ̇ 𝒋𝒋 𝒋𝒋 𝒋𝒋 𝒋𝒋 𝒋𝒋 𝑗𝑗−1 �𝑗𝑗−1 𝒋𝒋 𝑗𝑗 �𝑗𝑗 𝜽𝜽 = −𝑨𝑨 � 𝑞𝑞�𝜽𝜽 𝚽𝚽 � − 𝐾𝐾�𝜽𝜽 𝚽𝚽 �� 𝑩𝑩 ⋅ �𝑞𝑞� − 𝐾𝐾 � −+𝑩𝑩 ⋅ �𝑞𝑞� − 𝐾𝐾 � ′ ′ + − 𝜽𝜽̇ 𝒋𝒋+𝟏𝟏 −𝑨𝑨𝒋𝒋+𝟏𝟏� 𝑞𝑞�𝜽𝜽𝒋𝒋+𝟏𝟏𝚽𝚽 � − 𝐾𝐾�𝜽𝜽𝒋𝒋+𝟏𝟏𝚽𝚽 �� 𝑩𝑩𝒋𝒋+𝟏𝟏 ⋅ �𝑞𝑞�𝑗𝑗 − 𝐾𝐾�𝑗𝑗� −𝑩𝑩𝒋𝒋+𝟏𝟏 ⋅ �𝑞𝑞�𝑗𝑗+1 − 𝐾𝐾�𝑗𝑗+1�

It can be observed that each element has its own matrix, which indicates that the global matrix of all is block diagonal:

𝑨𝑨𝒋𝒋 𝑨𝑨𝒋𝒋 …

31 ′ ′ … 𝟏𝟏 𝟏𝟏 𝟏𝟏 𝑞𝑞�𝜽𝜽 𝚽𝚽 � − 𝐾𝐾�𝜽𝜽 𝚽𝚽 � 𝑨𝑨 𝟎𝟎 𝟎𝟎 ⎡ ′ ′ ⎤ 𝟐𝟐 𝟎𝟎 𝑨𝑨 … 𝟎𝟎 ⎢𝑞𝑞�𝜽𝜽𝟐𝟐𝚽𝚽 � − 𝐾𝐾�𝜽𝜽𝟐𝟐𝚽𝚽 �⎥ � � ⎢ ⎥ ⋮ ⋮ ⋱ ⋮ ′ ⋮ ′ 𝒏𝒏 ⎢ 𝒏𝒏 𝒏𝒏 ⎥ �𝟎𝟎��𝟎𝟎��𝑨𝑨�� ⎣𝑞𝑞�𝜽𝜽 𝚽𝚽 � − 𝐾𝐾�𝜽𝜽 𝚽𝚽 �⎦ ≡ 𝑨𝑨 Similarly, the global matrices of all and of all are also block diagonal: + − 𝑩𝑩𝒋𝒋 𝑩𝑩𝒋𝒋

0 1 … 0 … 1 1 2 + … 𝑞𝑞�1 − 𝐾𝐾� − … 𝑞𝑞�2 − 𝐾𝐾� 𝟏𝟏 ⎡ ⎤ 𝟏𝟏 ⎡ ⎤ 𝑩𝑩 𝟎𝟎+ 𝟎𝟎 𝑞𝑞2 − 𝐾𝐾2 𝑩𝑩 𝟎𝟎− 𝟎𝟎 𝑞𝑞3 − 𝐾𝐾3 ⎡ ⎤ ⎢ � � ⎥ 𝟐𝟐 ⎢� � ⎥ 𝟎𝟎 𝑩𝑩𝟐𝟐 … 𝟎𝟎 𝟎𝟎 𝑩𝑩 … 𝟎𝟎 ⎢ ⎥ ⎢ 𝑞𝑞� − 𝐾𝐾� ⎥ � � ⎢𝑞𝑞� − 𝐾𝐾� ⎥ ⎢ ⋮ ⋮ ⋱ ⋮+⎥ ⎢ 1 1⎥ ⋮ ⋮ ⋱ ⋮− ⎢ ⎥ 𝒏𝒏 �⎣ 𝟎𝟎��𝟎𝟎��𝑩𝑩�𝒏𝒏�⎦ ⎢ ⋮ ⎥ ��𝟎𝟎��𝟎𝟎��𝑩𝑩�� ⎢ ⋮ ⎥ + 𝑞𝑞�𝑛𝑛− − 𝐾𝐾�𝑛𝑛− 31 − 𝑞𝑞�𝑛𝑛 − 𝐾𝐾�𝑛𝑛 ≡ 𝑩𝑩 ⎣ ⎦ ≡ 𝑩𝑩 ⎣ ⎦ Since the numerical flux terms highlighted in same colors are shared between adjacent

elements except in the first and the last row (i.e. the two boundary flux terms), , + − and the numerical fluxes can also be conveniently expressed as the product of a𝑩𝑩 single𝑩𝑩

global matrix and a numerical flux vector:

0 … 0 1 + − … 𝑞𝑞�1 − 𝐾𝐾� 𝟏𝟏 𝟏𝟏 ⎡ ⎤ 𝑩𝑩 −𝑩𝑩 𝟎𝟎 𝟎𝟎 2 2 ⎡ + − ⎤ ⎢𝑞𝑞� − 𝐾𝐾� ⎥ 𝟎𝟎 𝑩𝑩𝟐𝟐 −…𝑩𝑩𝟐𝟐 𝟎𝟎 ⎢ ⎥ ⎢𝑞𝑞� − 𝐾𝐾� ⎥ ⎢ ⋮ ⋮ ⋱ ⋱+ ⋮ −⎥ ⎢ ⎥ �⎣ 𝟎𝟎��𝟎𝟎��𝑩𝑩�𝒏𝒏��−�𝑩𝑩�𝒏𝒏�⎦ ⎢ ⋮ ⎥ 𝑞𝑞�𝑛𝑛 − 𝐾𝐾�𝑛𝑛 ≡ 𝑩𝑩 ⎣ ⎦ Note that the numerical flux vector is one entry longer than the vector attached to the global matrix. In fact, the every first entry ( ) in the numerical flux is associated

0 0 with the𝑨𝑨 top boundary condition, and can be directly𝑞𝑞� − 𝐾𝐾� modified to account for sink/source

terms in the top boundary (see Chapter 3.3). In retrospect of Eq. (39), let be the global

vector of degrees of freedom; the global matrix equation is: 𝚯𝚯̇

0 0 ′ ′ 𝑞𝑞�1 − 𝐾𝐾�1 = 𝑞𝑞�𝜽𝜽𝟏𝟏𝚽𝚽 � − 𝐾𝐾�𝜽𝜽𝟏𝟏𝚽𝚽 � + ⎡ ⎤ (40) ⎡ ′ ′ ⎤ 𝑞𝑞2 − 𝐾𝐾2 𝟐𝟐 𝟐𝟐 ⎢� � ⎥ ̇ ⎢𝑞𝑞�𝜽𝜽 𝚽𝚽 � − 𝐾𝐾�𝜽𝜽 𝚽𝚽 �⎥ 𝚯𝚯 −𝑨𝑨 ⋅ ⎢ ⎥ 𝑩𝑩 ⋅ ⎢𝑞𝑞� − 𝐾𝐾� ⎥ ′ ⋮ ′ ⎢ ⎥ ⎢ 𝒏𝒏 𝒏𝒏 ⎥ ⋮ 𝑞𝑞�𝜽𝜽 𝚽𝚽 � − 𝐾𝐾�𝜽𝜽 𝚽𝚽 � ⎢ ⎥ ⎣ ⎦ 𝑞𝑞𝑛𝑛 − 𝐾𝐾�𝑛𝑛 in which can be numerically computed with known ⎣values� of⎦ the variables. This concludes 𝚯𝚯thė DG formulation of Eq. (7).

Before proceeding to Eq. (8), it is critical to point out that block-diagonal global

matrices are what allow distinct specification to be made element by element from a

computational standpoint. Taking as an example, different parameters can be assigned

to any of its diagonal entries ( ,𝑨𝑨 = 1, … , ) regardless of the others. Suppose a soil

𝑨𝑨𝒊𝒊 𝑖𝑖 32𝑛𝑛 column were made up of sand at the top, loam in the middle and clay at the bottom, and each layer is represented by one element, its global matrix would simply be

[ ] 𝑨𝑨 = [ ] 𝑨𝑨𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬 𝟎𝟎 𝟎𝟎 𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥 𝑨𝑨 � 𝟎𝟎 𝑨𝑨 𝟎𝟎 � 𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜 in which , and would𝟎𝟎 be based𝟎𝟎 on different�𝑨𝑨 � parameters for sand, loam

𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬 𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥 𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜 and clay, 𝑨𝑨respectively.𝑨𝑨 Conforming𝑨𝑨 FEM or FDM, on the other hand, would result in a tri-diagonal global iteration matrix indicating that same physical properties of the porous medium are shared throughout the entire domain of interest 9,25,26.

3.1.7 Hydraulic Conductivity and Diffusivity

Prior to formulating and Eq. (8), the diffusivity term needs to be transformed.

The Brooks and Corey soil-water𝑞𝑞 retention models are used in this model 7.26:

2 (41) ( ) = 3+ 𝜆𝜆 𝜃𝜃 − 𝜃𝜃𝑟𝑟 𝐾𝐾 𝜃𝜃 𝐾𝐾𝑠𝑠 ⋅ � � 𝜃𝜃𝑠𝑠 − 𝜃𝜃𝑟𝑟 1 (42) ( ) = 2+ ( 𝑏𝑏 ) 𝑟𝑟 𝜆𝜆 𝑠𝑠 𝜓𝜓 𝜃𝜃 − 𝜃𝜃 𝐷𝐷 𝜃𝜃 𝐾𝐾 ⋅ − 𝑠𝑠 𝑟𝑟 ⋅ � 𝑠𝑠 𝑟𝑟� where 𝜆𝜆 𝜃𝜃 − 𝜃𝜃 𝜃𝜃 − 𝜃𝜃

hydraulic conductivity [L T-1];

𝐾𝐾 soil water diffusivity [L2 T-1];

𝐷𝐷 volumetric soil moisture content [L3 L-3];

𝜃𝜃 hydraulic conductivity at saturation [L T-1];

𝑠𝑠 𝐾𝐾 air-entry pressure head [L];

𝜓𝜓𝑏𝑏 33 pore distribution index [-];

𝜆𝜆 residual volumetric soil moisture content [L3 L-3];

𝑟𝑟 𝜃𝜃 volumetric soil moisture content at saturation [L3 L-3].

𝑠𝑠 Procedures on𝜃𝜃 how to estimate the saturated hydraulic conductivity ( ), volumetric

𝑠𝑠 moisture content at saturation ( ) and residual volumetric moisture content𝐾𝐾 ( ) are

𝑠𝑠 𝑟𝑟 provided in Appendix B. Due to𝜃𝜃 lack of proper equipment, the air-entry pressure𝜃𝜃 ( )

𝑏𝑏 and the pore distribution index ( ) were obtained via curve fitting (see Chapter 5.1 𝜓𝜓for

more details). 𝜆𝜆

As regards the global matrix equation and Eq. (8), while can be directly

computed with known moisture values and parameters, is a bit tricky𝐾𝐾 due to having a non-linear term. An alternative form of Eq. (8) is necessary𝑞𝑞 for it to be formulated in a

way like Eq. (7).

