1. Numerical solution of first-order ordinary differential equations
1. First order Runge-Kutta method (Euler’s method)
Let’s start with the Taylor series of y(t) expanded at t=t0
We now restrict our solution to a short time step h.

2. 2. Second order Runge-Kutta method
Approximate derivative at t=t0
Intermediate estimate of y at t = t0+h/2
Estimate of derivative at t = t0+h/2
Estimate of y at t = t0+h

3. 3. Fourth order Runge-Kutta method
Approximate derivative at t=t0,
Intermediate estimate of y at t = t0+h/2 using k1,
Estimate of derivative at t = t0+h/2,
Intermediate estimate of y at t = t0+h/2 using k2,
Estimate of derivative at t = t0+h/2,
Intermediate Estimate of y at t = t0+h using k3,
Estimate of derivative at t = t0+h,

4. 4. Real application, vortex Rossby waves