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.