First, another auxiliary variable is introduced such that ∗ 𝐷𝐷 = (43) ∗ 𝜕𝜕𝜃𝜃 𝜕𝜕𝐷𝐷 𝐷𝐷 To simplify representation of parameters,𝜕𝜕𝜕𝜕 diffusivity𝜕𝜕𝜕𝜕 is rewritten as

( ) = ( ) (44) 𝛽𝛽 𝑟𝑟 where 𝐷𝐷 𝜃𝜃 𝛼𝛼 𝜃𝜃 − 𝜃𝜃

1 1 and = 2 + . = 𝛽𝛽 ( 𝑏𝑏 ) 𝑠𝑠 −𝜓𝜓 𝛼𝛼 𝐾𝐾 ⋅ 𝑠𝑠 𝑟𝑟 ⋅ � 𝑠𝑠 𝑟𝑟� 𝛽𝛽 By the chain rule, 𝜆𝜆 𝜃𝜃 − 𝜃𝜃 𝜃𝜃 − 𝜃𝜃 𝜆𝜆

= (45) ∗ ∗ 𝜕𝜕𝐷𝐷 𝜕𝜕𝐷𝐷 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

34 Combine Eq. (43), (44) and (45) and eliminate like terms to obtain:

= ( ) (46) ∗ 𝜕𝜕𝐷𝐷 𝛽𝛽 𝛼𝛼 𝜃𝜃 − 𝜃𝜃𝑟𝑟 Multiply both sides of Eq. (46) by 𝜕𝜕𝜕𝜕 and integrate on both sides to obtain an explicit

form for : 𝜕𝜕𝜕𝜕 ∗ 𝐷𝐷 = ( ) ∗ 𝛽𝛽 � 𝜕𝜕𝐷𝐷 � 𝛼𝛼 𝜃𝜃 − 𝜃𝜃𝑟𝑟 𝜕𝜕𝜕𝜕 = ( ) + + 1 ∗ 𝛼𝛼 𝛽𝛽+1 𝐷𝐷 ⋅ 𝜃𝜃 − 𝜃𝜃𝑟𝑟 𝛾𝛾 In expanded form, 𝛽𝛽

(47) ( ) = ( ) + 1 1 𝑠𝑠 𝑏𝑏 3+ ∗ ( −𝐾𝐾) ⋅ 𝜓𝜓 3 + 𝜆𝜆 1 𝑟𝑟 𝐷𝐷 𝜃𝜃 3+ ⋅ 𝜃𝜃 − 𝜃𝜃 𝛾𝛾 𝜆𝜆 𝜆𝜆 𝜃𝜃𝑠𝑠 − 𝜃𝜃𝑟𝑟 ⋅ � � where is a constant. As a result, Eq. (8) has been𝜆𝜆 transformed into

𝛾𝛾 = (48) ∗ 𝜕𝜕𝐷𝐷 𝑞𝑞 𝜕𝜕𝜕𝜕

3.1.8 Numerical Representation of Diffusivity

Multiply both sides of Eq. (48) by the same test function previously introduced

in Chapter 3.1.1 and integrate on both sides: 𝑣𝑣

(49) ( ) = ∗ 𝜕𝜕𝐷𝐷 � 𝑞𝑞𝑞𝑞 𝑑𝑑Ωj � � 𝑣𝑣� 𝑑𝑑Ωj Ωj Ωj 𝜕𝜕𝜕𝜕 Next, apply the integration-by-parts formula to the -derivative and represent approximations using “ ” subscripts: 𝑧𝑧

ℎ ( ( ) ) = ( ) ( ) (50) 𝑗𝑗 ∗ 𝑧𝑧 ∗ 𝑑𝑑𝑣𝑣ℎ � 𝑞𝑞 𝜃𝜃ℎ 𝑣𝑣ℎ 𝑑𝑑Ωj 𝐷𝐷 𝜃𝜃 𝑣𝑣│𝑧𝑧𝑗𝑗−1 − � �𝐷𝐷 𝜃𝜃ℎ � 𝑑𝑑Ωj Ωj Ωj 𝑑𝑑𝑑𝑑 35 To simplify representation, the following expressions are introduced:

( , ) = ( ( ) ) (51)

𝑗𝑗 ℎ ℎ ℎ ℎ j 𝑄𝑄 𝜃𝜃 𝑣𝑣 �Ωj 𝑞𝑞 𝜃𝜃 𝑣𝑣 𝑑𝑑Ω ( ), = ( ) (52) 𝑗𝑗 ∗ ∗ 𝑧𝑧 ⟨𝐷𝐷 𝜃𝜃ℎ 𝑣𝑣ℎ⟩𝑗𝑗 𝐷𝐷 𝜃𝜃 𝑣𝑣│𝑧𝑧𝑗𝑗−1 ( , ) = ( ) (53) ′ ∗ ℎ 𝑗𝑗 ℎ ℎ ℎ 𝜕𝜕𝑣𝑣 j 𝐺𝐺 𝜃𝜃 𝑣𝑣 �Ωj �𝐷𝐷 𝜃𝜃 � 𝑑𝑑Ω Additionally, the boundary term can be expressed in terms𝜕𝜕𝜕𝜕 of numerical flux:

( ), = ∗ ∗ + ∗ − ℎ ℎ 𝑗𝑗 �𝑗𝑗−1 ℎ 𝑗𝑗−1 �𝑗𝑗 ℎ 𝑗𝑗 Hence, Eq. (50) is equivalent⟨𝐷𝐷 𝜃𝜃 𝑣𝑣to ⟩ 𝐷𝐷 ⋅ 𝑣𝑣 �𝑧𝑧 � − 𝐷𝐷 ⋅ 𝑣𝑣 �𝑧𝑧 �

( , ) = ( , ) (54) ∗ + ∗ − ′ 𝑗𝑗 ℎ ℎ �𝑗𝑗−1 ℎ 𝑗𝑗−1 �𝑗𝑗 ℎ 𝑗𝑗 𝑗𝑗 ℎ ℎ in which ( 𝑄𝑄, 𝜃𝜃) is𝑣𝑣 linear� 𝐷𝐷in both⋅ 𝑣𝑣 � 𝑧𝑧and � − and𝐷𝐷 the⋅ 𝑣𝑣 remaining�𝑧𝑧 �� − 𝐺𝐺 terms𝜃𝜃 𝑣𝑣are linear only in

𝑗𝑗 ℎ ℎ ℎ ℎ . This allows𝑄𝑄 𝜃𝜃 Eq.𝑣𝑣 (54) to be numerically𝜃𝜃 represented𝑣𝑣 as a matrix equation vis-à-vis Eq.

ℎ (30)𝑣𝑣 with respect to the same basis polynomials used in Chapter 3.1.3:

( , ) ( , ) ( , )

𝑄𝑄𝑗𝑗(𝜙𝜙1, 𝜙𝜙1) 𝑄𝑄𝑗𝑗(𝜙𝜙2, 𝜙𝜙1) ⋯ 𝑄𝑄𝑗𝑗(𝜙𝜙𝑛𝑛, 𝜙𝜙1) 𝑞𝑞1 ⎡ ⎤ ⎡ ⎤ LHS: 𝑗𝑗 1 2 𝑗𝑗 2 2 𝑗𝑗 𝑛𝑛 2 2 ⎢𝑄𝑄 𝜙𝜙 𝜙𝜙 𝑄𝑄 𝜙𝜙 𝜙𝜙 ⋯ 𝑄𝑄 𝜙𝜙 𝜙𝜙 ⎥ ⎢𝑞𝑞 ⎥ ⎢ ⎥ ⎢ ( ⋮ , ) ( ⋮ , ) ⋱ ( ⋮ , )⎥ ⎢ ⋮ ⎥ ⎢ ⎥ ⎢ 𝑗𝑗 1 𝑛𝑛 𝑗𝑗 2 𝑛𝑛 𝑗𝑗 𝑛𝑛 𝑛𝑛 ⎥ 𝑛𝑛 �⎣𝑄𝑄��𝜙𝜙��𝜙𝜙��𝑄𝑄��𝜙𝜙��𝜙𝜙��⋯��𝑄𝑄��𝜙𝜙��𝜙𝜙��⎦ �⎣𝑞𝑞 ⎦ 𝒋𝒋 ≡ 𝑸𝑸 ≡ 𝒒𝒒 ( ) 𝒋𝒋 ( , ) ( + ) − ( , ′ ) 𝜙𝜙1 𝑧𝑧𝑗𝑗−1 𝜙𝜙1�𝑧𝑧𝑗𝑗 � 𝐺𝐺𝑗𝑗 𝜃𝜃ℎ 𝜙𝜙1 RHS: ⎡ + ⎤ ⎡ − ⎤ ⎡ ′ ⎤ ∗ ⎢𝜙𝜙2 𝑧𝑧𝑗𝑗−1 ⎥ ∗ ⎢𝜙𝜙2�𝑧𝑧𝑗𝑗 �⎥ 𝐺𝐺𝑗𝑗 𝜃𝜃ℎ 𝜙𝜙2 𝐷𝐷�𝑗𝑗−1 ⋅ ( ) − 𝐷𝐷�𝑗𝑗 ⋅ − ⎢ ( , )⎥ ⎢ ⎥ ⎢ ⎥ ⋮ ⎢ ⋮ + ⎥ ⋮ − ⎢ ′ ⎥ 𝑛𝑛 𝑗𝑗−1 ⎢ 𝑛𝑛 𝑗𝑗 ⎥ 𝑗𝑗 ℎ 𝑛𝑛 �⎣𝜙𝜙��𝑧𝑧 ��⎦ �⎣𝜙𝜙��𝑧𝑧��⎦ �⎣𝐺𝐺��𝜃𝜃��𝜙𝜙��⎦ + − 𝒋𝒋 𝑗𝑗−1 𝑗𝑗 ≡ 𝑮𝑮 ≡ 𝚽𝚽�𝑧𝑧 � ≡ 𝚽𝚽�𝑧𝑧 �

36 Further transformation shows that = :

𝑸𝑸𝒋𝒋 𝑴𝑴𝒋𝒋 ( , ) = ( ) ( ) 𝑧𝑧𝑗𝑗 𝑗𝑗 𝑘𝑘 𝑙𝑙 𝑘𝑘 𝑙𝑙 𝑄𝑄 𝜙𝜙 𝜙𝜙 �𝑧𝑧𝑗𝑗−1𝜙𝜙 𝑧𝑧 ⋅ 𝜙𝜙 𝑧𝑧 𝑑𝑑𝑑𝑑 = ( ) ( ) 1 𝑑𝑑𝑍𝑍𝑗𝑗 � 𝜙𝜙𝑘𝑘 𝜉𝜉 ⋅ 𝜙𝜙𝑙𝑙 𝜉𝜉 𝑑𝑑𝑑𝑑 −1 𝑑𝑑𝑑𝑑 ( ( ), ( )) 2 Δ𝑧𝑧𝑗𝑗 ≡ ⋅ 𝜙𝜙𝑘𝑘 𝜉𝜉 𝜙𝜙𝑙𝑙 𝜉𝜉 0 Likewise,

