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=t 0 We now restrict our solution to a short time step h.

2. 2. Second order Runge-Kutta method Approximate derivative at t=t 0 Intermediate estimate of y at t = t 0 +h/2 Estimate of derivative at t = t 0 +h/2 Estimate of y at t = t 0 +h

3. 3. Fourth order Runge-Kutta method Approximate derivative at t=t 0, Intermediate estimate of y at t = t 0 +h/2 using k 1, Estimate of derivative at t = t 0 +h/2, Intermediate estimate of y at t = t 0 +h/2 using k 2, Estimate of derivative at t = t 0 +h/2, Intermediate Estimate of y at t = t 0 +h using k 3, Estimate of derivative at t = t 0 +h,

4. 4. Real application, vortex Rossby waves