1. Sec 23: Runge–Kutta Methods
Trapezoidal rule
Heun’s method
This is an example of a Runge–Kutta method

2. Sec 23: Runge–Kutta Methods
2ed order RK method
Classical 4th order RK method

3. Sec 23: Runge–Kutta Methods
General RK
It is customary to arrange the coefficients in a so-called Runge-Kutta
or Butcher tableaux as follows:

4. Sec 23: Runge–Kutta Methods
2ed order RK method
Classical 4th order RK method

5. Derivation of second order RK
2ed order RK method
Taylor Expansion in 2d
We obtain general explicit second-order Runge-Kutta methods by assuming

6. Derivation of second order RK
Heun’s Method
Midpoint Method
Ralston’s Method
Example
A ball at 1200K is allowed to cool down in air at an ambient temperature of 300K. Assuming heat is lost only due to radiation, the differential equation for the temperature of the ball is given by
where y is in K and t in seconds. Find the temperature at t = 480 seconds

7. Sec 23: Runge–Kutta Methods
General RK
Expand all f around (x,y) and then choose constants such that the equation match as many terms in Taylor expansion.