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Publisher:

Princeton University Press

Publication Date:

2011

Number of Pages:

240

Format:

Paperback

Price:

45.00

ISBN:

9780691145143

Category:

Textbook

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by , on ]

William J. Satzer

06/30/2011

If you’ve ever taught a differential equations course and yearned for fresh examples, this is definitely a book for you. Wouldn’t it be great to give some of those terribly overworked examples a rest? Even some of the more recent examples from chemistry and biology are starting to look a bit fatigued.

This book was written by two earth scientists with the aim of providing what they regard as essential skills for graduate students and advanced undergraduates in their field. Those skills include the ability to translate chemical and physical systems into mathematical and computational models that provide insight into dynamical processes on the earth’s surface and inside. All the models treated here are based on ordinary and partial differential equations.

What kinds of examples do the authors use? One of the first is a model of the radiocarbon content of the biosphere. In simplest form, this leads to exponential decay; with periodic forcing (determined perhaps by the sunspot cycle), more complicated solutions arise. As the book proceeds, the examples get more complex. Other examples include: dissolved species in an aquifer, evolution of a sandy coastline, and pollutant transport in a confined aquifer. One amazing pictorial example from the introduction shows a simulated time sequence of an iron bolide asteroid, one kilometer in diameter, hitting the ocean at a 45 degree angle. Two of the most interesting worked out examples are analysis of complicated circulation patterns in Lake Ontario and modeling a lahar (water and pyroclastic debris flowing down the side of a volcano).

Although the authors concentrate on earth science, pretty much everything is accessible to anyone having some knowledge of basic physics and chemistry. The early chapters of the book pay a lot of attention to the details of developing a model. It is clear that the authors have a good deal of expertise in modeling. Yet they note that their book is intended as a primer, and students are not expected to have any modeling experience. By the nature of their interests, most of the discussion revolves around partial differential equations — nearly all the problems of interest have time and at least one spatial dimension as independent variables. One consequence of this is a secondary focus in the book on finite difference methods for solving partial differential equations.

Besides being a wonderful source of examples, this short book is a pleasing and well-organized introduction to modeling with differential equations.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

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

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