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1. Useful calculus
1.1. Taylor series
1.2. Trigonometry
1.3. Absolute values
1.4. Complex numbers
1.5. The Euler formula
1.6. Fourier transforms
1.8. The Formal definition of the derivative
1.9. Differential equations
2. Geophysical Fluid Dynamics (GFD)
2.1. The Primitive equations
2.2. The Advection equation
2.3. The diffusion equation
2.4. Inertial oscillations
2.5. The Shallow water equations
2.6. The wave equation
3. The finite difference method (FDM)
3.1. Algebraic approximation of the first derivative
3.2. Truncation Error
3.3. Estimates of the truncation error
3.4. Finite difference formulas for the 1st derivative
3.5. Finite difference formulas for the 2nd derivative
4. Stability of numerical schemes
4.1. Concepts and definitions
4.2. Domain of Dependence (DoD)
4.3. Courant-Friedrich-Lewy, CFL criterion
4.4. Total Variation (TV) and Total Variation Diminishing (TVD) schemes (part 1)
4.5. Von Neumann Stability Analysis
5. Grids
5.1. Staggered grids
6. Implicit schemes vs explicit schemes
6.1. The Crank-Nicholson Scheme for linear advection
6.2. The Implicit Scheme
7. Boundary conditions
7.1. Mathematical descriptions of boundary conditions
7.2. Physical descriptions of boundary conditions
8. The advection equation
8.1. The exact equation
8.2. The difference equation
8.3. Solution of the difference equation
8.4. Analysis of the solution of the difference equation
8.5. Schemes with stabilizing terms
8.6. Predictor-corrector schemes
8.7. Half-step schemes
8.8. The Lax-Wendroff 1-step approach from Taylor expansions
9. Total Variation Diminishing schemes
9.1. Godunov’s method
9.2. Flux Limiters
9.3. Godunov schemes with TVD flux limiters
10. Diffusion
10.1. The diffusion equation
10.2. The Crank-Nicholson Scheme for diffusion
10.3. von Neumann stability analysis for the diffusion equation
11. Waves
11.1. The classical wave equation and its solution
11.2. von Neumann analysis of the wave equation
12. References
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