( ) ( ) 1 ( ) ( ) ( , ) ∗ ′ ⎡� 𝐷𝐷 �𝜃𝜃ℎ 𝜉𝜉 � ⋅ 𝜙𝜙1 𝜉𝜉 𝑑𝑑𝑑𝑑⎤ ∗ ′ ( ′ ) −1 ( ) 1( ) 0 𝑗𝑗 ℎ, 1 ( ) ( ) �𝐷𝐷 𝜉𝜉 ⋅ 𝜙𝜙 𝜉𝜉 � = 𝐺𝐺 𝜃𝜃 𝜙𝜙 ⎢ 1 ⎥ ⎡ ∗ ′ ⎤ ⎡ ′ ⎤ 2 ∗ ′ 2 2 𝑗𝑗 ℎ 2 𝑗𝑗 ⎢ ℎ 2 ⎥ 𝑗𝑗 ⎢�𝐷𝐷 𝜉𝜉 ⋅ 𝜙𝜙 𝜉𝜉 �0⎥ 𝒋𝒋 𝐺𝐺 (𝜃𝜃 , 𝜙𝜙 ) Δ𝑧𝑧 � 𝐷𝐷 �𝜃𝜃 𝜉𝜉 � ⋅ 𝜙𝜙 𝜉𝜉 𝑑𝑑𝑑𝑑 Δ𝑧𝑧 𝑮𝑮 ⎢ ⎥ ↦ ⎢ −1 ⎥ ≡ ⎢ ( ) ( ) ⎥ ⎢ ⋮ ′ ⎥ ⎢ ⎥ ⎢ ⋮ ⎥ 𝑗𝑗 ℎ 𝑛𝑛 ( )⋮ ( ) ⎣𝐺𝐺 𝜃𝜃 𝜙𝜙 ⎦ ⎢ 1 ⎥ ⎢ ∗ ′ ⎥ ∗ ′ �𝐷𝐷 𝜉𝜉 ⋅ 𝜙𝜙𝑛𝑛 𝜉𝜉 �0 ⎢ ℎ 𝑛𝑛 ⎥ ⎣ ⎦ �−1𝐷𝐷 �𝜃𝜃 𝜉𝜉 � ⋅ 𝜙𝜙 𝜉𝜉 𝑑𝑑𝑑𝑑 In quadrature form, ⎣ ⎦

= [( ) ] ( ) (55) 2 Δ𝑧𝑧𝑗𝑗 ′ T ∗ ′ 𝑮𝑮𝒋𝒋 𝚽𝚽 𝑾𝑾𝒏𝒏 𝐷𝐷 𝜽𝜽𝒋𝒋𝚽𝚽 Therefore, Eq. (8) can be numerically represented by the following matrix equation:

[ ] = [( ) ] ( ) 2 2 (56) Δ𝑧𝑧𝑗𝑗 T ∗ + ∗ − Δ𝑧𝑧𝑗𝑗 ′ T ∗ ′ 𝚽𝚽 𝑾𝑾𝒏𝒏𝚽𝚽 ⋅ 𝒒𝒒𝒋𝒋 𝐷𝐷�𝑗𝑗−1 ⋅ 𝚽𝚽�𝒛𝒛𝒋𝒋−𝟏𝟏� − 𝐷𝐷�𝑗𝑗 ⋅ 𝚽𝚽�𝒛𝒛𝒋𝒋 � − 𝚽𝚽 𝑾𝑾𝒏𝒏 𝐷𝐷 𝜽𝜽𝒋𝒋𝚽𝚽 �� 𝒋𝒋 𝑫𝑫 𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩𝑩 𝑻𝑻𝑻𝑻𝑻𝑻𝑻𝑻 𝒋𝒋 ≡ 𝑴𝑴 ≡ 𝑮𝑮

or equivalently:

= + (57) ∗ ′ + ∗ − ∗ 𝒋𝒋 𝒋𝒋 𝒋𝒋 𝒋𝒋 �𝑗𝑗−1 𝒋𝒋 �𝑗𝑗 where 𝒒𝒒 −𝑨𝑨 ⋅ 𝐷𝐷 �𝜽𝜽 𝚽𝚽 � 𝑩𝑩 ⋅ 𝐷𝐷 − 𝑩𝑩 ⋅ 𝐷𝐷

= [ ] [( ) ] T −1 ′ T 𝑨𝑨𝒋𝒋 𝚽𝚽 𝑾𝑾𝒏𝒏𝚽𝚽37 ⋅ 𝚽𝚽 𝑾𝑾𝒏𝒏 = [ ] 2 −1 + Δ𝑧𝑧𝑗𝑗 T + 𝑩𝑩𝒋𝒋 � 𝚽𝚽 𝑾𝑾𝒏𝒏𝚽𝚽 � ⋅ 𝚽𝚽�𝒛𝒛𝒋𝒋−𝟏𝟏� = [ ] 2 −1 − Δ𝑧𝑧𝑗𝑗 T − 𝑩𝑩𝒋𝒋 � 𝚽𝚽 𝑾𝑾𝒏𝒏𝚽𝚽 � ⋅ 𝚽𝚽�𝒛𝒛𝒋𝒋 � The global matrix equation is therefore:

( ) 0 ∗ ∗( ′) 𝐷𝐷�1 (58) 𝒒𝒒𝟏𝟏 = 𝟏𝟏 + ⎡ ∗ ⎤ 𝐷𝐷∗ 𝜽𝜽 𝚽𝚽′ 2 𝒒𝒒𝟐𝟐 𝟐𝟐 ⎢𝐷𝐷� ⎥ 𝐷𝐷 (𝜽𝜽 𝚽𝚽 ) ∗ 𝒒𝒒 ≡ � � −𝑨𝑨 ⋅ � � 𝑩𝑩 ⋅ ⎢𝐷𝐷� ⎥ ⋮ ∗ ⋮ ′ 𝒏𝒏 ⎢ ⎥ 𝒒𝒒 𝒏𝒏 ⎢ ⋮∗ ⎥ 𝐷𝐷 𝜽𝜽 𝚽𝚽 𝑛𝑛 This concludes the DG formulation of Eq. (8). Going through⎣𝐷𝐷� ⎦ the same formulation procedures may seem tedious, but because the same basis polynomials and global matrices are used, computation of can be easily done in a MATLAB script with a few additional lines. 𝒒𝒒

For convenience of reference, the formulated global equations and their components are summarized in Chapter 3.1.9.

38 3.1.9 Equation Summary

PDE Form Numerical Form

0 0 ′ ′ 𝑞𝑞�1 − 𝐾𝐾�1 + = = 𝑞𝑞�𝜽𝜽𝟏𝟏𝚽𝚽 � − 𝐾𝐾�𝜽𝜽𝟏𝟏𝚽𝚽 � + ⎡ ⎤ ⎡ ′ ′ ⎤ 𝑞𝑞2 − 𝐾𝐾2 𝟐𝟐 𝟐𝟐 ⎢� � ⎥ 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 ̇ ⎢𝑞𝑞�𝜽𝜽 𝚽𝚽 � − 𝐾𝐾�𝜽𝜽 𝚽𝚽 �⎥ 𝚯𝚯 −𝑨𝑨 ⋅ ⎢ ⎥ 𝑩𝑩 ⋅ ⎢𝑞𝑞� − 𝐾𝐾� ⎥ 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 ′ ⋮ ′ ⎢ ⎥ ⎢ 𝒏𝒏 𝒏𝒏 ⎥ ⋮ ⎣𝑞𝑞�𝜽𝜽 𝚽𝚽 � − 𝐾𝐾�𝜽𝜽 𝚽𝚽 �⎦ ⎢ 𝑛𝑛⎥ ⎣𝑞𝑞�𝑛𝑛 − 𝐾𝐾� ⎦ ( ) 0 ∗ ∗( ′) 𝐷𝐷�1 = = 𝒒𝒒𝟏𝟏 = 𝟏𝟏 + ⎡ ∗ ⎤ ∗ 𝐷𝐷∗ 𝜽𝜽 𝚽𝚽′ 2 𝒒𝒒𝟐𝟐 𝟐𝟐 ⎢𝐷𝐷� ⎥ 𝜕𝜕𝜕𝜕 𝜕𝜕𝐷𝐷 𝐷𝐷 (𝜽𝜽 𝚽𝚽 ) ∗ 𝑞𝑞 𝐷𝐷 � � −𝑨𝑨 ⋅ � � 𝑩𝑩 ⋅ ⎢𝐷𝐷� ⎥ 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 ⋮ ∗ ⋮ ′ 𝒏𝒏 ⎢ ⎥ 𝒒𝒒 𝐷𝐷 𝜽𝜽𝒏𝒏𝚽𝚽 ⎢ ⋮∗ ⎥ Components ⎣𝐷𝐷�𝑛𝑛⎦

= [ ] 2 𝑗𝑗 T 𝒋𝒋 Δ𝑧𝑧 𝒏𝒏 𝑴𝑴 𝚽𝚽 𝑾𝑾 𝚽𝚽 … … [ ] [( ) ] 𝟏𝟏 = = 𝑨𝑨 𝟎𝟎 𝟎𝟎 T −1 ′ T 𝟐𝟐 𝒋𝒋 𝒏𝒏 𝒏𝒏 𝟎𝟎 𝑨𝑨 … 𝟎𝟎 𝑨𝑨 𝚽𝚽 𝑾𝑾 𝚽𝚽 ⋅ 𝚽𝚽 𝑾𝑾 𝑨𝑨 � � ⋮ ⋮ ⋱ ⋮ 𝟎𝟎 𝟎𝟎 𝑨𝑨𝒏𝒏 = [ ] … 2 −1 + Δ𝑧𝑧𝑗𝑗 T + + − … 𝒋𝒋 𝒏𝒏 𝒋𝒋−𝟏𝟏 = 𝟏𝟏 𝟏𝟏 𝑩𝑩 � 𝚽𝚽 𝑾𝑾 𝚽𝚽 � ⋅ 𝚽𝚽�𝒛𝒛 � 𝑩𝑩 −𝑩𝑩+ 𝟎𝟎 − 𝟎𝟎 ⎡ 𝟐𝟐 𝟐𝟐 ⎤ ⎢ 𝟎𝟎 𝑩𝑩 −…𝑩𝑩 𝟎𝟎 ⎥ = [ ] −1 𝑩𝑩 2𝑗𝑗 ⎢ ⋮ ⋮ ⋱ ⋱+ ⋮ −⎥ − Δ𝑧𝑧 T − 𝒏𝒏 𝒏𝒏 𝑩𝑩𝒋𝒋 � 𝚽𝚽 𝑾𝑾𝒏𝒏𝚽𝚽 � ⋅ 𝚽𝚽�𝒛𝒛𝒋𝒋 � ⎣ 𝟎𝟎 𝟎𝟎 𝑩𝑩 −𝑩𝑩 ⎦ Lax-Friedrichs Numerical Flux Bassi-Rebay Numerical Flux

Hydraulic Conductivity: Diffusivity: 1 = + 1 2 = + − + 2 𝑗𝑗 1 𝑗𝑗 𝑗𝑗 − + − + 𝑞𝑞� = �𝑞𝑞�𝜃𝜃 � +𝑞𝑞�𝜃𝜃 �� 𝑗𝑗 𝑗𝑗 𝑗𝑗 𝑗𝑗 𝑗𝑗 2 𝐾𝐾 �𝐾𝐾�𝜃𝜃 � 𝐾𝐾�𝜃𝜃 � − 𝛼𝛼 ⋅ �𝜃𝜃 − 𝜃𝜃 �� ∗ ∗ − ∗ + �𝑗𝑗 𝑗𝑗 𝑗𝑗 𝐷𝐷 �𝐷𝐷 �𝜃𝜃 � 𝐷𝐷 �𝜃𝜃 ��

39 3.2 Temporal Discretization

Time stepping schemes are commonly divided into two categories: explicit and

implicit. Explicit methods allow solutions of the current time step be directly computed

based on solutions of past times steps, whereas implicit methods require iteration. 56:

Explicit: = ( , , )

𝒏𝒏+𝟏𝟏 𝑛𝑛 𝑛𝑛−1 Implicit: 𝒖𝒖 = 𝑓𝑓(𝑢𝑢 𝑢𝑢, ,⋯ , )

𝒏𝒏+𝟏𝟏 𝒏𝒏+𝟏𝟏 𝑛𝑛 𝑛𝑛−1 This is, however, not indicative that𝒖𝒖 explicit𝑓𝑓 𝒖𝒖 methods𝑢𝑢 𝑢𝑢 will ⋯always outpace implicit

methods in computation, because explicit methods face more restriction on time step sizes in order to maintain stability which potentially demands more time step; implicit

methods can be unconditionally stable, allowing time step sizes to be more flexible 36,56.

