Preface xi

Chapter 1: Modeling and Mathematical Concepts 1

Pros and Cons of Dynamical Models 2

An Important Modeling Assumption 4

Some Examples 4

Example I: Simulation of Chicxulub Impact and Its Consequences 5

Example II: Storm Surge of Hurricane Ivan in Escambia Bay 7

Steps in Model Building 8

Basic Definitions and Concepts 11

Nondimensionalization 13

A Brief Mathematical Review 14

Summary 22

Chapter 2: Basics of Numerical Solutions by Finite Difference 23

First Some Matrix Algebra 23

Solution of Linear Systems of Algebraic Equations 25

General Finite Difference Approach 26

Discretization 27

Obtaining Difference Operators by Taylor Series 28

Explicit Schemes 29

Implicit Schemes 30

How Good Is My Finite Difference Scheme? 33

Stability Is Not Accuracy 35

Summary 37

Modeling Exercises 38

Chapter 3: Box Modeling: Unsteady, Uniform Conservation of Mass 39

Translations 40

Example I: Radiocarbon Content of the Biosphere as a One-Box Model 40

Example II: The Carbon Cycle as a Multibox Model 48

Example III: One-Dimensional Energy Balance Climate Model 53

Finite Difference Solutions of Box Models 57

The Forward Euler Method 57

Predictor-Corrector Methods 59

Stiff Systems 60

Example IV: Rothman Ocean 61

Backward Euler Method 65

Model Enhancements 69

Summary 71

Modeling Exercises 71

Chapter 4: One-Dimensional Diffusion Problems 74

Translations 75

Example I: Dissolved Species in a Homogeneous Aquifer 75

Example II: Evolution of a Sandy Coastline 80

Example III: Diffusion of Momentum 83

Finite Difference Solutions to 1-D Diffusion Problems 86

Summary 86

Modeling Exercises 87

Chapter 5: Multidimensional Diffusion Problems 89

Translations 90

Example I: Landscape Evolution as a 2-D Diffusion Problem 90

Example II: Pollutant Transport in a Confined Aquifer 96

Example III: Thermal Considerations in Radioactive Waste Disposal 99

Finite Difference Solutions to Parabolic PDEs and Elliptic Boundary Value Problems 101

An Explicit Scheme 102

Implicit Schemes 103

Case of Variable Coefficients 107

Summary 108

Modeling Exercises 109

Chapter 6: Advection-Dominated Problems 111

Translations 112

Example I: A Dissolved Species in a River 112

Example II: Lahars Flowing along Simple Channels 116

Finite Difference Solution Schemes to the Linear Advection Equation 122

Summary 126

Modeling Exercises 128

Chapter 7: Advection and Diffusion (Transport) Problems 130

Translations 131

Example I: A Generic 1-D Case 131

Example II: Transport of Suspended Sediment in a Stream 134

Example III: Sedimentary Diagenes Influence of Burrows 138

Finite Difference Solutions to the Transport Equation 143

QUICK Scheme 144

QUICKEST Scheme 146

Summary 147

Modeling Exercises 147

Chapter 8: Transport Problems with a Twist: The Transport of Momentum 151

Translations 152

Example I: One-Dimensional Transport of Momentum in a Newtonian Fluid (Burgers' Equation) 152

An Analytic Solution to Burgers' Equation 157

Finite Difference Scheme for Burgers' Equation 158

Solution Scheme Accuracy 160

Diffusive Momentum Transport in Turbulent Flows 163

Adding Sources and Sinks of Momentum: The General Law of Motion 165

Summary 166

Modeling Exercises 167

Chapter 9: Systems of One-Dimensional Nonlinear Partial Differential Equations 169

Translations 169

Example I: Gradually Varied Flow in an Open Channel 169

Finite Difference Solution Schemes for Equation Sets 175

Explicit FTCS Scheme on a Staggered Mesh 175

Four-Point Implicit Scheme 177

The Dam-Break Problem: An Example 180

Summary 183

Modeling Exercises 185

Chapter 10: Two-Dimensional Nonlinear Hyperbolic Systems 187

Translations 188

Example I: The Circulation of Lakes, Estuaries, and the Coastal Ocean 188

An Explicit Solution Scheme for 2-D Vertically Integrated Geophysical Flows 197

Lake Ontario Wind-Driven Circulation: An Example 202

Summary 203

Modeling Exercises 206

Closing Remarks 209

References 211

Index 217