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I.   Introduction
       A. Overview
       B. The Art of Computational Sciences

II.   Basic Tools
       A. Introduction
       B. Essential Concepts
            1. Discrete Representations
            2. Interpolation/Extrapolation
            3. Types of Error
            4. Discretization Error
            5. Algorithmic Error
       C. Interpolation/Extrapolation
            1. Lagrange Interpolation
            2. Piecewise Interpolation
            3. Rational Function Approximation
            4. Fourier and Chebyshev Interprolation
       D. Numerical Differentiation
            1. Introduction to Finite Differences
            2. Low-order Approximation
            3. High-order Approximation
       E. Numerical Integration
            1. General-purpose Quadrature
            2. High Efficiency Quadrature
            3. Stochastic Quadrature
       F. Basic Tools Projects

III.   Ordinary Differential Equations
       A. Overview
            1. Introduction
            2. IVPs verses BVPs
            3.  Summary of ODE-IVP Methods
       B. Initial Values Problems -- Basic Methods and Issues
            1. Euler’s Method
            2. Stability
            3. Backward Euler & Centered Schemes
            4. Handling Implicitness
            5. Simulation Examples
            6. Basic Multi-Step Methods
       C. Initial Value Problems -- One-Step Methods
            1. Runge-Kutta Methods
            2. Bulirsch-Stoer Method
            3. Simulation Examples
            4. Exponential Fitting Methods
       D. Initial Value Problems  -- Multi-Step Methods
            1. Introduction
            2. Adams-Bashforth-Moulton Methods
       E. ODE Systems
            1. Solving ODE Systems
            2. Stiffness
            3. Stiff Bulirsch-Stoer Method
            4. Gear’s Methods
            5. Simulation Example

IV.  Partial Differential Equations: Finite-
    Difference Methods
       A. Overview
       B. Parabolic PDEs
            1. Diffusion Equation
            2. A Low-Order Method
            3. Stability
            4. Crank-Nicolson
            5. Compact Methods
            6. Simulation examples: 1-D
            7. Scalar Implicit Methods
            8. Splitting Methods
            9. Simulation examples: 2-D
       C. Elliptic PDEs
            1. Introduction
            2. Relaxation Methods
            3. Simulation Examples
            4. Nonlinear Problems
            5. Multi-Grid Approach
            6. Conjugate-Gradient Method
            7. Fast Direct Methods
            8. Non-Uniform Grids
       D. Hyperbolic PDEs
            1. Wave Equation
            2. Numerical Diffusion and Dispersion
            3. Low-Order Explicit Methods
            4. High-Order Schemes
            5. Flux-Corrected Transport
                Algorithm
            6. Simulation example
            7. Nonlinear Problems
        E. PDE Projects
           
V.  Partial Differential Equations: short introduction to Basis-Function Expansion Methods
       A. Basics Function expansion Methods
       B. Finite Element Method: Elliptic PDEs
            1. Finite Elements in One Dimension      
       C. Spectral Method: Hyperbolic PDEs
            1. Introduction to the Spectral Method

VI.  References

VII.  Appendices
       A. Tridiagonal Systems
             1. Algorithms
             2. Examples
       B. Vector Algebra and Vector Calculus
       C. Input/Output
       D. Project Guidelines