In this model, the Crank-Nicolson method (aka the “trapezoidal rule”) is used,

which is implicit and second-order accurate in time 56:

= + [ ( ) + ( )] 2 Δ𝑡𝑡 𝚯𝚯𝒏𝒏+𝟏𝟏 𝚯𝚯𝒏𝒏 ⋅ 𝑓𝑓 𝚯𝚯𝒏𝒏+𝟏𝟏 𝑓𝑓 𝚯𝚯𝒏𝒏 where stands for solution of the -th time step and is of the subsequent.

𝒏𝒏 𝒏𝒏+𝟏𝟏 Convergence𝚯𝚯 criterion is met when 𝑛𝑛 𝚯𝚯

CC 10 × , , −6 𝒏𝒏 𝒎𝒎+𝟏𝟏 𝒏𝒏 𝒎𝒎 where abbreviates “convergence≤ � criterion”,�𝚯𝚯 ,− 𝚯𝚯 is �the� solution for the ( +1)-th

𝒏𝒏 𝒎𝒎+𝟏𝟏 iteration𝐶𝐶𝐶𝐶 level in the -th time step, and , is 𝚯𝚯of the -th. That is, the model proceeds𝑚𝑚

𝒏𝒏 𝒎𝒎 to the next time step 𝑛𝑛when absolute error𝚯𝚯 of iteration falls𝑚𝑚 below 0.0001% of the current

iteration result.

To begin time stepping, the initial moisture profile ( ) and a first guess on the

𝟎𝟎 moisture profile after the first time step ( ) are necessary𝚯𝚯. As changes of moisture

𝟏𝟏 40𝚯𝚯 profile over a very brief period could be extremely unnoticeable, throughout the entire

time stepping process the moisture profile determined for a time step is used as the first

guess for its subsequent time step, i.e.

⋯

𝐟𝐟𝐟𝐟𝐟𝐟𝐟𝐟𝐟𝐟 𝐠𝐠𝐠𝐠𝐠𝐠𝐠𝐠𝐠𝐠 = , , , 𝑪𝑪𝑪𝑪 𝐢𝐢𝐢𝐢 𝐦𝐦𝐦𝐦𝐦𝐦

�𝒏𝒏 𝒏𝒏+𝟏𝟏 𝟏𝟏 𝒏𝒏+𝟏𝟏 𝟐𝟐 𝒏𝒏+𝟏𝟏 𝒎𝒎 ��𝒏𝒏+�𝟏𝟏 𝚯𝚯 𝚯𝚯 → 𝚯𝚯 → ⋯ → 𝚯𝚯 → ⋯ → 𝚯𝚯

= , , ,

�𝚯𝚯�𝒏𝒏+�𝟏𝟏 𝚯𝚯𝒏𝒏+𝟐𝟐 𝟏𝟏 → 𝚯𝚯𝒏𝒏+𝟐𝟐 𝟐𝟐 → ⋯ → 𝚯𝚯𝒏𝒏+𝟐𝟐 𝒎𝒎 → ⋯ → �𝚯𝚯�𝒏𝒏+�𝟐𝟐 𝐟𝐟𝐟𝐟𝐟𝐟𝐟𝐟 𝐠𝐠𝐠𝐠𝐠𝐠𝐠𝐠𝐠𝐠 𝑪𝑪𝑪𝑪 𝐢𝐢𝐢𝐢 𝐦𝐦𝐦𝐦𝐦𝐦

⋯

3.2.1 Initial Condition

The initial moisture profile ( ) is specified by assigning a value to every

𝟎𝟎 element. In the verification process of𝚯𝚯 this model (as will be explained𝜃𝜃 in Chapter 4),

moisture sensors were used to obtain moisture readings in certain locations in a sand

column. These point-wise readings were used to create a uniform initial profile for every

element, as illustrated in Fig. 4 by the asterisks and blue line segments. Also shown is a

polynomial curve fitted to the sensor readings. While it could be plausible to fill the gaps

between readings using this polynomial curve, considerable error could be introduced at

locations relatively far from the known values, such as in the upper part of the top layer.

Figure 4: Vertical Discretization and Moisture Content. The asterisks indicate depths and readings of sensors. The red dashed lines are center dividing lines of every pair of adjacent sensor readings. The blue line segments indicate uniform profiles in layers based on sensor readings. The curve is a cubic polynomial fitted to the asterisks, which may be used to approximate unknown values near a series of known points but could lead to considerable error further away.

42 3.3 Boundary Conditions and Sink/Source Terms

Boundary conditions are specified via modifying the flux vectors. The bottom

boundary flux for drainage, , is computed based on , and of the bottom 𝑛𝑛 �𝑛𝑛 layer and takes care of itself�𝑞𝑞�. The− 𝐾𝐾 top� boundary flux, 𝜃𝜃 𝐾𝐾, is set𝑞𝑞 equal to the

0 0 difference between rates of input (e.g. rainfall intensity) �and𝑞𝑞� − output𝐾𝐾� � (e.g. evaporation),

which can be either a specific value or a net equation describing a dynamic relation.

No lateral flows or vegetation intake of water were assumed in this model. If such processes were to be modeled, a global sink/source term, ( ), would be introduced on the RHS of the RE and formulated together with the flux term𝑠𝑠 𝜃𝜃:

+ ( ) ( ) = ( ) 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕𝜕𝜕 �𝐾𝐾 𝜃𝜃 − 𝐷𝐷 𝜃𝜃 � 𝑠𝑠 𝜃𝜃 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 + ( ) = ( ) + ( ) 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕 𝜕𝜕𝜕𝜕 → 𝐾𝐾 𝜃𝜃 �𝐷𝐷 𝜃𝜃 � 𝑠𝑠 𝜃𝜃 Multiplying ( ) by a smooth𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 test function𝜕𝜕𝜕𝜕 and 𝜕𝜕𝜕𝜕approximating the solution using

polynomials belonging𝑠𝑠 𝜃𝜃 to the space of would result in the following: 𝑝𝑝 ℙ ( ) ( ( ) ) ( , )

ℎ ℎ j 𝑗𝑗 ℎ ℎ 𝑠𝑠 𝜃𝜃 ↦ �Ωj 𝑠𝑠 𝜃𝜃 ⋅ 𝑣𝑣 𝑑𝑑Ω ≡ 𝑆𝑆 𝜃𝜃 𝑣𝑣 As regards Eq. (8), ( , ) can be formulated like and . Similarly, and are

𝑗𝑗 ℎ ℎ 𝑗𝑗 𝑗𝑗 ℎ ℎ expanded using the 𝑆𝑆basis𝜃𝜃 polynomials𝑣𝑣 given in Chapter𝑄𝑄 3.1.3𝐺𝐺; entries are mapped𝜃𝜃 𝑣𝑣to the

master element given in Chapter 3.1.4 and rewritten in quadrature forms:

43 ( , ) 𝑛𝑛 ( , )

𝑆𝑆𝑗𝑗 𝜃𝜃ℎ 𝑣𝑣ℎ ↦ � 𝑆𝑆𝑗𝑗 𝜃𝜃ℎ 𝜙𝜙𝑖𝑖 𝑖𝑖=1 ( ) ( ) 1 ⎡� 𝑠𝑠�𝜃𝜃ℎ 𝜉𝜉 � ⋅ 𝜙𝜙1 𝜉𝜉 𝑑𝑑𝑑𝑑⎤ −1 ( ) ( ) ⎢ 1 ⎥ 2𝑗𝑗 ⎢ ⎥ Δ𝑧𝑧 ℎ 2 ↦ ⎢�−1𝑠𝑠�𝜃𝜃 𝜉𝜉 � ⋅ 𝜙𝜙 𝜉𝜉 𝑑𝑑𝑑𝑑⎥ ⎢ ( ) ( ) ⎥ ⎢ 1 ⋮ ⎥ ⎢� 𝑠𝑠�𝜃𝜃ℎ 𝜉𝜉 � ⋅ 𝜙𝜙𝑛𝑛 𝜉𝜉 𝑑𝑑𝑑𝑑⎥ ⎣ −1( ) ( ) ⎦ ( ) ( ) �𝑠𝑠 𝜉𝜉 ⋅ 𝜙𝜙1 𝜉𝜉 �0 ⎡ ⎤ 2 2 Δ𝑧𝑧𝑗𝑗 ⎢�𝑠𝑠 𝜉𝜉 ⋅ 𝜙𝜙 𝜉𝜉 �0⎥ ↦ ⎢ ( ) ( ) ⎥ ⎢ ⋮ ⎥ ⎢ 𝑛𝑛 ⎥ ⎣�𝑠𝑠 𝜉𝜉 ⋅ 𝜙𝜙 𝜉𝜉 �0⎦ [ ] ( ) 2 Δ𝑧𝑧𝑗𝑗 T ′ ↦ 𝚽𝚽 𝑾𝑾𝒏𝒏 ⋅ 𝑠𝑠 𝜽𝜽𝒋𝒋𝚽𝚽 Hence, the global matrix equation with ( ) is as follows:

𝑠𝑠 𝜃𝜃 0 0 ( ) 1 ′ ′ 𝑞𝑞�1 − 𝐾𝐾� ( ′) = 𝑞𝑞 𝜽𝜽𝟏𝟏𝚽𝚽 − 𝐾𝐾 𝜽𝜽𝟏𝟏𝚽𝚽 + ⎡ ⎤ + 𝟏𝟏 � � � � 2 2 𝑠𝑠 𝜽𝜽 𝚽𝚽 ⎡ ′ ′ ⎤ ⎢𝑞𝑞� − 𝐾𝐾� ⎥ ′ ⎢𝑞𝑞�𝜽𝜽𝟐𝟐𝚽𝚽 � − 𝐾𝐾�𝜽𝜽𝟐𝟐𝚽𝚽 �⎥ 𝑠𝑠(𝜽𝜽𝟐𝟐𝚽𝚽 ) 𝚯𝚯̇ −𝑨𝑨 ⋅ 𝑩𝑩 ⋅ ⎢𝑞𝑞 − 𝐾𝐾� ⎥ 𝑪𝑪 ⋅ � � ⎢ ⎥ � ⋮ ⎢ ′ ⋮ ′ ⎥ ⎢ ⎥ ′ 𝑞𝑞 𝜽𝜽𝒏𝒏𝚽𝚽 − 𝐾𝐾 𝜽𝜽𝒏𝒏𝚽𝚽 ⎢ ⋮ ⎥ 𝑠𝑠 𝜽𝜽𝒏𝒏𝚽𝚽 ⎣ � � � �⎦ 𝑛𝑛 𝑛𝑛 where ⎣𝑞𝑞� − 𝐾𝐾� ⎦

= [ ] [ ] 2 T −1 Δ𝑧𝑧𝑗𝑗 T 𝑪𝑪 𝚽𝚽 𝑾𝑾𝒏𝒏𝚽𝚽 ⋅ 𝚽𝚽 𝑾𝑾𝒏𝒏 Note that the expression of ( ) could vary element by element depending on the natures of the sink/source terms. 𝑠𝑠 𝜃𝜃

Chapter 4: Materials and Methods

The goal was to prove that this DG model is computationally efficient as well as

consistent in runoff partitioning. Chapter 4.1 explains how real-time moisture data was

collected. Chapter 4.2 compares this DG model with another Richards’ equation (RE)

solver based on a Continuous Galerkin formulation.

