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Table of Contents
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 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 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 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 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 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 7 Advection and Diffusion (Transport) Problems
Translations
Example I: A Generic 1-DCase
Example II: Transport of Suspended Sediment in a Stream
Example III: Sedimentary Diagenesis
Finite Difference Solutions to the Transport Equation
QUICK Scheme
QUICKEST Scheme 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 9 Systems of One-Dimensional Non linear 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 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.
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 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 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 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 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 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 7 Advection and Diffusion (Transport) Problems
Translations
Example I: A Generic 1-DCase
Example II: Transport of Suspended Sediment in a Stream
Example III: Sedimentary Diagenesis
Finite Difference Solutions to the Transport Equation
QUICK Scheme
QUICKEST Scheme 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 9 Systems of One-Dimensional Non linear 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 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.