__Basic differential equation formulations__

__Solving Dx/Dt=f(x,t)__

- Examples of time differencing schemes
- Overview on Euler, Runge-Kutta, and multistep methods
- A comparison of RK versus Leapfrog. Note the errors in phase speed and amplitude. Also note Leapfrog has a (fictional) computation mode. This is why WRF uses a high-order Runge-Kutta.
- Centered-in-time, centered-in-space scheme (Leapfrog scheme) example for advection equation
- Semi-Lagrangian schemes
- Supplementary material:

2) CK12.org material on Euler's Method and Runge-Kutta

3) Saylor Academy notes on Runge-Kutta

__Trajectories__

- Pertinent equations and basics
- Example applications for computing trajectories

2) Dr. Fitz's simulation of the Deepwater Horizon oil spill

3) Inflow trajectories into hurricanes

4) a. Cross-ocean balloon flight planning b. More on the first Transpacific balloon flight c. More on Steve Fossett d. Article on Steve Fossett

__Modeling overview__

- Overview on weather modeling process (dated document on modeling, but conceptually correct)
- Stull's overview

__Spectral method (used in global models and climate models)__

- Pseudo-spectral example
- Advantage and disadvantage of the spectral method
- Comparison of grid point and spectral methods
- Example with spherical harmonics. What are spherical harmonics?: 1) The classic paper on spherical harmonics by Baer. 2) Advanced graduate school material from CSU professor David Randall.

__Example solutions__

__Spreadsheet examples__

- Euler compared to 2nd-order Runge-Kutta
- 3rd-order Runge-Kutta
- 4th-order Runge-Kutta
- Euler compared to improved Euler with VBA function
- Euler compared to 4th-order Runge-Kutta with VBA function

__FORTRAN basics__

- What is FORTRAN?
- Compiling and running FORTRAN code
- Supplementary material - Dr. Paul Tackley's FORTRAN notes and additional tutorials
- Supplementary material - website on FORTRAN

__FORTRAN examples (to be used as part of final exam)__

- Basic Leapfrog example
- Basic Leapfrog sensitivity to Courant number. What happens if the Courant number>1, =1, and <1?
- Forward-in-time, backward-in-space (FTBS). How well does this actually work?
- 4th-order Runge-Kutta example

__Other computer algorithm examples__

- Matlab example with ODE45. More information on the ODE45 algorithm.
- More MATLAB examples. Accompanying code is below:

2) HeunODE

3) Function RK4

4) Adams-Bashforth

5) Adams-Bashforth-Moulton predictor-corrrector

- FORTRAN, First order ODE - Euler, Predictor-Corrector, Runge-Kutta 4th order
- FORTRAN, Second order single ODE - Euler, Predictor-Corrector, Runge-Kutta 4th order
- Mathematica ODE solver

__"Simple" model codes__

- Dr. Fitz's barotropic model
- Kerry Emanuel's balanced vortex model. The README.
- A barotropic/ reduced gravity 1.5 layers QG model. Courtesy of Dr. Tziperman
- ENSO delay oscillator model. Courtesy of Dr. Tziperman. Documented in this paper.

__Advanced material on stability, amplitude errors, phase speed errors, and computational mode__

- Dr. Roger Smith's notes
- Dr. Fitz's detailed analysis of FTCS scheme
- Dr. Fitz's detailed analysis of Leapfrog scheme
- Dr. Fitz's detailed analysis of implicit scheme
- Dr. Fitz's von Neumann analysis on a filter scheme
- Dr. Roger Pielke's analysis plots
- Wicker and Skamarock (2002) paper on Runka-Kutta
- Dr. Eugenia Kalnay's' summary analysis of time-differencing schemes
- Dr. Eugenia Kalnay's detailed lectures on the subject. Part 1. Part 2.
- Positive definite advection schemes
- Review on positive definite advection schemes
- ECMWF lecture on Semi-Lagrangian schemes

__Recorded lectures__