4.1 Data Collection

The experiment was conducted in Room 750G of the Biological Sciences

Greenhouse at the Ohio State University. The type of soil selected was play sand

because of its high hydraulic conductivity, allowing more trials to be conducted in a

given time frame than loamy or clayey soils. Real time volumetric soil moisture content

was measured using four batches of six 5TE soil moisture, temperature, & electrical

conductivity sensors and retrieved with a data logger 16; the 5TE sensors measure

dielectric permittivity ( ) in the range of 1 to 80, and convert dielectric permittivity to

𝑎𝑎 volumetric moisture content𝜖𝜖 ( ) using the Topp equation to an accuracy of ±3% and a resolution of 0.0008 m3/m3. Scanning𝜃𝜃 interval was set to 120 seconds in order for the data logger to fully register all sensor readings. The use of 4 sensors per layer was to account

45 for hysteresis and discrepancies. For more details on configuration, please refer Appendix

Data collection consisted of two major processes. The saturation-drainage

process conditioned the sand column and produced dry-down curves for parameter fitting; the rainfall simulation process produced moisture time series for model

verification. Descriptions of the processes are as follows:

1. Saturation-Drainage:

. Bottom-up saturation: A running hose was connected to the faucet, and

the sand column was allowed to saturation from the bottom until ponding

occurred at its surface. Ideally, this would force all air out of the sand

column, achieved a uniform saturated moisture profile, and reduce the

possibility of forming preferential flow paths in later experiment.

. Constant-head drainage: After ponding occurred, the running hose was

removed from the faucet and place at the sand surface to maintain a

constant ponding level while allowing drainage from the bottom.

Meanwhile, the moisture profile obtained after bottom-up saturation was

expected to remain constant.

. Free drainage: The running hose was removed from the sand surface

after some period, and the sand column was allowed to drain freely.

Ideally, this would produce the necessary dry-down curves for parameter

fitting ( and ). To elaborate, a layer of the sand column would be

𝑏𝑏 selected,𝜓𝜓 and the𝜆𝜆 model would be run with arbitrary combinations of

𝑏𝑏 and to produce simulated dry-down curves. The values of and𝜓𝜓

𝜆𝜆 46 𝜓𝜓𝑏𝑏 𝜆𝜆 would be adjusted (within the range provided in [Rawls et al, 1982]) and

the model re-run until a simulated dry-down curve closely resembles the

actual dry-down curve.

2. Rainfall Simulation:

. Rainfall and hiatus: Sprinklers above the rain barrel were turned on to

simulate precipitation. Intensities of the rains were adjusted to include

heavy (> 10 mm/hr) and extreme (> 40 mm/hr) events. A hiatus period

was inserted between every two rainfall events. The rainfall-hiatus regime

designed was entirely random.

In terms of simulation, the DG model was operated using 6 (1 element per layer),

12 (2 elements per layer) and 18 elements (3 elements per layer). The simulated time series was directly compared to the sensor readings on plots of vs. time, as well as on point-to-point scattered plots. Coefficient of determination ( 𝜃𝜃) of each point-to-point 2 plot with regard to the 1:1 line was computed to reflect 𝑅𝑅how well the DG model represented real data and whether or not improvement in accuracy was achieved with more elements.

Due to lack of suitable equipment, evaporation rate in the greenhouse and at the sand surface was undetermined. Data analysis was first done without an evaporation scheme. An ad-hoc evaporation scheme was later added to the model, and the re- simulated results were compared to their early counterparts. More details on the ad-hoc evaporation scheme are provided in Chapter 5.2.

47 4.2 Model Comparison

Ability of the DG model to converge under extreme event and to consistently

partition runoff was studied via comparison with a continuous Galerkin (CG) RE solver

incorporated in the tRIBS+VEGGIE model – a physically-based comprehensive

hydrologic model developed in the Bras Lab at Massachusetts Institute of Technology

that simulates hydrologic processes including surface energy balance, infiltration, and

overland flow 26. The CG solver formulated the 1D -based RE like in this thesis. A

major difference between these two solvers is that in contrast𝜃𝜃 to the block-diagonal global

matrices resulted from DG formulation, and CG formulation generates a tridiagonal

global iteration matrix. For more details on the CG formulation, please refer to Appendix

D in the work of Ivanov 25.

An 11-year rainfall dataset of the Kendall Basin in the Walnut Gulch Experiment

Watershed of Arizona, USA, was retrieved for simulation and comparison. The dataset

consists of a total 96409 hours of recorded rainfall events and hiatuses. Out of the 1908

hours of rainfall events, about 7% had intensities over 5 mm/hr, 2% had over 10 mm/hr,

and 0.1% had over 40 mm/hr.

Two simulations were conducted. Settings for Simulation I and II are summarized

respectively in Table 2 and Table 3. The objective of Simulation I was to check if both

solvers would be able to converge under intense rainfalls and successfully finish the

entire 96409-hour span. The objective of Simulation II was to compare runoff partitioning results for different vertical discretization using the first 2000 hours of the

96409-hour span. No extreme rainfall event was recorded in the first 2000 hours; both

48 models were expected to converge without any potential adjustment made. Clayey soil

( 1 mm/hr) was chosen to ensure runoff would occur during simulation.

𝑠𝑠 𝐾𝐾 ≈

Table 2: Default Settings of Simulation I

Soil Type: Sandy 48 Initial Condition: 0.5 × uniformly throughout the domain

Soil Depth: 1000 mm𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠 Number of Elements: 20 * (uniform )

Time Step Size: 1 min * Δ𝑧𝑧 Convergence Criterion: Error within 0.0001% of current time step estimation Win 7 x64, Intel i5-6600K, CPU 2x 3.50GHz, RAM 16.0 Operating System: GB

*: Would be adjusted if model failed to meet convergence criterion.

Table 3: Default Settings of Simulation II

Soil Type: Clayey 48 Initial Condition: 0.5 × uniformly throughout the domain

Soil Depth: 1000 mm𝜃𝜃𝑠𝑠𝑠𝑠 𝑠𝑠 Number of Elements: 10, 20, 30, 40, 50 (uniform )

Time Step Size: 1 min Δ𝑧𝑧 Convergence Criterion: Error within 0.0001% of current time step estimation Win 7 x64, Intel i5-6600K, CPU 2x 3.50GHz, RAM 16.0 Operating System: GB

*: Would be adjusted if model failed to meet convergence criterion.

Chapter 5: Results and Discussion

Parameter fitting is explained in Chapter 5.1, followed by discussion on rainfall

time series in Chapter 5.2. Note that simulation results in Chapter 5.1 and 5.2 were

generated using 6 elements. Mass conservation and model sensitivity using more

elements are discussed later in Chapter 5.2.3 alongside the influence of vertical

discretization and top boundary specification on accuracy. Discussed in Chapter 5.3 is the

capability of this DG model to handle extreme rainfall events and consistently partition runoff versus a Continuous Galerkin (CG) model.

Due to persistent technical difficulties, Batch 4 sensors frequently malfunctioned.

Analyses used only readings of sensors from Batch 1, 2 and 3.

5.1 Parameter Fitting

After the saturation-drainage process, soil moisture readings of all six layers were

retrieved and plots. Please refer to Fig. 5, Fig. 6 and Fig. 7 for still images; a web-link to

an animated time series plot is provided in the description of Fig. 7.

Fig. 5 depicts an instant during constant-head drainage when the moisture profile

was stabilized, in which noticeable standard deviation was seen in almost all layers.

Ideally, the saturated moisture profile would have been more uniform. Discrepancies

50 could have been caused by inconsistent compaction pressure applied during the sensor

installation process, which led to a change in porosity in certain parts of the sand column.

Also observed is that almost not standard deviation is seen in Layer 2, implying that

Layer 2 was more uniformly compacted than the other layers. Hence, for parameter

fitting, the dry-down curve of Layer 2 was used.

Fig. 6 shows the dry-down time series of Layer 2. The average soil moisture

readings (of Batch 1, 2 and 3) over time were plotted with standard deviations, resulting

in a band indicative of the range of hysteresis in Layer 2. In the DG model, dry-down in

Layer 2 was simulated using different combinations of and with adherence to their

𝑏𝑏 typical ranges provided in Rawls et al. 48: 𝜓𝜓 𝜆𝜆

[mm]: Avg. = 72.6 (Range = 13.6 ~ 387.4)

𝑏𝑏 𝜓𝜓 [-]: Avg. = 0.592 (Range = 0.334 ~ 1.051)

A simulated dry𝜆𝜆-down curve that fell within the band was obtained. The estimated soil

parameters are summarized in Table 4. Procedures to estimate , and are

𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠 𝑟𝑟𝑟𝑟𝑟𝑟 provided in Appendix B. 𝐾𝐾 𝜃𝜃 𝜃𝜃

Table 4: Soil Parameters for Play Sand

Soil Type [--] [--] [mm/hr] [mm] [--]

Commercial 𝒔𝒔 𝒓𝒓 𝒔𝒔 𝒃𝒃 0𝜽𝜽.3799 0𝜽𝜽.00066 𝑲𝑲 205.78 𝝍𝝍 50 𝝀𝝀0.64 Play Sand −

With reference to Fig. 7 and the animated time series, soil moisture level in lower

layers interestingly underwent minimal change over the drying period. Moisture time

series for pulse inputs are shown in Fig. 8. Consistently observed in Layer 6 is a slight

51 decrease in moisture content briefly after each pulse, which replenishes in about 30 minutes (differing from sudden spikes that recover after one scanning period). This was possibly due to an upward pressure exerted by the fabric that countered the downward movement of soil moisture, even though the fabric was assured to be permeable prior to installation.

Prior to rainfall simulation, a pulse input sequence was designed to validate the above speculation. Each pulse was created by pouring approximately 2.5 × 10 mm3 of 6 water (i.e. 5-inch depth of water in a bucket 10 inches in diameter) onto the sand surface, with a 24-hour interval between every two pulses. Presumably, within a short period after a pulse air in the sand column would be compressed. The upward pressure would be temporarily overcome, allowing a minimal amount of water to be “squeezed” out from the bottom.

A video documentation of a pulse input was made (see description under Fig. 8).

Shortly after a pulse, a small amount of water was seen flowing out from the faucet. This could not possibly be the newly added water as regards time needed for it to pass through the entire sand column. In conjunction with Fig. 8, the speculation was confirmed.

Figure 5: Soil Depth vs. Moisture Content at Saturation. The six round markers stand for the average readings of six layers, and the error bars represent the respective standard deviations. Of all six layers, Layer 2 had all sensor readings almost perfectly agreed with one another.

