Basic differential equation formulations Solving Dx/Dt=f(x,t)
2) CK12.org material on Euler's Method and Runge-Kutta
3) Saylor Academy notes on Runge-Kutta
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
2) HeunODE
3) Function RK4
4) Adams-Bashforth
5) Adams-Bashforth-Moulton predictor-corrrector
- 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
- 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.
- 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
- What is FORTRAN?
- Compiling and running FORTRAN code
- Supplementary material - Dr. Paul Tackley's FORTRAN notes and additional tutorials
- Supplementary material - website on FORTRAN
- 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
- 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
- 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.
- 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