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

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

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:

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

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

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

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.