Figure 6: Dry-down Time Series of Layer 2. A dry-down period of over 5000 minutes was allowed. The orange band was plotted using standard deviation of the three batches of sensor reading, which is also indicative of hysteresis of wetting and drying in Layer 2. The estimated parameters are: , . 𝝍𝝍𝒃𝒃 ≈ −𝟔𝟔𝟔𝟔 𝒎𝒎𝒎𝒎 𝝀𝝀 ≈ 𝟓𝟓𝟓𝟓

53 54

Figure 7: Dry-down Times Series of All Layers. Fluctuation in the curves during the period of 1000 – 3500 minutes was likely due to some instability of voltage in the data logger, which also resulted in sharp spikes as can be seen in other time series plots. Note that this figure does not include the standard deviation of the sensor readings, which was illustrated in the animated plot of Chapter 5.1. Link to animated plot: https://drive.google.com/file/d/0B3MVaYQhsYmCeDFFbDBOWHlEb2c/view?usp=sharing) 54 55

Figure 8: Moisture Time Series of Pulse Inputs. Enclosed in black, dashed-line circles are the minor but consistent changes in moisture reading of the bottom layer. It was observed that upon every pulse input, a small amount of water was released from the bottom. Link to footage: https://drive.google.com/open?id=0B3MVaYQhsYmCOXAtajlaWGIxTEE

55 5.2 Rainfall Simulation

The rainfall-hiatus regime is summarized in Table 5. Three rainfall events were

successfully simulated over a period of 4477 minutes (~3 days) with intensities of 10+,

30+ and 40+ mm/hr. Accuracy of the DG model is evaluated based on time series comparison and point-to-point comparison among measured and simulated results.

Table 5: Rainfall/Hiatus Regime

Rainfall Duration (min) Hiatus (min) Intensity (mm/hr) 103 30.10 60 - 276 42.75 1072 - 1440 13.57 1526 -

5.2.1 Times Series

Times series plots of all layers across the entire rainfall-hiatus period were created

(Fig. 9 thru 15). Discussion on each layer is as follows:

Layer 1. Fig. 9 shows good agreement between the simulation result and the

sensor readings during precipitation. However, also prominent are the

discrepancies in dry-down curves during the second and third hiatuses, during

which the DG model over-predicted moisture in the first layer possibly due to

the lack of an evaporation scheme (sink term).

In an attempt to show how a sink term in the top boundary could affect

model prediction, an ad-hoc linear evaporation scheme was added to modify 56 the top boundary flux (please refer back to Chapter 3 for numerical fluxes and

specification of boundary conditions):

0.5 × ( ), if no rain = − rainfall𝜃𝜃𝑡𝑡𝑡𝑡 intensity𝑡𝑡 − 𝜃𝜃𝑟𝑟𝑟𝑟𝑟𝑟 , if rains 𝑞𝑞�0 − 𝐾𝐾�0 � where 0.5 is an arbitrary rate constant and is the moisture content of the

𝑡𝑡𝑡𝑡𝑡𝑡 top layer. Basically this scheme depletes 𝜃𝜃moisture in the top layer linearly

down to during hiatuses. The DG model was re-run, and simulation

𝑟𝑟𝑟𝑟𝑟𝑟 results with𝜃𝜃 the evaporation scheme were overlain on the original (Fig. 15).

Noticeably, gaps between the measured and simulated dry-down curves closed

up. It ought to be highlighted that surface evaporation is a far more

complicated process – involving ground heat flux, air temperature, humidity,

and many other factors – than this ad-hoc scheme can describe.

Layer 2. Similar to Layer 1, good agreement was achieved during rains

while disagreement lied mainly in dry-down periods (Fig. 10). Because

moisture content in the first layer was over-predicted due to lack of

evaporation, more moisture was transmitted from the top layer into lower

layers. On a side note, the Batch 1 sensor in Layer 2 could be experiencing

voltage instability during the experimental process, producing a band instead

of a steady line.

Layer 3. Having no evaporation in Layer 1 clearly affected Layer 3 as well,

as can be inferred from the wide gaps between the simulated and measured

dry-down curves (Fig. 11). Otherwise, good agreement is seen during rainfall

events.

57 Layer 4. Gaps between simulated and measured dry-down curves are still

visible but apparently much narrower than in the upper layers (Fig. 12). This

highlights the importance of boundary condition specification in accurately

predicting overall soil moisture profiles.

Layer 5. Sensor readings appear less smooth than those in the upper layers

likely due to voltage issues (Fig. 13). Moisture readings of Batch 2 and 3

sensors agree with the model prediction in general trends. However, the Batch

1 curve (red) behaves very differently than not just the model but also Batch 2

and 3, possibly due to a combination of hysteresis and inconsistency in

compaction.

Layer 6. Of all layers, sensor readings of Layer 6 deviate from model

prediction to the greatest extent, most likely due to directly experiencing the

upward pressure from the fabric (Fig. 14). More irregularity can be seen in all

sensor curves during dry-down periods. The model almost under-predicted

moisture content throughout the entire simulation.

58 59

Figure 9: Rainfall Times Series of Layer 1. Overestimation of the profile during hiatuses is likely due to the lack of an evaporation scheme in the model. Otherwise, the simulation curve reflects trends in the measurement well.

59 60

Figure 10: Rainfall Time Series of Layer 2. Similar to Layer 1, over-prediction is observed during dry-down period. The simulation curve adheres more closely to the Batch 1 curve than to the other two. However, possibly due to voltage issue, Batch 1 curve appears as a band rather than a smooth line, which brings question to how accurate the readings of Batch 1 were.

60 61

Figure 11: Rainfall Time Series of Layer 3. Influence of the lack of an evaporation scheme is seen during dry-down periods. Otherwise, the model did a good job at predicting.

61 62

Figure 12: Rainfall Time Series of Layer 4. Even though gaps during dry-down periods are still noticeable, they appear narrower than in upper layers. The model somehow over-predicted during the third rain.

62 63

Figure 13: Rainfall Time Series of Layer 5. Some instability of readings is observed in all batches. Nonetheless, the simulation curve follows trends of Batch 2 and 3, whereas the Batch 1 curve is somewhat random.

63 64

Figure 14: Rainfall Time Series of Layer 6. Irregularity is prominent in all three measurement curves, likely due to the combined effect of voltage, inconsistent compaction and the fabric. The model was unable to predict moisture content of this layer well.

64 65

Figure 15: Rainfall Time Series of Layer 1 (with Evaporation). Gaps during dry-down periods are narrowed with the addition of an ad-hoc evaporation scheme. However, because evaporation is a much more complicated and non-linear process than the ad-hoc scheme can describe, shape of the green dry-down curve appears less similar to not only the measurement curves but also the simulation curve without evaporation (orange).

65 5.2.2 Point-to-Point

Visual comparison of time series falls short in quantifying how representative the

model can be. In conjunction, a point-to-point comparison figure (Fig. 16) was created

and the coefficient of determination (R2) of every sensor batch with respect to the 1:1 line

calculated.

Fig. 16 consists of six subplots, each representing a layer. Every subplot is halved

into two triangular regions (grey and yellow) by a 1:1 line. If a point falls in the yellow

region, the model over-predicted that point; if a point appears in the grey region, the

model under-predicted that point. Perfect prediction occurs where points fall exactly on

the 1:1 line.

As regards Layer 1 thru 4, data points generally stay close to the 1:1 line near

both ends – around the coordinates (0.1, 0.1) and (0.3, 0.3) – while having more

separation in between. Overall, more points are seen in the yellow region, indicating over-prediction by the model which is in agreement with discussion on the time series plot (Chapter 5.2.1). Moreover, interestingly shown in these subplots are “pockets”

formed by data points. These “pockets” are an indication of soil hysteresis. For example,

in the subplot of Layer 1, the string of red data points in the grey region follows a

wetting/drying path different from the other string of red data points in the yellow region,

despite both strings meet at almost the same starting and ending points. The smaller the

“pockets,” the weaker the hysteretic effect (e.g. less hysteresis is seen in Layer 4 than

Layer 1).

On the other hand, much more randomness of data points is seen in Layer 5 and 6

than in the upper layers. In Layer 5 and 6, more points appear in the grey region,

66 concurring with the earlier observation of moisture retention near the bottom (Chapter

5.1).

In addition to the point-to-point comparison, the coefficients of determination

(R2) of all sensor batches with reference to the 1:1 line were calculated and summarized

in Table 6. Additional R2 values with the ad-hoc evaporation scheme are also included.

As indicated by the column of “+/-” in R2, the addition of the ad-hoc evaporation scheme led to improvement in model prediction, especially in Layer 1 where R2 on average

increased by almost 0.200. Negative R2 were obtained in Layer 5 and Layer 6, implying

that the model prediction was entirely unrepresentative of the actual measurement, as can

be inferred from their time series (Fig. 13 and Fig. 14).

Table 6: Coefficients of Determination

2 Sensor R

Batch Without Evap. With Evap. +/– 1 0.750 0.879 +0.129 Layer 1 2 0.662 0.891 +0.230 3 0.676 0.901 +0.225 1 0.873 0.897 +0.024 Layer 2 2 0.853 0.932 +0.079 3 0.852 0.951 +0.099 1 0.740 0.913 +0.173 Layer 3 2 0.724 0.914 +0.190 3 0.680 0.884 +0.204 1 0.494 0.627 +0.133 Layer 4 2 0.738 0.875 +0.138 3 0.593 0.711 +0.118 1 –2.074 –1.462 +0.612 Layer 5 2 0.819 0.864 +0.045 3 0.831 0.907 +0.076 1 0.818 0.799 –0.019 Layer 6 2 0.462 0.589 +0.128 3 –0.604 –0.148 +0.455

Average 0.723 0.846 +0.123 ± Standard Deviation ±0.125 ±0.109 ±0.148

68 69

Figure 16: Point-to-point Comparison between Measurement and Simulation. Plots were created by plotting sensor readings (y-axis) against model prediction (x-axis). Each plot area is divided into two regions by a 1:1 line. If a point falls in the yellow region, that point is over-predicted by the mode. Conversely, a point is under-predicted if it appears in the grey region. Perfect prediction is when a point falls on the boundary of the two regions. Also shown are hysteretic effects. 69 5.2.3 Mass Conservation and Sensitivity Study

Mass balance errors were computed by the following equation:

Error = 1 × 100% 𝐴𝐴model � − sensor� in which is the area under the model𝐴𝐴 prediction curve in the times series plots and

model is𝐴𝐴 the area under a sensor batch curve on the same time series plot. Should the

sensor mass𝐴𝐴 balance error be negative, the model over-predicted; should it be positive, the model under-predicted. On the other hand, model sensitivity was studied via simulation using 12 and 18 elements (N = 12, 18) in addition to all results based on 6 elements presented thus far. Summarized in Table 7 are R2 and mass balance error (%) of 6, 12 and 18 elements

with and without the ad-hoc evaporation scheme. Fig. 17 thru 20 are visual

representations of the numbers and trends embedded in Table 7. Note that in Fig. 18 and

20, absolute errors are used to facilitate comparison; the negative signs are retained in

Table 7 to indicate over- and under-prediction.

. Fig. 17 compares R2 values of simulation without the ad-hoc evaporation

scheme. Consistently shown is that R2 values of N=6 are the lowest in all

layers and batches. While R2 values of N=18 are generally the highest,

they separate minimally from their N=12 counterparts.

. Fig. 18 compares the absolute errors of simulation results without the ad-

hoc evaporation scheme. Errors of N=6 consistently appear above their

N=12 and N=18 counterparts except in Layer 2 of Batch 1 and Batch 2

where N=6 produced the lowest errors despite the small discrepancies

among all three data series.

70 . Fig. 19 compares R2 values of simulation results with the ad-hoc

evaporation scheme. While N=6 have generally the lowest in values

(except in Layer 5 and 6), the three data series are much closer to one

another in contrast to Fig. 16. It should be noted that while N=18 data

points are above all in Layer 1, N=12 data points are the highest in Layer

2 thru 4.

. Fig. 20 compares the absolute errors of simulation results with the ad-hoc

evaporation scheme. Regardless of Layer 5 and 6, N=6 show the greatest

error in Layer 1, 3 and 4, but the lowest in Layer 2. Except in Layer 1

where relatively significant gaps are shown, data series of N=12 and N=18

differ minimally.

