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• Cover
• Title
• Copyright
• Contents
• Preface
• 1 Modeling and Mathematical Concepts
• Pros and Cons of Dynamical Models
• An Important Modeling Assumption
• Some Examples
• Example I: Simulation of Chicxulub Impact and Its Consequences
• Example II: Storm Surge of Hurricane Ivan in Escambia Bay
• Steps in Model Building
• Basic Definitions and Concepts
• Nondimensionalization
• A Brief Mathematical Review
• Summary
• 2 Basics of Numerical Solutions by Finite Difference
• First Some Matrix Algebra
• Solution of Linear Systems of Algebraic Equations
• General Finite Difference Approach
• Discretization
• Obtaining Difference Operators by Taylor Series
• Explicit Schemes
• Implicit Schemes
• How Good Is My Finite Difference Scheme?
• Stability Is Not Accuracy
• Summary
• Modeling Exercises
• 3 Box Modeling: Unsteady, Uniform Conservation of Mass
• Translations
• Example I: Radiocarbon Content of the Biosphere as a One-Box Model
• Example II: The Carbon Cycle as a Multibox Model
• Example III: One-Dimensional Energy Balance Climate Model
• Finite Difference Solutions of Box Models
• The Forward Euler Method
• Predictor–Corrector Methods
• Stiff Systems
• Example IV: Rothman Ocean
• Backward Euler Method
• Model Enhancements
• Summary
• Modeling Exercises
• 4 One-Dimensional Diffusion Problems
• Translations
• Example I: Dissolved Species in a Homogeneous Aquifer
• Example II: Evolution of a Sandy Coastline
• Example III: Diffusion of Momentum
• Finite Difference Solutions to 1-D Diffusion Problems
• Summary
• Modeling Exercises
• 5 Multidimensional Diffusion Problems
• Translations
• Example I: Landscape Evolution as a 2-D Diffusion Problem
• Example II: Pollutant Transport in a Confined Aquifer
• Example III: Thermal Considerations in Radioactive Waste Disposal
• Finite Difference Solutions to Parabolic PDEs and Elliptic Boundary Value Problems
• An Explicit Scheme
• Implicit Schemes
• Case of Variable Coefficients
• Summary
• Modeling Exercises
• 6 Advection-Dominated Problems
• Translations
• Example I: A Dissolved Species in a River
• Example II: Lahars Flowing along Simple Channels
• Finite Difference Solution Schemes to the Linear Advection Equation
• Summary
• Modeling Exercises
• 7 Advection and Diffusion (Transport) Problems
• Translations
• Example I: A Generic 1-D Case
• Example II: Transport of Suspended Sediment in a Stream
• Example III: Sedimentary Diagenesis: Influence of Burrows
• Finite Difference Solutions to the Transport Equation
• QUICK Scheme
• QUICKEST Scheme
• Summary
• Modeling Exercises
• 8 Transport Problems with a Twist: The Transport of Momentum
• Translations
• Example I: One-Dimensional Transport of Momentum in a Newtonian Fluid (Burgers’ Equation)
• An Analytic Solution to Burgers’ Equation
• Finite Difference Scheme for Burgers’ Equation
• Solution Scheme Accuracy
• Diffusive Momentum Transport in Turbulent Flows
• Adding Sources and Sinks of Momentum: The General Law of Motion
• Summary
• Modeling Exercises
• 9 Systems of One-Dimensional Nonlinear Partial Differential Equations
• Translations
• Example I: Gradually Varied Flow in an Open Channel
• Finite Difference Solution Schemes for Equation Sets
• Explicit FTCS Scheme on a Staggered Mesh
• Four-Point Implicit Scheme
• The Dam-Break Problem: An Example
• Summary
• Modeling Exercises
• 10 Two-Dimensional Nonlinear Hyperbolic Systems
• Translations
• Example I: The Circulation of Lakes, Estuaries, and the Coastal Ocean
• An Explicit Solution Scheme for 2-D Vertically Integrated Geophysical Flows
• Lake Ontario Wind-Driven Circulation: An Example
• Summary
• Modeling Exercises
• Closing Remarks
• References
• Index