Overall, simulation using 12 and 18 elements generated higher R2 values and

smaller errors than using 6 elements. For a fixed number of elements, the addition of an ad-hoc evaporation scheme reduced error by approximately 10% in Layer 1 and about

5% in Layer 3 and 4; insignificant improvement was achieved in Layer 2. Simulation

with 18 elements produced the highest average R2 of 0.766 (without evaporation) and

0.858 (with ad-hoc evaporation), whereas using 6 elements resulted in the lowest average

R2 of 0.723 (without evaporation) and 0.846 (with ad-hoc evaporation). That the addition of even a simple, unrealistic evaporation scheme improved R2 from ~0.75 to ~0.85

emphasizes the importance of correctly specifying the top boundary condition.

Table 7: Comparison of R2 and Error

Without Evaporation Scheme With Ad-Hoc Evaporation Scheme Sensor N = 6 N = 12 N = 18 N = 6 N = 12 N = 18 Batch Error Error Error Error Error Error R2 R2 R2 R2 R2 R2 (%) (%) (%) (%) (%) (%) 1 0.750 –37.3 0.803 –30.6 0.803 –30.0 0.879 –24.7 0.894 –20.4 0.921 –16.3 Layer 2 0.662 –22.5 0.747 –16.3 0.762 –16.2 0.891 –11.8 0.938 –6.13 0.942 –3.93 1 3 0.676 –20.8 0.752 –14.7 0.764 –14.5 0.901 –10.2 0.931 –4.48 0.941 –2.46 1 0.873 –1.30 0.869 2.32 0.873 3.02 0.897 3.97 0.916 4.80 0.901 5.33 Layer 2 0.853 –1.52 0.858 2.10 0.868 2.80 0.932 3.74 0.941 4.54 0.931 5.08 2 3 0.852 –4.95 0.871 –1.20 0.884 –0.45 0.951 0.29 0.959 0.70 0.951 1.38

72 1 0.740 –13.2 0.768 –11.2 0.780 –11.4 0.913 –6.48 0.936 –5.17 0.924 –4.82 Layer 2 0.724 –15.4 0.770 –13.4 0.775 –13.6 0.914 –8.61 0.939 –7.24 0.940 –6.92 3 3 0.680 –17.5 0.721 –15.4 0.730 –15.6 0.884 –10.5 0.911 –9.09 0.904 –8.76 1 0.494 –21.6 0.575 –18.6 0.582 –18.4 0.627 –18.9 0.665 –17.0 0.653 –17.8 Layer 2 0.738 –13.1 0.796 –10.2 0.804 –9.9 0.875 –9.06 0.891 –7.40 0.888 –8.08 4 3 0.593 –14.2 0.653 –11.4 0.663 –11.2 0.711 –10.7 0.738 –9.11 0.727 –9.74 1 –2.074 17.7 –2.049 17.4 –2.067 17.4 –1.462 17.6 –1.641 17.6 –1.540 –17.4 Layer 2 0.819 3.81 0.837 3.41 0.827 3.39 0.864 –0.85 0.838 0.42 0.861 –0.38 5 3 0.831 4.84 0.850 4.39 0.840 4.39 0.907 –0.50 0.878 1.65 0.905 –0.91 1 0.818 3.63 0.823 2.47 0.819 2.53 0.799 –0.46 0.769 0.19 0.775 –0.61 Layer 2 0.462 7.40 0.485 6.25 0.481 6.27 0.589 4.64 0.533 4.77 0.558 –4.19 6 3 –0.604 16.6 –0.499 15.5 –0.506 15.5 –0.148 16.7 –0.272 15.8 –0.190 –15.7 Average 0.723 13.2* 0.761 10.9* 0.766 10.9* 0.846 8.87* 0.855 7.58 0.858 7.21

*: Calculated using absolute values of entries in the same column. 72 73

Figure 17: Comparison of R2 (without Evaporation). In all three batches the red data points are noticeably lower than the blue and black except in Layer 2. The black data points are generally the highest, even though differences between the blue and black data points are insignificant.

73 74

Figure 18: Comparison of Absolute Error (without Evaporation). Red data points are noticeably higher than the blue and the black except in Layer 5 and 6, while the blue and black are close to each other.

74 75

Figure 19: Comparison of R2 (with Evaporation). As compared to Fig. 17, even though red data points lie below the blue and black, overall the difference is smaller. N=18 resulted in the highest R2 in Layer 1, but N=12 had the highest in Layer 2 thru 4.

75 76

Figure 20: Comparison of Absolute Error (with Evaporation). Relatively huge discrepancies are seen in Layer 1 for all three batches. N=6 interestingly resulted in error least in Layer 2 but greatest in 1, 3 and 4.

76 5.3 Model Comparison: DG vs. CG

Under default simulation settings, the DG model successfully completed

Simulation I with an elapsed time of approximately 1200 seconds; the CG model failed

to converge during one of the extreme-intensity rainfall events (56.9 mm/hr), and was

manually terminated after extensive iterations. The CG model was subsequently re-run

with its infiltration depth (top boundary condition) doubled, and completed simulation

with an elapsed time of about 1000 seconds. Simulation I showed that the DG model is

capable of converging where the CG model could fail while maintaining a computational

speed comparable to that of the CG model.

Results of Simulation II are summarized in Table 8. The DG model predicted a

13.75-mm runoff volume per unit area regardless of vertical discretization. The CG

model failed to converge for 10 and 20 elements and otherwise produced less runoff

volume per unit area as the number of elements increased. Huge difference in runoff

volume generated was seen between the models (> 12 mm3/mm2), but accuracy was

unclear due to lack of data. Simulation II showed that the DG model is consistent in

runoff partitioning regardless of change in .

Δ𝑧𝑧

77 Table 8: Summary of Simulation II

Surface runoff per unit area (mm3/mm2) # of elements DG CG 10 13.75 N.A. 20 13.75 N.A. 30 13.75 1.072 40 13.75 0.8986 50 13.75 0.7925 Soil Parameters: = 1.0 mm/hr = 0.385 = 0.090 = 0.150 = 370 mm 𝐾𝐾𝑠𝑠 𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃𝑟𝑟𝑟𝑟𝑟𝑟 𝑏𝑏 Total volume𝜆𝜆 of rainwater per𝜓𝜓 unit area:− 40.75 mm3/mm2 (Intensity, Hour) of recorded rainfall events in the simulation period:

(1.02, 70) (1.02, 621) (4.07, 1398) (2.04, 74) (1.02, 622) (3.05, 1399) (1.02, 75) (3.05, 1042) (1.02, 1400) (1.02, 132) (1.02, 1043) (1.02, 1406) (5.08, 205) (1.02, 1180) (1.02, 1407) (1.02, 206) (1.02, 1393) (1.02, 1408) (1.02, 310) (1.02, 1395) (1.02, 1412) (2.04, 332) (1.02, 1396) (1.02, 1417) (1.02, 375) (1.02, 1397) (1.02, 1858)

Chapter 6: Conclusion

In this thesis, a DG-FEM solution of the -based 1D Richards’ equation is presented. DG simulation results of upper sand layers𝜃𝜃 (Layer 1 thru 4) achieved good agreement ( 0.75) with the moisture sensor readings, with hysteresis and the lack of 2 an evaporation𝑅𝑅 ≈ scheme contributing to errors. Better prediction was made with the addition of an ad-hoc evaporation scheme ( 0.85). Comparison between the DG 2 model and a conforming CG model showcased𝑅𝑅 ≈the capability of DG-FEM in handling high-intensity rainfall events as well as consistency in runoff partitioning.

Likely due to undesirable equipment setup, simulation results of the lower sand layers (Layer 5 and 6) were particularly questionable. The ad-hoc evaporation scheme improved model prediction to an extent but lacked the complexity in accounting for natural soil surface environments. Moreover, the lack of site runoff record jeopardized evaluation of accuracy in runoff partitioning. Future work should include the addition of a well-defined evaporation scheme and verification of model in deeper soil using archived field data containing soil moisture, precipitation and surface runoff measurements. Incorporation of this DG model in other hydrologic and ecologic models to capture vegetation intake throughout the soil column can be achieved, as sink/sources terms can be easily introduced via direct modification of the numerical flux vector. DG

79 models of higher dimensions (2D and 3D) can be developed to account for more spatial complexity.

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88 Appendix A: Setup 89

Figure 21: Rain Barrel Configuration. A piece of fabric is used to separate the two layers, allowing the flow of water while preventing sand from leaking out. Porosity of the gravel is significantly higher than that of the sand. The photo on the far left shows the actual setup. The image at the center shows different layers of materials in the barrel and the six layers of sensors; sensors are inserted at the depths of 110 mm, 220 mm, 320 mm, 410 mm, 510 mm, and 620 mm (4 sensors per depth; see Table 9). The image on the rights shows how sensors are positioned in each layer.

Table 9: Location of Sensors in Sand Column

Depth from Layer Batch 1 Batch 2 Batch 3 Batch 4 sand surface (mm)

- - - - 0 1 A B C D* 110 2 E F G H* 220 3 I J K L* 320 4 M N O P* 410 5 Q R S T* 510 6 U V W X* 620

- - - - - 660

*: Malfunctioned

90 Appendix B: Procedures for Determining Soil Parameters

Materials:

. Play sand (naturally dried)

. Oven

. Analytical balance

. Long open-ended polymer tube ( > 20 cm in length; 1 in quantity)

. Short open-ended polymer tubes (~ 10 cm in length; at least 3 in quantity)

. Fabric pieces (easily permeable for water but not for the sand)

. Cable ties

. Ruler (at least as long as the polymer tubes; width of ruler < diameter of tube)

. Graduate cylinder and beakers

. Oven-safe weighting plates (at least 3 in quantity)

. Container (tall enough to submerge a polymer tube entirely with water)

. Stopwatch

91 To estimate hydraulic conductivity at saturation, :

𝒔𝒔 1. Measure dimensions of the long polymer 𝑲𝑲tube.

2. Secure a piece of fabric to one end of the long polymer tube using a cable tie.

Fill the polymer with sand up to about half of its depth from the other end.

Measure the depth from the open end to the sand surface.

3. Fill the container with water to a height higher than the sand in the polymer

tube but lower than the tube.

4. Place the tube in the container with its open end facing up (ensure that no

water enters the tube from its open end). Allow the tube to sit in the container

until ponding occurs on the sand surface. During the process, one should see

bubbles escaping from the tube bottom thru the fabric.

5. Remove the tube from the container. Slightly dry its bottom so that no

apparently dripping of water occurs on the tube wall or the wet fabric.

6. Position the tube over the graduated cylinder. Fill the tube from its open end

with water using a beaker. Start the stopwatch. Over time, one should observe

droplets of water dripping from the bottom of the tube. Maintain the water

level in the tube at its maximum during the process.

7. After a certain level of water is collected in the graduated cylinder, stop the

stopwatch and record the time spent.

8. Use Darcy’s Law to compute .

𝑠𝑠 𝐾𝐾

92 To estimate residual moisture content, :

𝒓𝒓𝒓𝒓𝒓𝒓 1. Weigh a certain amount of sand𝜽𝜽 on each of the weighing plates using the

analytical balance.

2. Spread the sand well on the weighing plates and place them into the oven. Set

the temperature to ~80°C.

3. In about 30 minutes, remove the plates and sand from the oven. Weigh them

using the analytical balance and record the decrease in mass.

4. Some of the required parameters for calculation will be obtained in the

procedures to get .

𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃

To estimate moisture content at saturation, :

𝒔𝒔𝒔𝒔𝒔𝒔 1. Measure dimensions of each short polymer𝜽𝜽 tube.

2. Assign two pieces of fabric and two cable locks to every short polymer tube.

Weigh every piece of fabric, cable lock and short polymer tube. Afterwards,

DO NOT MIX UP THE ASSIGNED MATERIALS!

3. Secure a piece of fabric to one end of each short polymer tube using a cable

tie. Fill up each short polymer tube fully with sand and secure the other end

likewise.

4. Weigh the secured tubes (filled with sand). This gives the required mass to

estimate density of the sand, which can be used toward calculating .

𝑟𝑟𝑟𝑟𝑟𝑟 5. Fill up the container with water to a height that will submerge𝜃𝜃 the tubes

entirely.

93 6. Place a tube into the container of water and allow the sand to saturate. In the

process, one should see air bubbles escaping from both ends of the tubes. Wait

for another 5 to 10 minutes after no more bubbles can be seen.

7. Remove the tube from the container. Dry its exterior until no apparent

dripping occurs. Weight the tube on the analytical balance.

8. Remove the fabric and locks from the tube and weigh them on the analytical

balance. Thus far, one should have enough mass data for dry fabric, dry cable

lock, wet fabric, wet cable lock, polymer tube, dry sand, wet sand.

9. Repeat Step 5 thru 8 for the remaining tubes.

10. Calculate based on the recorded mass data.

𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠

94 Appendix C: Pseudo-Code

%% Create 1D mesh: = [ , … , ]; % Nodal coordinates = ( ) ; % Number of elements 𝟎𝟎 𝒏𝒏 𝒛𝒛 𝒛𝒛 = 𝒛𝒛 _ ( , ); % Create mesh 𝑵𝑵 𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥 𝒛𝒛 − 𝟏𝟏 %%𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴 Basis𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜 polynomials:𝐦𝐦𝐦𝐦𝐦𝐦𝐦𝐦 𝒛𝒛 𝑵𝑵 = (… ); % Degree of basis polynomial = _ ( ); % Basis polynomial 𝒑𝒑 =𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢 _ _ ( , ); % Global matrix of 𝝓𝝓 𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 𝒑𝒑 𝚽𝚽 𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜𝐜 𝐠𝐠𝐠𝐠𝐠𝐠𝐠𝐠𝐠𝐠𝐠𝐠 𝐛𝐛𝐛𝐛𝐛𝐛𝐛𝐛𝐛𝐛 𝝓𝝓 𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴 𝝓𝝓 %% Mass matrix (symmetric): for from 1 to ( + ) for from 1 to ( + ) 𝒊𝒊 𝒑𝒑 𝟏𝟏 ( , ) = ( ) × ( ); 𝒋𝒋 (𝒊𝒊, )𝟏𝟏= ( , ); end 𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴 𝒊𝒊 𝒋𝒋 𝝓𝝓 𝒊𝒊 𝝓𝝓 𝒋𝒋 end 𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴 𝒋𝒋 𝒊𝒊 𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴 𝒊𝒊 𝒋𝒋

%% Assemble global matrices: [ , ] = _ ( ); % Gauss points and weights for from 1 to 𝝃𝝃 𝒘𝒘 Compute𝐪𝐪𝐪𝐪𝐪𝐪𝐪𝐪𝐪𝐪𝐪𝐪𝐪𝐪𝐪𝐪𝐪𝐪𝐪𝐪 ( ) at𝐭𝐭𝐭𝐭𝐭𝐭𝐭𝐭𝐭𝐭 the 𝒑𝒑 points; % Derivative of end𝒊𝒊 𝒑𝒑 ′ = ( ); 𝚽𝚽 𝝃𝝃 % Weight matrix,𝚽𝚽 diagonal = ( ) ( ) ; 𝑾𝑾 =𝐝𝐝𝐝𝐝𝐝𝐝𝐝𝐝( 𝒘𝒘 )−𝟏𝟏 ( ′ 𝐓𝐓); 𝑨𝑨𝒋𝒋 𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴 ⋅ 𝚽𝚽 ⋅ 𝑾𝑾 + = ( )−𝟏𝟏 ( +); 𝑩𝑩𝒋𝒋 𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴 ⋅ 𝚽𝚽 −, ] = _−𝟏𝟏 − ( , , , , ) [ 𝒋𝒋 ; 𝑩𝑩 𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴 ⋅ 𝚽𝚽 + − 𝒋𝒋 𝒋𝒋 𝒋𝒋 %%𝑨𝑨 𝑩𝑩 Initial𝐠𝐠𝐠𝐠𝐠𝐠 profile𝐠𝐠𝐠𝐠𝐠𝐠 𝐚𝐚𝐚𝐚𝐚𝐚𝐚𝐚𝐚𝐚𝐚𝐚𝐚𝐚𝐚𝐚 and variables:𝑨𝑨 𝑩𝑩 𝑩𝑩 𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴 𝑴𝑴𝑴𝑴𝑴𝑴𝑴𝑴 = (… ); % Initial value for each element [ , , , , ] = [… ]; % Soil parameters 𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕𝒕= @( )(𝐢𝐢𝐢𝐢…𝐢𝐢)𝐢𝐢𝐢𝐢 % Hydraulic𝜽𝜽 conductivity 𝒔𝒔 𝒔𝒔𝒔𝒔𝒔𝒔 𝒓𝒓𝒓𝒓𝒓𝒓 𝒃𝒃 𝑲𝑲 =𝜽𝜽@( 𝜽𝜽)(… )𝝀𝝀 𝝍𝝍 % Diffusivity 𝑲𝑲∗ 𝜽𝜽 95 𝑫𝑫 𝜽𝜽 %% Initialize Variables for Time-Stepping Loop = ; = ; % Beware of unit 𝑻𝑻 = 𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓 𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬 𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬;𝐬𝐬 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩 % Starts at 1 𝚫𝚫𝒕𝒕 = 𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓 𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬 𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬; % Starts at 1 𝒏𝒏 𝐓𝐓𝐓𝐓𝐓𝐓𝐓𝐓 𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬 𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥 %%𝒎𝒎 𝐈𝐈Time𝐈𝐈𝐈𝐈𝐈𝐈𝐈𝐈𝐈𝐈𝐈𝐈𝐈𝐈𝐈𝐈-Stepping𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥 𝐥𝐥Loop: while total simulation period = ; % First guess 𝒕𝒕 ≤ = . × [ ( ) + ( )]; % Bassi-Rebay flux 𝒉𝒉 𝚯𝚯∗ =𝚽𝚽 ⋅ 𝚯𝚯 ( )∗+ − ; ∗ + % Compute the approx. � 𝑫𝑫 = 𝟎𝟎 𝟓𝟓 ∗; 𝑫𝑫 𝚯𝚯 ∗𝑫𝑫 𝚯𝚯 % Compute 𝒉𝒉 𝒒𝒒 = 𝑨𝑨. ⋅ ×𝑫𝑫[ 𝚯𝚯( )𝑩𝑩+⋅ 𝑫𝑫�( )]; % Bassi-Rebay flux 𝒒𝒒 𝒉𝒉 𝒒𝒒 =𝚽𝚽.⋅ 𝒒𝒒× [ ( − ) + ( + ) ( )]; % Lax-Fredrichs𝒒𝒒 flux if𝒒𝒒� 𝟎𝟎 𝟓𝟓 𝒒𝒒>𝚯𝚯 − 𝒒𝒒 𝚯𝚯 + − + 𝒔𝒔 𝑲𝑲� 𝟎𝟎 if𝟓𝟓 𝑲𝑲 𝚯𝚯 >𝑲𝑲 𝚯𝚯 − 𝑲𝑲 ⋅ 𝚯𝚯 − 𝚯𝚯 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 𝒊𝒊𝒊𝒊𝒊𝒊 𝟎𝟎( ) = ; = ; 𝒔𝒔 else𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 𝒊𝒊𝒊𝒊𝒊𝒊 𝑲𝑲 � 𝒋𝒋 𝒔𝒔 𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫 𝒔𝒔 𝑲𝑲 (𝟏𝟏) = 𝑲𝑲 ; 𝒓𝒓𝒓𝒓𝒓𝒓𝒆𝒆 = 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊; 𝒊𝒊𝒊𝒊𝒊𝒊 − 𝑲𝑲 end 𝒋𝒋 𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫 else 𝑲𝑲� 𝟏𝟏 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 𝒓𝒓𝒓𝒓𝒓𝒓𝒆𝒆 𝟎𝟎 ( ) = .; = ; end � 𝒋𝒋 𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞 𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫 ( 𝑲𝑲) =𝟏𝟏 ( 𝒓𝒓𝒓𝒓𝒓𝒓)𝒆𝒆; 𝒓𝒓𝒓𝒓𝒓𝒓𝒆𝒆 % Drainage𝟎𝟎 from bottom = × ( ) + ; % Increment over 𝑲𝑲� 𝒋𝒋 𝒆𝒆𝒆𝒆𝒆𝒆 𝑲𝑲 𝜽𝜽𝒆𝒆𝒆𝒆𝒆𝒆 = + ; % Second guess Δ𝚯𝚯 Δ𝑡𝑡 �𝑨𝑨 ⋅ 𝒒𝒒 − 𝑲𝑲 𝑩𝑩 ⋅ �𝒒𝒒� − 𝑲𝑲�� Δ𝒕𝒕 %𝚯𝚯 Check𝚯𝚯 Δ convergence𝚯𝚯 if convergence criterion is fulfilled Update to the beginning of next time step; = + ; = +𝚯𝚯 ; % Proceed in time 𝒕𝒕 =𝒕𝒕 ; 𝚫𝚫𝒕𝒕 % Reset iteration level 𝒏𝒏 𝒏𝒏= 𝟏𝟏 + × ; % Cumulative input 𝒎𝒎 𝟏𝟏 = + × ; % Cumulative runoff 𝐢𝐢𝐢𝐢 𝐢𝐢𝐢𝐢 𝒗𝒗𝒗𝒗𝒗𝒗 . 𝒗𝒗=𝒗𝒗𝒗𝒗 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊. + 𝒊𝒊𝒊𝒊𝒊𝒊 𝚫𝚫. ×𝒕𝒕 ; % Cumulative evaporation 𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫 𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫 𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫𝐫 𝒗𝒗𝒗𝒗𝒗𝒗 = 𝒗𝒗𝒗𝒗𝒗𝒗 + 𝒓𝒓𝒓𝒓𝒓𝒓( 𝒆𝒆 ) × 𝚫𝚫𝒕𝒕 𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞 𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞 𝐞𝐞𝐞𝐞𝐞𝐞𝐞𝐞 ; % Cumulative drainage 𝒗𝒗𝒗𝒗𝒗𝒗 = 𝒗𝒗𝒗𝒗𝒗𝒗 + 𝒓𝒓𝒓𝒓𝒓𝒓( 𝒆𝒆) 𝚫𝚫𝒕𝒕 𝐝𝐝𝐝𝐝𝐝𝐝𝐝𝐝𝐝𝐝 𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅𝒅 𝒋𝒋 ; % Total moisture in soil else 𝒗𝒗𝒗𝒗𝒗𝒗 𝒗𝒗𝒗𝒗𝒗𝒗 𝑲𝑲� 𝒆𝒆𝒆𝒆𝒆𝒆 𝚫𝚫𝒕𝒕 𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬 𝐬𝐬𝐬𝐬𝐬𝐬𝐬𝐬 𝒗𝒗𝒗𝒗𝒗𝒗= +𝒗𝒗𝒗𝒗; 𝒗𝒗 𝐬𝐬𝐬𝐬𝐬𝐬 𝚯𝚯 % Iteration level +1 Update to the second guess; end 𝒎𝒎 𝒎𝒎 𝟏𝟏 end 𝚯𝚯